problem solving examples in math grade 10

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Grade 10 math questions and solutions.

The Smarter Balanced Assessment Consortium (SBAC) is a standardized test that includes a variety of new technology-enhanced questions.

Some of them are Multiple choice-single correct responses, Multiple choice-multiple correct responses, Matching Tables, Drag and Drop, Hot text, Table Fill in, Graphing, Equation/numeric, Extended constructed response, Short answer, and many more.

This page contains several sample questions along with practice test links for Grade 10 Math that gives you an idea of questions that your students are likely to see on the test. After each sample question, an answer explanation follows. The explanation includes essential aspects of the task that you may need to consider for the skills, processes, and information your students need to know.

Domain: Grade 10 >> Number and Quantity – The Real Number System

Sample Question: Evaluate 9 150/300

Answer Explanation: 9 150/300 = 9 1/2 = square root of 9 = 3. In a problem with a rational exponent, the numerator tells you the power, and the denominator the root. However, in this problem the exponent can be reduced, so we should reduce that first. The exponent 150/300 = 1/2. So the problem becomes 9 to the 1/2 power. The denominator is 2 so we take the square root of 9 which is 3. The numerator is 1 so we raise 3 to the 1st power and the answer is 3.

Standards: HSN.RN.A.1

Click here to practice: Number and Quantity – The Real Number System Questions on Grade 10 Math

Domain: Grade 10 >> Number and Quantity – Quantities

Sample Question: Rewrite x 1/2 in radical form.

Answer Explanation: In a problem with a rational exponent, the numerator tells you the power, and the denominator the root. Since the problem is, x 1/2 , the denominator is 2 indicating we should take a square root and the numerator is 1 so we would raise that to the first power or there will be no exponent since an exponent of 1 is rarely used. That makes the answer the square root of x, written as √x.

Click here to practice: Grade 10 Number and Quantity – Quantities Questions

Domain: Grade 10 >> Number and Quantity – The Complex Number System

Sample Question: Simplify completely i(7−i)

Answer Explanation: i(7−i)=i*7−i*i=7i−i 2 =7i−(−1)=7i+1=1+7i

Start by using the distributive method. Now simplify −i 2 =1 by definition. Now rearrange and put the real part first and the imaginary part last so that it looks like this a+bi.

Standards: HSN.CN.A.2

Click here to practice: Grade 10 Math Number and Quantity – The Complex Number System Questions

Domain: Grade 10 >> Number and Quantity – Vector & Matrix Quantities

Sample Question: A vector in standard form has components . What is the initial point?

• Not enough information given

Answer Explanation: Since the vector is in standard position, we know that the initial point is (0, 0) or the origin.

Standards: HSN.VM.A.2

Click here to practice: Number and Quantity – Vector & Matrix Quantities Questions for Grade 10 Math

Domain: Grade 10 >> Algebra – Seeing Structure in Expressions

Sample Question: Which expression is equivalent to 9x 2 – 16y 2 ?

• (3x – 4y) (3x – 4y)
• (3x + 4y) (3x + 4y)
• (3x + 4y) (3x – 4y)
• (3x – 4y) 2

Answer Explanation: Student must recognize the expression is the difference of two perfect squares

Standards: HSA.SSE.A.2

Click here to practice: Algebra – Seeing Structure in Expressions Questions for Grade 10 Math

Domain: Grade 10 >> Algebra – Arithmetic with Polynomials & Rational Expressions

Sample Question: Evaluate f(x)=−a 3 +6a−7 at a = – 1 and state the remainder.

Answer Explanation: student must substitute – 1 into the function as follows −(−1) 3 +6(−1)−7=−12 and find the value to get the remainder

Standards: HSA.APR.B.2

Click here to practice: Algebra – Arithmetic with Polynomials & Rational Expressions Questions for Grade 10 Math

Domain: Grade 10 >> Algebra – Creating Equations

Sample Question: The ratio of staff to guests at the gala was 3 to 5. There were a total of 576 people in the ballroom. How many guests were at the gala?

Answer Explanation: Setup a proportion of guests to the total number of people, 8/5 = x/576. Solve by cross multiplying. 8x = 2880. Divide both sides by 8. So x=360.

Standards: HSA.CED.A.3

Click here to practice: Algebra – Creating Equations Questions for Grade 10 Math

Domain: Grade 10 >> Algebra – Reasoning with Equations & Inequalities

Sample Question: Solve the quadratic x 2 +10x=−25.

Answer Explanation: This problem can be easily solved by rearranging the equation so that it is solved for zero and then factoring as shown:

x 2 +10x=−25

x 2 +10x+25=0

(x+5)(x+5)=0

Since both factors are exactly the same, you will only have one solution to this problem.

Standards: HSA.REI.B.4

Click here to practice: Algebra – Reasoning with Equations & Inequalities Questions for Grade 10 Math

Domain: Grade 10 >> Functions – Interpreting Functions

Sample Question: Which graph could represent the graph of f(x)=sin(x)?

Answer Explanation: The sin function always graphs to look like a wave. The only one that could be the sin function is D.

Standards: HSF.IF.C.7

Click here to practice: Functions – Interpreting Functions Questions for Grade 10 Math

Domain: Grade 10 >> Functions – Building Functions

Sample Question: Describe how the graph of g(x)=x 3 – 5 can be obtained by shifting f(x) = x 3 + 2.

• Shift right 7 units
• Shift left 7 units
• Shift up 7 units
• Shift down 7 units

Answer Explanation: The only thing that changed in the two equations is the y-intercept which controls the vertical shift (up or down). To get the graph of g(x) by shifting the graph of f(x), you would shift f(x) down 7 units to change from +2 to -5.

Standards: HSF.BF.B.3

Click here to practice: Functions – Building Functions Questions for Grade 10 Math

Sample Question: Solve 3 x =12 using logarithmic form.

• x = ln12/ln3
• None of these

Answer Explanation: Solve using logs as follows 3 x =12 x=log(base 3) 12 x=ln12/ln3

Standards: HSF.LE.A.4

Domain: Grade 10 >> Functions – Trigonometric Functions

Sample Question: In the unit circle, one can see that tan(5π/4)=1 . What is the value of cos(5π/4)?

Standards: HSF.TF.A.2

Click here to practice: Functions – Trigonometric Functions Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Congruence

Sample Question: What would be the coordinates of point S after applying the following rule: (x+3, y -2)?

Answer Explanation: Answer: B Explanation: The transformation rule that is give is to translate the point 3 units to the right and 2 units down as shown by the following diagram:

Click here to practice: Geometry – Congruence Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Similarity, Right Triangles, & Trigonometry

Sample Question: By what property can the angles BAX and TSX be found to be congruent?

• Corresponding angles
• Vertical angles
• Alternate interior angles
• Congruent angles

Answer Explanation: Answer: A

Although they are congruent angles, the question is asking for the property. Since they are in corresponding locations with the transversal (AX) the correct answer is A

Standards: HSG.SRT.A.3

Click here to practice: Geometry – Similarity, Right Triangles, & Trigonometry Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Circles

Sample Question: What is the translation rule and the scale factor of the dilation as Circle F→Circle F′ ?

• (x,y)→1/4(x,y+10)
• (x,y)→4(x,y+10)
• (x,y)→1/4(x+10,y)
• (x,y)→1/4(x,y−10)

Answer Explanation: The original circle F has its center at the point (−5,−6) with a radius of 4 units. The translated/dilated circle F’ has its center at the point (−5,4) with a radius of 1 units. This means the center was translated up 10 units. As a transformation, this translation is written as (x,y)→(x,y+10). Circle F was also dilated by a factor of 1/4 because the radius was reduced from 4 units to 1 units. As a transformation, this dilation is written as (x,y)→1/4(x,y). Putting the translation and dilation together, the rule is (x,y)→1/4(x,y+10).

Click here to practice: Geometry – Circles Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Expressing Geometric Properties with Equations

Sample Question: What value on the number line in the figure below divides segment EF into two parts having a ratio of their lengths of 3:1?

Answer Explanation: Point E is at -7 on the number line in the figure, and pointF is at 1. Thus, the length of segment EF is 8. To divide the segment into two parts with a ratio of their lengths of 3:1, change the ratio to 3x:1x to allow variation in the location on the number line. Next, set the sum of the two parts equal to 8 and solve for x. 3x+1x=8;4x=8;x=2.Now, that you know that x=2, find 3x, which equals 6. Find the value on the number line by adding 6 to the position of point E. −7+6=−1.The value on the number line that divides segment EF in a ratio of 3:1 is -1.

Standards: HSG.GPE.B.6

Click here to practice: Geometry – Expressing Geometric Properties with Equations Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Geometric Measurement & Dimension

Sample Question: What is the volume of the prism shown below?

Answer Explanation: Use the formula for volume of a pyramid:

V=1/2.a.c.h

In this case the length is 15cm, the base is 10 cm in length, and the height is 9 cm. Therefore :

V=1/2.15.10.9=675cm 3

Standards: HSG.GMD.A.3

Click here to practice: Geometry – Geometric Measurement & Dimension Questions for Grade 10 Math

Domain: Grade 10 >> Geometry – Modeling with Geometry

Sample Question: A company ships spherical paperweights in cubic boxes. The circumference of the paperweight is 9π cm. If the box fits the sphere exactly with the sides of the sphere touching the box, what is the volume of the smallest box the company can use for shipping.

• 1009 π cm 3

Standards: HSG.MG.A.3

Click here to practice: Geometry – Modeling with Geometry Questions for Grade 10 Math

Domain: Grade 10 >> Statistics & Probability – Interpreting Categorical & Quantitative Data

Sample Question: Given the scatter plot below, what type of function expresses the correlation between the two variables?

• Exponential

Answer Explanation: Notice that the trend of the graph (in red) between the data points forms a line.

Standards: HSS.ID.A.4

Click here to practice: Statistics & Probability – Interpreting Categorical & Quantitative Data Questions for Grade 10 Math

Domain: Grade 10 >> Statistics & Probability – Making Inferences & Justifying Conclusions

Sample Question: In a research project about pet behavior, a random sample of 400 cats was chosen. The study showed that 60% of the cats preferred to sleep inside the house. Chicken was the favorite food for 35% of those cats. The study also showed that 85% of the cats that preferred to sleep outside the house had a different favorite dish. How many cats in the sample liked chicken the best and preferred to sleep inside?

Answer Explanation: If the sample has 400 cats and 60% of the cats preferred to sleep inside, then 400.0.60=240 cats preferred to sleep inside and 160 cats preferred to sleep outside. Next, if the favorite dish of 35% of those cats that preferred to sleep inside was chicken, then, 240.0.35=84 cats in the sample preferred to sleep inside and had chicken as their favorite dish.

Standards: HSS.IC.B.6

Click here to practice: Statistics & Probability – Making Inferences & Justifying Conclusions Questions for Grade 10 Math

Domain: Grade 10 >> Statistics & Probability – Conditional Probability & the Rules of Probability

Sample Question: A student council has one upcoming vacancy. The school is holding an election and has eight equally likely candidates. The AP Statistics class wants to simulate the results of the election, so the students have to choose an appropriate simulation method. They intend to do trials with the simulation. Which of these methods would be the most appropriate?

• Spin a wheel with eight equal spaces
• Toss a coin eight times for each election
• Throw a dice
• Throw four die

Answer Explanation: The question states that there are eight equally likely candidates. This means that each candidate has the same chance of winning the election. Only the spinning wheel with eight equal spaces could simulation this situation because the wheel has an equal chance of landing on each space.

Standards: HSS.IC.A.1

Click here to practice: Statistics & Probability – Conditional Probability & the Rules of Probability Questions for Grade 10 Math

Domain: Grade 10 >> Statistics & Probability – Using Probability to Make Decisions

Answer Explanation: Simply count the data points in circles C and E. There are 8 of them out of 24 total data points and by reducing we get 8/24=1/3.

Standards: HSS.CP.B.7

Click here to practice: Statistics & Probability – Using Probability to Make Decisions Questions for Grade 10 Math

Sample Question: A statistician is working for Sweet Shop USA and has been given the task to find out what the probability is that the fudge machine malfunctions messing up a whole batch of fudge in the process. Each malfunction of the machine costs the company $250. The statistician calculates the probability is 1 in 20 batches of fudge will be lost due to machine malfunction. What is the expected value of this loss for one month if the company produces 20 batches of fudge each day?

Answer Explanation: Since most months have 30 days we will assume 30 days in a month. We can use E(x)=x1p1+x2p2+…+xipi or simply calculate as follows E(X)=.05*250*30=$375

Standards: HSS.MD.A.4

Looking for online practice tests? Here is the link to practice more of SBAC Grade 10 Math questions.

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Jenny Watson

2 thoughts on “ grade 10 math questions and solutions ”.

Pingback: Create 10 questions for Grade 1 Algebra with explainations and answer choice - Parent.wiki

https://www.lumoslearning.com/llwp/sample-test-questions/sbac-sample-questions-grade-10-math.html

8/5 = x/576 should read 5/8 = x/576 The rest of the solution is correct so it’s a simple typo.

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4.5 Word problems

4.5 word problems (ema3d).

To solve word problems we need to write a set of equations that represent the problem mathematically. The solution of the equations is then the solution to the problem.

Problem solving strategy (EMA3F)

Read the whole question.

What are we asked to solve for?

Assign a variable to the unknown quantity, for example, \(x\).

Translate the words into algebraic expressions by rewriting the given information in terms of the variable.

Set up an equation or system of equations to solve for the variable.

Solve the equation algebraically using substitution.

Check the solution.

The following video shows two examples of working with word problems.

Video: 2FDX

Worked example 11: Solving word problems

A shop sells bicycles and tricycles. In total there are \(\text{7}\) cycles (cycles include both bicycles and tricycles) and \(\text{19}\) wheels. Determine how many of each there are, if a bicycle has two wheels and a tricycle has three wheels.

Assign variables to the unknown quantities

Let \(b\) be the number of bicycles and let \(t\) be the number of tricycles.

Set up the equations

Rearrange equation \(\left(1\right)\) and substitute into equation \(\left(2\right)\), calculate the number of tricycles \(t\), write the final answer.

There are \(\text{5}\) tricycles and \(\text{2}\) bicycles.

Worked example 12: Solving word problems

Bongani and Jane are friends. Bongani takes Jane's maths test paper and will not tell her what her mark is. He knows that Jane dislikes word problems so he decides to tease her. Bongani says: “I have \(\text{2}\) marks more than you do and the sum of both our marks is equal to \(\text{14}\). What are our marks?”

We have two unknown quantities, Bongani's mark and Jane's mark. Let Bongani's mark be \(b\) and Jane's mark be \(j\).

Set up a system of equations

Bongani has \(\text{2}\) more marks than Jane.

Both marks add up to \(\text{14}\).

Use equation \(\left(1\right)\) to express \(b\) in terms of \(j\)

Substitute into equation \(\left(2\right)\), rearrange and solve for \(j\), substitute the value for \(j\) back into equation \(\left(1\right)\) and solve for \(b\), check that the solution satisfies both original equations.

Bongani got \(\text{8}\) for his test and Jane got \(\text{6}\).

Worked example 13: Solving word problems

A fruitshake costs \(\text{R}\,\text{2,00}\) more than a chocolate milkshake. If \(\text{3}\) fruitshakes and \(\text{5}\) chocolate milkshakes cost \(\text{R}\,\text{78,00}\), determine the individual prices.

Let the price of a chocolate milkshake be \(x\) and let the price of a fruitshake be \(y\).

Substitute equation \(\left(1\right)\) into \(\left(2\right)\)

Rearrange and solve for \(x\), substitute the value of \(x\) back into equation \(\left(1\right)\) and solve for \(y\), write final answer.

One chocolate milkshake costs \(\text{R}\,\text{9,00}\) and one fruitshake costs \(\text{R}\,\text{11,00}\).

Worked example 14: Solving word problems

The product of two consecutive negative integers is \(\text{1 122}\). Find the two integers.

Let the first integer be \(n\) and let the second integer be \(n + 1\)

Set up an equation

Expand and solve for \(n\), find the sign of the integers.

It is given that both integers must be negative.

The two consecutive negative integers are \(-\text{34}\) and \(-\text{33}\).

Two jets are flying towards each other from airports that are \(\text{1 200}\) \(\text{km}\) apart. One jet is flying at \(\text{250}\) \(\text{km·h$^{-1}$}\) and the other jet at \(\text{350}\) \(\text{km·h$^{-1}$}\). If they took off at the same time, how long will it take for the jets to pass each other?

Let distance \(d_{1} = \text{1 200} - x\text{ km}\) and distance \(d_{2} = x\text{ km}\).

Speed \(s_{1}= \text{250}\text{ km·h$^{-1}$}\) and speed \(s_{2}= \text{350}\text{ km·h$^{-1}$}\).

Time is found by dividing distance by speed.

When the jets pass each other:

Now we know the distance travelled by the second jet when it passes the first jet, we can find the time:

It will take take the jets 2 hours to pass each other.

Two boats are moving towards each other from harbours that are \(\text{144}\) \(\text{km}\) apart. One boat is moving at \(\text{63}\) \(\text{km·h$^{-1}$}\) and the other boat at \(\text{81}\) \(\text{km·h$^{-1}$}\). If both boats started their journey at the same time, how long will they take to pass each other?

Notice that the sum of the distances for the two boats must be equal to the total distance when the boats meet: \(d_{1} + d_{2} = d_{\text{total}} \longrightarrow d_{1} + d_{2} = \text{144}\text{ km}\).

This question is about distances, speeds, and times. The equation connecting these values is \[\text{speed } = \frac{\text{distance }}{\text{time}} \quad \text{- or -} \quad \text{distance } = \text{speed } \times \text{time}\]

You want to know the amount of time needed for the boats to meet - let the time taken be \(t\). Then you can write an expression for the distance each of the boats travels: \begin{align*} \text{For boat 1:} \quad d_{1} &= s_{1} t \\ &= \text{63}t \\ \text{For boat 2:} \quad d_{2} &= s_{2} t \\ &= \text{81}t \end{align*}

Now we can substitute the two expressions for the distances into the expression for the total distance:

The boats will meet after \(\text{1}\) hour.

Zwelibanzi and Jessica are friends. Zwelibanzi takes Jessica's civil technology test paper and will not tell her what her mark is. He knows that Jessica dislikes word problems so he decides to tease her. Zwelibanzi says: “I have \(\text{12}\) marks more than you do and the sum of both our marks is equal to \(\text{148}\). What are our marks?”

Let Zwelibanzi's mark be \(z\) and let Jessica's mark be \(j\). Then \begin{align*} z &= j+\text{12} \\ z+j &= \text{148} \end{align*}

Substitute the first equation into the second equation and solve: \begin{align*} z+j &= \text{148} \\ (j+\text{12})+j &= \text{148} \\ 2j &= 148 - \text{12}\\ \therefore j &= \frac{\text{136}}{\text{2}}\\ &= \text{68} \end{align*}

Substituting this value back into the first equation gives: \begin{align*} z &= j+\text{12} \\ &= \text{68}+\text{12} \\ &= \text{80} \end{align*} Zwelibanzi achieved \(\text{80}\) marks and Jessica achieved \(\text{68}\) marks.

Kadesh bought \(\text{20}\) shirts at a total cost of \(\text{R}\,\text{980}\). If the large shirts cost \(\text{R}\,\text{50}\) and the small shirts cost \(\text{R}\,\text{40}\), how many of each size did he buy?

Let \(x\) be the number of large shirts and \(20 − x\) the number of small shirts.

Next we note the following:

• He bought \(x\) large shirts for \(\text{R}\,\text{50}\)
• He bought \(20 - x\) small shirts for \(\text{R}\,\text{40}\)
• He spent \(\text{R}\,\text{980}\) in total

We can represent the cost as:

Therefore Kadesh buys \(\text{18}\) large shirts and \(\text{2}\) small shirts.

The diagonal of a rectangle is \(\text{25}\) \(\text{cm}\) more than its width. The length of the rectangle is \(\text{17}\) \(\text{cm}\) more than its width. What are the dimensions of the rectangle?

Let length \(= l\), width \(= w\) and diagonal \(= d\). \(\therefore d = w + 25\) and \(l = w + 17\).

By the theorem of Pythagoras:

The width must be positive, therefore: width \(w = \text{28}\text{ cm}\) length \(l = (w + 17) = \text{45}\text{ cm}\) and diagonal \(d = (w + 25) = \text{53}\text{ cm}\).

The sum of \(\text{27}\) and \(\text{12}\) is equal to \(\text{73}\) more than an unknown number. Find the unknown number.

Let the unknown number \(= x\).

The unknown number is \(-\text{34}\).

A group of friends is buying lunch. Here are some facts about their lunch:

• a milkshake costs \(\text{R}\,\text{7}\) more than a wrap
• the group buys 8 milkshakes and 2 wraps
• the total cost for the lunch is \(\text{R}\,\text{326}\)

Let a milkshake be \(m\) and a wrap be \(w\). From the given information we get the following equations:

Substitute the first equation into the second equation and solve for \(w\):

Substitute the value of \(w\) into the first equation and solve for \(m\):

Therefore a milkshake costs \(\text{R}\,\text{34}\) and a wrap costs \(\text{R}\,\text{27}\).

The two smaller angles in a right-angled triangle are in the ratio of \(1:2\). What are the sizes of the two angles?

Let \(x =\) the smallest angle. Therefore the other angle \(= 2x\).

We are given the third angle \(=90°\).

The sizes of the angles are \(30°\) and \(60°\).

The length of a rectangle is twice the breadth. If the area is \(\text{128}\) \(\text{cm$^{2}$}\), determine the length and the breadth.

We are given length \(l = 2b\) and \(A = l \times b = 128\).

Substitute the first equation into the second equation and solve for \(b\):

But breadth must be positive, therefore \(b = 8\).

Substitute this value into the first equation to solve for \(l\):

Therefore \(b = \text{8}\text{ cm}\) and \(l = 2b = \text{16}\text{ cm}\).

If \(\text{4}\) times a number is increased by \(\text{6}\), the result is \(\text{15}\) less than the square of the number. Find the number.

Let the number \(= x\). The equation that expresses the given information is:

We are not told if the number is positive or negative. Therefore the number is \(\text{7}\) or \(-\text{3}\).

The length of a rectangle is \(\text{2}\) \(\text{cm}\) more than the width of the rectangle. The perimeter of the rectangle is \(\text{20}\) \(\text{cm}\). Find the length and the width of the rectangle.

Let length \(l = x\), width \(w = x - 2\) and perimeter \(= p\).

\(l = \text{6}\text{ cm}\) and \(w = l - 2 = \text{4}\text{ cm}\).

Stephen has 1 litre of a mixture containing \(\text{69}\%\) salt. How much water must Stephen add to make the mixture \(\text{50}\%\) salt? Write your answer as a fraction of a litre.

The new volume (\(x\)) of mixture must contain \(\text{50}\%\) salt, therefore:

The volume of the new mixture is \(\text{1,38}\) litre The amount of water (\(y\)) to be added is:

Therefore \(\text{0,38}\) litres of water must be added. To write this as a fraction of a litre: \(\text{0,38} = \frac{38}{100} = \frac{19}{50} \text{ litres}\)

Therefore \(\frac{19}{50} \text{ litres}\) must be added.

The sum of two consecutive odd numbers is \(\text{20}\) and their difference is \(\text{2}\). Find the two numbers.

Let the numbers be \(x\) and \(y\).

Then the two equations describing the constraints are:

Add the first equation to the second equation:

Substitute into first equation:

Therefore the two numbers are 9 and 11.

The denominator of a fraction is \(\text{1}\) more than the numerator. The sum of the fraction and its reciprocal is \(\frac{5}{2}\). Find the fraction.

Let the numerator be \(x\). So the denominator is \(x + 1\).

Solve for \(x\):

From this the fraction could be \(\frac{1}{2}\) or \(\frac{-2}{-1}\). For the second solution we can simplify the fraction to \(\text{2}\) and in this case the denominator is not 1 less than the numerator.

So the fraction is \(\frac{1}{2}\).

Masindi is \(\text{21}\) years older than her daughter, Mulivhu. The sum of their ages is \(\text{37}\). How old is Mulivhu?

Let Mulivhu be \(x\) years old. So Masindi is \(x + 21\) years old.

Mulivhu is \(\text{8}\) years old.

Tshamano is now five times as old as his son Murunwa. Seven years from now, Tshamano will be three times as old as his son. Find their ages now.

Let Murunwa be \(x\) years old. So Tshamano is \(5x\) years old.

In \(\text{7}\) years time Murunwa's age will be \(x + 7\). Tshamano's age will be \(5x + 7\).

So Murunwa is 7 years old and Tshamano is 35 years old.

If adding one to three times a number is the same as the number, what is the number equal to?

Let the number be \(x\). Then:

If a third of the sum of a number and one is equivalent to a fraction whose denominator is the number and numerator is two, what is the number?

Rearrange until we get a trinomial and solve for \(x\):

A shop owner buys 40 sacks of rice and mealie meal worth \(\text{R}\,\text{5 250}\) in total. If the rice costs \(\text{R}\,\text{150}\) per sack and mealie meal costs \(\text{R}\,\text{100}\) per sack, how many sacks of mealie meal did he buy?

There are 100 bars of blue and green soap in a box. The blue bars weigh \(\text{50}\) \(\text{g}\) per bar and the green bars \(\text{40}\) \(\text{g}\) per bar. The total mass of the soap in the box is \(\text{4,66}\) \(\text{kg}\). How many bars of green soap are in the box?

Lisa has 170 beads. She has blue, red and purple beads each weighing \(\text{13}\) \(\text{g}\), \(\text{4}\) \(\text{g}\) and \(\text{8}\) \(\text{g}\) respectively. If there are twice as many red beads as there are blue beads and all the beads weigh \(\text{1,216}\) \(\text{kg}\), how many beads of each type does Lisa have?

• 10th Grade Math

10th grade math topics are planned and covered all the lessons in different segments. 10th grade math help is provided for the 10th grade students in all segments to cover all the math lesson plans which are categorized into Arithmetic, Algebra, Geometry, Mensuration and Trigonometry.

All types of solved examples on different topics are explained along with the step-by-step solutions so that students can attempt how to solve these problems quickly in your own convenient methods. 10th grade math practice sheets are presented in such a way that students can learn math while practicing math problems.

Keeping in mind the mental level of student in tenth grade, every efforts has been made to introduce new concepts in a simple and easy language, so that the students can understand the problems easily.

The difficulty level of the 10th grade math problems has emphasized the theoretical as well as the numerical aspects of the mathematics course. Each topic contains a large number of examples to understand the applications of concepts.

To get prepared for 10th grade math test or exams students need to learn problems ratio and proportion, profit & loss and interest (simple and compound), ratio, proportion and variation, inequation, surds, graphing lines on the coordinate plane, solving literal equations, compound inequalities, graphing inequalities in two variables, multiplying binomials, polynomials, factoring techniques for trinomials, solving systems of equations, algebra word problems, variation, rational expressions, rational equations, graphs & functions, theorems on circle, theorems on tangent to a circle, theorems on similarity, constructions and appendix, triangle theorems & proofs, properties of polygons, transformations, trigonometric ratios, trigonometric ratios of complementary angles, application of trigonometric ratios in practical problems and etc……

If student follow math-only-math they can improve their knowledge by practicing the worksheets for 10th graders which will help them to score in their exam.

Tenth Grade Math Lessons – Table of Contents

Commercial mathematics.

●  Compound Interest

• Compound Interest Definition
• Compound Interest
• Compound Interest with Growing Principal
• Compound Interest with Periodic Deductions
• Compound Interest by Using Formula
• Compound Interest when Interest is Compounded Yearly
• Compound Interest when Interest is Compounded Half-Yearly
• Compound Interest when Interest is Compounded Quarterly
• Problems on Compound Interest
• Variable Rate of Compound Interest
• Difference of Compound Interest and Simple Interest
• Practice Test on Compound Interest
• Uniform Rate of Growth
• Uniform Rate of Depreciation
• Uniform Rate of Growth and Depreciation

● Sales Tax and Value Added Tax

• Calculation of Sales Tax
• Sales Tax in a Bill
• Mark-ups and Discounts Involving Sales Tax
• Profit Loss Involving Tax
• Value Added Tax
• Problems on Value Added Tax (VAT)
• Worksheet on Printed Price, Rate of Sales Tax and Selling Price
• Worksheet on Profit/Loss Involving Sales Tax
• Worksheet on Sales Tax and Value-added Tax
• Worksheet on Mark-ups and Discounts Involving Sales Tax

● Shares and Dividends

• Share and Value of Shares
• Dividend and Rate of Dividend
• Calculation of Income, Return and Number of Shares
• Problems on Income and Return from Shares
• Problems on Shares and Dividends
• Worksheet on Basic Concept on Shares and Dividends
• Worksheet on Income and Return from Shares
• Worksheet on Share and Dividend

Algebra/Linear Algebra

● Linear Inequations in One Variable (Unknown)

• Linear Inequation in One Variable
• Laws of Inequality
• Replacement Set and Solution Set in Set Notation
• Difference Between Equation and Inequation
• Solving a Linear Inequation Algebraically
• Problems on Law of Inequality
• Problems on Linear Inequation
• Worksheet on Laws of Inequality
• Worksheet on Solution of a Linear Inequation in One Variable

●  Quadratic Equation

• Introduction to Quadratic Equation
• Formation of Quadratic Equation in One Variable
• Solving Quadratic Equations
• General Properties of Quadratic Equation
• Methods of Solving Quadratic Equations
• Roots of a Quadratic Equation
• Examine the Roots of a Quadratic Equation
• Problems on Quadratic Equations
• Quadratic Equations by Factoring
• Word Problems Using Quadratic Formula
• Examples on Quadratic Equations
• Word Problems on Quadratic Equations by Factoring
• Worksheet on Formation of Quadratic Equation in One Variable
• Worksheet on Quadratic Formula
• Worksheet on Nature of the Roots of a Quadratic Equation
• Worksheet on Word Problems on Quadratic Equations by Factoring

●  Factorization

• Polynomial Equation and its Roots
• Division Algorithm
• Remainder Theorem
• Problems on Remainder Theorem
• Factors of a Polynomial
• Worksheet on Remainder Theorem
• Factor Theorem
• Application of Factor Theorem

● Ratio and proportion

• Basic Concept of Ratios
• Important Properties of Ratios
• Ratio in Lowest Term
• Types of Ratios
• Comparing Ratios
• Arranging Ratios
• Dividing into a Given Ratio
• Divide a Number into Three Parts in a Given Ratio
• Dividing a Quantity into Three Parts in a Given Ratio
• Problems on Ratio
• Worksheet on Ratio in Lowest Term
• Worksheet on Types of Ratios
• Worksheet on Comparison on Ratios
• Worksheet on Ratio of Two or More Quantities
• Worksheet on Dividing a Quantity in a Given Ratio
• Word Problems on Ratio
• Definition of Continued Proportion
• Mean and Third Proportional
• Word Problems on Proportion
• Worksheet on Proportion and Continued Proportion
• Worksheet on Mean Proportional
• Properties of Ratio and Proportion

●  Matrix

• Definition of a Matrix
• Order of a Matrix
• Position of an Element in a Matrix
• Classification of Matrices
• Problems on Classification of Matrices
• Square Matrix
• Column Matrix
• Null Matrix
• Equal Matrices
• Identity (or Unit) Matrix
• Triangular Matrix
• Addition of Matrices
• Addition of Two Matrices
• Properties of Addition of Matrices
• Negative of a Matrix
• Subtraction of Matrices
• Subtraction of Two Matrices
• Scalar Multiplication of a Matrix
• Multiplication of a Matrix by a Number
• Properties of Scalar Multiplication of a Matrix
• Multiplication of Matrices
• Multiplication of Two Matrices
• Problems on Understanding Matrices
• Worksheet on Understanding Matrix
• Worksheet on Addition of Matrices
• Worksheet on Matrix Multiplication
• Worksheet on Matrix

Reflection and Coordinate Geometry

● Reflection

• Position of a Point in a Plane
• Reflection of a Point in a Line
• Reflection of a Point in the x-axis
• Reflection of a Point in the y-axis
• Reflection of a Point in the Origin
• Reflection of a Point in a Line Parallel to the x-axis
• Reflection of a Point in a Line Parallel to the y-axis
• Problems on Reflection in the x-axis or y-axis
• Invariant Points for Reflection in a Line
• Reflection in Lines Parallel to Axes
• Worksheet on Reflection in the Origin

● Distance and Section Formulae

• Distance Formula
• Distance Properties in some Geometrical Figures
• Conditions of Collinearity of Three Points
• Problems on Distance Formula
• Distance of a Point from the Origin
• Distance Formula in Geometry
• Section Formula
• Midpoint Formula
• Centroid of a Triangle
• Worksheet on Distance Formula
• Worksheet on Collinearity of Three Points
• Worksheet on Finding the Centroid of a Triangle
• Worksheet on Section Formula

● Equation of a Straight Line

• Inclination of a Line
• Slope of a Line
• Intercepts Made by a Straight Line on Axes
• Slope of the Line Joining Two Points
• Equation of a Straight Line
• Slope-intercept Form of a Straight Line
• Point-slope Form of a Line
• Two-point Form of a Line
• Equally Inclined Lines
• Slope and Y-intercept of a Line
• Condition of Perpendicularity of Two Straight Lines
• Condition of parallelism
• Problems on Condition of Perpendicularity
• Worksheet on Slope and Intercepts
• Worksheet on Slope Intercept Form
• Worksheet on Two-point Form
• Worksheet on Point-slope Form
• Worksheet on Collinearity of 3 Points
• Worksheet on Equation of a Straight Line

Geometry and Measurement

●   Loci

• Concept of loci
• Theorems on Locus of a Point which is Equidistant from Two Fixed Points

●  Properties of Tangents

• Secant and Tangent
• Common Tangents to Two Circles
• Tangent - Perpendicular to Radius
• Two Circles Touch each Other
• Two Tangents from an External Point
• Angles between the Tangent and the Chord
• Problems on Properties on Tangents
• Chord and Tangent Intersect Externally
• Direct Common Tangents
• Important Properties of Direct Common Tangents
• Transverse Common Tangents
• Important Properties of Transverse Common Tangents
• Examples of Loci Based on Circles Touching Straight Lines or Other Circles
• Circumcircle of a Triangle
• Incircle of a Triangle
• Circumcentre and Incentre of a Triangle
• Solved on the Basic Properties of Tangents
• Problems on Two Tangents to a Circle from an External Point
• Two Tangents are Drawn to a Circle from an External Point
• Two Parallel Tangents of a Circle Meet a Third Tangent
• Contact of Two Circles
• Tangent is Parallel to a Chord
• Measure of the Angles of the Cyclic Quadrilateral
• Problems on Relation Between Tangent and Secant
• Problems on Common Tangent to Two Circles

●  Area and Perimeter of a Circle

• Area and Perimeter of a Circle
• Area and Perimeter of a Sector of a Circle
• Area and Perimeter of a Semicircle and Quadrant of a Circle
• Area of Combined Figures
• Area of the Shaded Region
• Find the Area of the Shaded Region
• Application Problems on Area of a Circle

Trigonometry

●  Trigonometrical Ratios and Identities

• Basic Trigonometric Ratios : Learn the determination of trigonometrical ratios and their names.
• Relations Between the Trigonometric Ratios : Learn how the trigonometric ratios related between each other.
• Properties of Trigonometrical Ratios: Learn how to prove the properties of basic formula of trigonometrical ratios.
• Problems on Trigonometric Ratios :
• Reciprocal Relations of Trigonometric Ratios :
• Trigonometrical Identity :
• Problems on Trigonometric Identities :
• Elimination of Trigonometric Ratios :
• Eliminate Theta between the equations :
• Problems on Eliminate Theta :
• Trig Ratio Problems :
• Proving Trigonometric Ratios :
• Trig Ratios Proving Problems :
• Verify Trigonometric Identities :
• Trigonometric Identities
• Elimination of Unknown Angles
• Finding the Unknown Angle
• Worksheet on Trigonometric Identities
• Worksheet on Evaluation Using Trigonometric Identities
• Worksheet on Establishing Conditional Results
• Worksheet on Elimination of Unknown Angle(s)
• Worksheet on Finding the Unknown Angle
• Problems on Relation between the Rations: Learn how to solve the problems based on the relation between the trigonometrical ratios.

●   Trigonometric Ratios of Complementary Angles

• Complementary Angles and their Trigonometric Ratios
• Worksheet on Evaluation using Trigonometric Ratios of Complementary Angles
• Worksheet on Establishing Equality using Trigonometric Ratios of Complementary Angles
• Worksheet on Establishing Identities and Simplification Using Trigonometric Ratios of Complementary Angles

●   Height and Distance

• Angle of Elevation
• Angle of Depression
• Height and Distance with Two Angles of Elevation
• Finding sin Value from Trigonometric Table
• Finding cos Value from Trigonometric Table
• Finding tan Value from Trigonometric Table
• Worksheet on Heights and Distances

Statistics and Probability 

●   Graphical Representation

• Frequency Polygon
• Method of Constructing a Frequency Polygon with the Help of a Histogram
• Method of Constructing Frequency Polygon with the Help of Class Marks
• Cumulative-Frequency Curve
• Problems on Histogram
• Problems on Frequency Polygon
• Problems on Cumulative-Frequency Curve

●   Measures of Central Tendency

• Mean of Ungrouped Data
• Problems on Mean of Ungrouped Data
• Mean of Grouped Data
• Mean of Classified Data
• Step-deviation Method
• Finding the Mean from Graphical Representation
• Worksheet on Finding the Mean of Raw Data
• Worksheet on Finding the Mean of Arrayed Data
• Median of Raw Data
• Problems on Median of Raw Data
• Finding the Median of Grouped Data
• Lower Quartile and the Method of Finding it for Raw Data
• Upper Quartile and the Method of Finding it for Raw Data
• Find the Quartiles for Arrayed Data
• Range and Interquartile Range
• Median Class
• Estimate Median, Quartiles from Ogive
• Worksheet on Finding the Median of Raw Data
• Worksheet on Finding the Median of Arrayed Data
• Worksheet on Finding the Quartiles & Interquartile Range of Raw Data
• Worksheet on Estimating Median and the Quartiles using Ogive

●   Probability

• Probability
• Definition of Probability
• Random Experiments
• Experimental Probability
• Events in Probability
• Empirical Probability
• Coin Toss Probability
• Probability of Tossing Two Coins
• Probability of Tossing Three Coins
• Complimentary Events
• Mutually Exclusive Events
• Mutually Non-Exclusive Events
• Conditional Probability
• Theoretical Probability
• Odds and Probability
• Playing Cards Probability
• Probability and Playing Cards
• Probability Rolling a Die
• Probability for Rolling Two Dice
• Probability for Rolling Three Dice
• Solved Probability Problems
• Probability Questions Answers
• Coin Toss Probability Worksheet
• Worksheet on Playing Cards
• 10th Grade Worksheet on Probability

Pre-calculus

Mathematical models with applications.

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Recent Articles

Multiplication by Ten, Hundred and Thousand |Multiply by 10, 100 &1000

Mar 29, 24 06:35 PM

Properties of Multiplication | Multiplicative Identity | Whole Numbers

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Multiply a Number by a 2-Digit Number | Multiplying 2-Digit by 2-Digit

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Mar 26, 24 04:14 PM

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Math Problem Solving Strategies

In these lessons, we will learn some math problem solving strategies for example, Verbal Model (or Logical Reasoning), Algebraic Model, Block Model (or Singapore Math), Guess & Check Model and Find a Pattern Model.

Related Pages Solving Word Problems Using Block Models Heuristic Approach to Problem-Solving Algebra Lessons

Problem Solving Strategies

The strategies used in solving word problems:

• What do you know?
• What do you need to know?
• Draw a diagram/picture

Solution Strategies Label Variables Verbal Model or Logical Reasoning Algebraic Model - Translate Verbal Model to Algebraic Model Solve and Check.

Solving Word Problems

Step 1: Identify (What is being asked?) Step 2: Strategize Step 3: Write the equation(s) Step 4: Answer the question Step 5: Check

Problem Solving Strategy: Guess And Check

Using the guess and check problem solving strategy to help solve math word problems.

Example: Jamie spent $40 for an outfit. She paid for the items using $10, $5 and $1 bills. If she gave the clerk 10 bills in all, how many of each bill did she use?

Problem Solving : Make A Table And Look For A Pattern

• Identify - What is the question?
• Plan - What strategy will I use to solve the problem?
• Solve - Carry out your plan.
• Verify - Does my answer make sense?

Example: Marcus ran a lemonade stand for 5 days. On the first day, he made $5. Every day after that he made $2 more than the previous day. How much money did Marcus made in all after 5 days?

Find A Pattern Model (Intermediate)

In this lesson, we will look at some intermediate examples of Find a Pattern method of problem-solving strategy.

Example: The figure shows a series of rectangles where each rectangle is bounded by 10 dots. a) How many dots are required for 7 rectangles? b) If the figure has 73 dots, how many rectangles would there be?

a) The number of dots required for 7 rectangles is 52.

b) If the figure has 73 dots, there would be 10 rectangles.

Example: Each triangle in the figure below has 3 dots. Study the pattern and find the number of dots for 7 layers of triangles.

The number of dots for 7 layers of triangles is 36.

Example: The table below shows numbers placed into groups I, II, III, IV, V and VI. In which groups would the following numbers belong? a) 25 b) 46 c) 269

Solution: The pattern is: The remainder when the number is divided by 6 determines the group. a) 25 ÷ 6 = 4 remainder 1 (Group I) b) 46 ÷ 6 = 7 remainder 4 (Group IV) c) 269 ÷ 6 = 44 remainder 5 (Group V)

Example: The following figures were formed using matchsticks.

a) Based on the above series of figures, complete the table below.

b) How many triangles are there if the figure in the series has 9 squares?

c) How many matchsticks would be used in the figure in the series with 11 squares?

b) The pattern is +2 for each additional square.   18 + 2 = 20   If the figure in the series has 9 squares, there would be 20 triangles.

c) The pattern is + 7 for each additional square   61 + (3 x 7) = 82   If the figure in the series has 11 squares, there would be 82 matchsticks.

Example: Seven ex-schoolmates had a gathering. Each one of them shook hands with all others once. How many handshakes were there?

Total = 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakes.

The following video shows more examples of using problem solving strategies and models. Question 1: Approximate your average speed given some information Question 2: The table shows the number of seats in each of the first four rows in an auditorium. The remaining ten rows follow the same pattern. Find the number of seats in the last row. Question 3: You are hanging three pictures in the wall of your home that is 16 feet wide. The width of your pictures are 2, 3 and 4 feet. You want space between your pictures to be the same and the space to the left and right to be 6 inches more than between the pictures. How would you place the pictures?

The following are some other examples of problem solving strategies.

Explore it/Act it/Try it (EAT) Method (Basic) Explore it/Act it/Try it (EAT) Method (Intermediate) Explore it/Act it/Try it (EAT) Method (Advanced)

Finding A Pattern (Basic) Finding A Pattern (Intermediate) Finding A Pattern (Advanced)

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Trigonometry Problems and Questions with Solutions - Grade 10

Grade 10 trigonometry problems and questions with answers and solutions are presented.

• The area of a right triangle is 50. One of its angles is 45°. Find the lengths of the sides and hypotenuse of the triangle.
• In a right triangle ABC, tan(A) = 3/4. Find sin(A) and cos(A).
• In a right triangle ABC with angle A equal to 90°, find angle B and C so that sin(B) = cos(B).
• A rectangle has dimensions 10 cm by 5 cm. Determine the measures of the angles at the point where the diagonals intersect.
• The lengths of side AB and side BC of a scalene triangle ABC are 12 cm and 8 cm respectively. The size of angle C is 59°. Find the length of side AC.
• From the top of a 200 meters high building, the angle of depression to the bottom of a second building is 20 degrees. From the same point, the angle of elevation to the top of the second building is 10 degrees. Calculate the height of the second building.
• Karla is riding vertically in a hot air balloon, directly over a point P on the ground. Karla spots a parked car on the ground at an angle of depression of 30°. The balloon rises 50 meters. Now the angle of depression to the car is 35 degrees. How far is the car from point P?
• If the shadow of a building increases by 10 meters when the angle of elevation of the sun rays decreases from 70° to 60°, what is the height of the building?

Solutions to the Above Problems

• x = 10 / tan(51°) = 8.1 (2 significant digits) H = 10 / sin(51°) = 13 (2 significant digits)
• Area = (1/2)(2x)(x) = 400 Solve for x: x = 20 , 2x = 40 Pythagora's theorem: (2x) 2 + (x) 2 = H 2 H = x √(5) = 20 √(5)
• BH perpendicular to AC means that triangles ABH and HBC are right triangles. Hence tan(39°) = 11 / AH or AH = 11 / tan(39°) HC = 19 - AH = 19 - 11 / tan(39°) Pythagora's theorem applied to right triangle HBC: 11 2 + HC 2 = x 2 solve for x and substitute HC: x = √ [ 11 2 + (19 - 11 / tan(39°) ) 2 ] = 12.3 (rounded to 3 significant digits)
• Since angle A is right, both triangles ABC and ABD are right and therefore we can apply Pythagora's theorem. 14 2 = 10 2 + AD 2 , 16 2 = 10 2 + AC 2 Also x = AC - AD = √( 16 2 - 10 2 ) - √( 14 2 - 10 2 ) = 2.69 (rounded to 3 significant digits)
• Use right triangle ABC to write: tan(31°) = 6 / BC , solve: BC = 6 / tan(31°) Use Pythagora's theorem in the right triangle BCD to write: 9 2 + BC 2 = BD 2 Solve above for BD and substitute BC: BD = √ [ 9 + ( 6 / tan(31°) ) 2 ] = 13.4 (rounded to 3 significant digits)
• The triangle is right and the size one of its angles is 45°; the third angle has a size 45° and therefore the triangle is right and isosceles. Let x be the length of one of the sides and H be the length of the hypotenuse. Area = (1/2)x 2 = 50 , solve for x: x = 10 We now use Pythagora to find H: x 2 + x 2 = H 2 Solve for H: H = 10 √(2)
• Let a be the length of the side opposite angle A, b the length of the side adjacent to angle A and h be the length of the hypotenuse. tan(A) = opposite side / adjacent side = a/b = 3/4 We can say that: a = 3k and b = 4k , where k is a coefficient of proportionality. Let us find h. Pythagora's theorem: h 2 = (3k) 2 + (5k) 2 Solve for h: h = 5k sin(A) = a / h = 3k / 5k = 3/5 and cos(A) = 4k / 5k = 4/5
• Let b be the length of the side opposite angle B and c the length of the side opposite angle C and h the length of the hypotenuse. sin(B) = b/h and cos(B) = c/h sin(B) = cos(B) means b/h = c/h which gives c = b The two sides are equal in length means that the triangle is isosceles and angles B and C are equal in size of 45°.
• Let x be the length of side AC. Use the cosine law 12 2 = 8 2 + x 2 - 2 · 8 · x · cos(59°) Solve the quadratic equation for x: x = 14.0 and x = - 5.7 x cannot be negative and therefore the solution is x = 14.0 (rounded to one decimal place).

More References and links on Trigonometry

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10 Strategies for Problem Solving in Math

Created: May 19, 2022

Last updated: January 6, 2024

When faced with problem-solving, children often get stuck. Word puzzles and math questions with an unknown variable, like x, usually confuse them. Therefore, this article discusses math strategies and how your students may use them since instructors often have to lead students through this problem-solving maze.

What Are Problem Solving Strategies in Math?

If you want to fix a problem, you need a solid plan. Math strategies for problem solving are ways of tackling math in a way that guarantees better outcomes. These strategies simplify math for kids so that less time is spent figuring out the problem. Both those new to mathematics and those more knowledgeable about the subject may benefit from these methods.

There are several methods to apply problem-solving procedures in math, and each strategy is different. While none of these methods failsafe, they may help your student become a better problem solver, particularly when paired with practice and examples. The more math problems kids tackle, the more math problem solving skills they acquire, and practice is the key.

Strategies for Problem-solving in Math

Even if a student is not a math wiz, a suitable solution to mathematical problems in math may help them discover answers. There is no one best method for helping students solve arithmetic problems, but the following ten approaches have shown to be very effective.

Understand the Problem

Understanding the nature of math problems is a prerequisite to solving them. They need to specify what kind of issue it is ( fraction problem , word problem, quadratic equation, etc.). Searching for keywords in the math problem, revisiting similar questions, or consulting the internet are all great ways to strengthen their grasp of the material. This step keeps the pupil on track.

Math for Kids

Guess and Check

One of the time-intensive strategies for resolving mathematical problems is the guess and check method. In this approach, students keep guessing until they get the answer right.

After assuming how to solve a math issue, students should reintroduce that assumption to check for correctness. While the approach may appear cumbersome, it is typically successful in revealing patterns in a child’s thought process.

Work It Out

Encourage pupils to record their thinking process as they go through a math problem. Since this technique requires an initial comprehension of the topic, it serves as a self-monitoring method for mathematics students. If they immediately start solving the problem, they risk making mistakes.

Students may keep track of their ideas and fix their math problems as they go along using this method. A youngster may still need you to explain their methods of solving the arithmetic questions on the extra page. This confirmation stage etches the steps they took to solve the problem in their minds.

Work Backwards

In mathematics, a fresh perspective is sometimes the key to a successful solution. Young people need to know that the ability to recreate math problems is valuable in many professional fields, including project management and engineering.

Students may better prepare for difficulties in real-world circumstances by using the “Work Backwards” technique. The end product may be used as a start-off point to identify the underlying issue.

In most cases, a visual representation of a math problem may help youngsters understand it better. Some of the most helpful math tactics for kids include having them play out the issue and picture how to solve it.

One way to visualize a workout is to use a blank piece of paper to draw a picture or make tally marks. Students might also use a marker and a whiteboard to draw as they demonstrate the technique before writing it down.

Find a Pattern

Kids who use pattern recognition techniques can better grasp math concepts and retain formulae. The most remarkable technique for problem solving in mathematics is to help students see patterns in math problems by instructing them how to extract and list relevant details. This method may be used by students when learning shapes and other topics that need repetition.

Students may use this strategy to spot patterns and fill in the blanks. Over time, this strategy will help kids answer math problems quickly.

When faced with a math word problem, it might be helpful to ask, “What are some possible solutions to this issue?” It encourages you to give the problem more thought, develop creative solutions, and prevent you from being stuck in a rut. So, tell the pupils to think about the math problems and not just go with the first solution that comes to mind.

Draw a Picture or Diagram

Drawing a picture of a math problem can help kids understand how to solve it, just like picturing it can help them see it. Shapes or numbers could be used to show the forms to keep things easy. Kids might learn how to use dots or letters to show the parts of a pattern or graph if you teach them.

Charts and graphs can be useful even when math isn’t involved. Kids can draw pictures of the ideas they read about to help them remember them after they’ve learned them. The plan for how to solve the mathematical problem will help kids understand what the problem is and how to solve it.

Trial and Error Method

The trial and error method may be one of the most common problem solving strategies for kids to figure out how to solve problems. But how well this strategy is used will determine how well it works. Students have a hard time figuring out math questions if they don’t have clear formulas or instructions.

They have a better chance of getting the correct answer, though, if they first make a list of possible answers based on rules they already know and then try each one. Don’t be too quick to tell kids they shouldn’t learn by making mistakes.

Review Answers with Peers

It’s fun to work on your math skills with friends by reviewing the answers to math questions together. If different students have different ideas about how to solve the same problem, get them to share their thoughts with the class.

During class time, kids’ ways of working might be compared. Then, students can make their points stronger by fixing these problems.

Check out the Printable Math Worksheets for Your Kids!

There are different ways to solve problems that can affect how fast and well students do on math tests. That’s why they need to learn the best ways to do things. If students follow the steps in this piece, they will have better experiences with solving math questions.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

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120 Math Word Problems To Challenge Students Grades 1 to 8

Written by Marcus Guido

Hey teachers! 👋

Use Prodigy to spark a love for math in your students – including when solving word problems!

• Teaching Tools
• Subtraction
• Multiplication
• Mixed operations
• Ordering and number sense
• Comparing and sequencing
• Physical measurement
• Ratios and percentages
• Probability and data relationships

You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.

A jolt of creativity would help. But it doesn’t come.

Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.

This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.

There are 120 examples in total.

The list of examples is supplemented by tips to create engaging and challenging math word problems.

120 Math word problems, categorized by skill

Addition word problems.

Best for: 1st grade, 2nd grade

1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total?

2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store to get 3 more pieces of gum. How many pieces of gum does Adrianna have now?

3. Adding to 100: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store and got 70 pieces of strawberry gum and 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

4. Adding Slightly over 100: The restaurant has 175 normal chairs and 20 chairs for babies. How many chairs does the restaurant have in total?

5. Adding to 1,000: How many cookies did you sell if you sold 320 chocolate cookies and 270 vanilla cookies?

6. Adding to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In June, the hobby store sold 15,498 more trading cards than normal. In total, how many trading cards did the hobby store sell in June?

7. Adding 3 Numbers: Billy had 2 books at home. He went to the library to take out 2 more books. He then bought 1 book. How many books does Billy have now?

8. Adding 3 Numbers to and over 100: Ashley bought a big bag of candy. The bag had 102 blue candies, 100 red candies and 94 green candies. How many candies were there in total?

Subtraction word problems

Best for: 1st grade, second grade

9. Subtracting to 10: There were 3 pizzas in total at the pizza shop. A customer bought 1 pizza. How many pizzas are left?

10. Subtracting to 20: Your friend said she had 11 stickers. When you helped her clean her desk, she only had a total of 10 stickers. How many stickers are missing?

11. Subtracting to 100: Adrianna has 100 pieces of gum to share with her friends. When she went to the park, she shared 10 pieces of strawberry gum. When she left the park, Adrianna shared another 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

Practice math word problems with Prodigy Math

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12. Subtracting Slightly over 100: Your team scored a total of 123 points. 67 points were scored in the first half. How many were scored in the second half?

13. Subtracting to 1,000: Nathan has a big ant farm. He decided to sell some of his ants. He started with 965 ants. He sold 213. How many ants does he have now?

14. Subtracting to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In July, the hobby store sold a total of 20,777 trading cards. How many more trading cards did the hobby store sell in July compared with a normal month?

15. Subtracting 3 Numbers: Charlene had a pack of 35 pencil crayons. She gave 6 to her friend Theresa. She gave 3 to her friend Mandy. How many pencil crayons does Charlene have left?

16. Subtracting 3 Numbers to and over 100: Ashley bought a big bag of candy to share with her friends. In total, there were 296 candies. She gave 105 candies to Marissa. She also gave 86 candies to Kayla. How many candies were left?

Multiplication word problems

Best for: 2nd grade, 3rd grade

17. Multiplying 1-Digit Integers: Adrianna needs to cut a pan of brownies into pieces. She cuts 6 even columns and 3 even rows into the pan. How many brownies does she have?

18. Multiplying 2-Digit Integers: A movie theatre has 25 rows of seats with 20 seats in each row. How many seats are there in total?

19. Multiplying Integers Ending with 0: A clothing company has 4 different kinds of sweatshirts. Each year, the company makes 60,000 of each kind of sweatshirt. How many sweatshirts does the company make each year?

20. Multiplying 3 Integers: A bricklayer stacks bricks in 2 rows, with 10 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?

21. Multiplying 4 Integers: Cayley earns $5 an hour by delivering newspapers. She delivers newspapers 3 days each week, for 4 hours at a time. After delivering newspapers for 8 weeks, how much money will Cayley earn?

Division word problems

Best for: 3rd grade, 4th grade, 5th grade

22. Dividing 1-Digit Integers: If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?

23. Dividing 2-Digit Integers: If you have 80 tickets for the fair and each ride costs 5 tickets, how many rides can you go on?

24. Dividing Numbers Ending with 0: The school has $20,000 to buy new computer equipment. If each piece of equipment costs $50, how many pieces can the school buy in total?

25. Dividing 3 Integers: Melissa buys 2 packs of tennis balls for $12 in total. All together, there are 6 tennis balls. How much does 1 pack of tennis balls cost? How much does 1 tennis ball cost?

26. Interpreting Remainders: An Italian restaurant receives a shipment of 86 veal cutlets. If it takes 3 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible?

Mixed operations word problems

27. Mixing Addition and Subtraction: There are 235 books in a library. On Monday, 123 books are taken out. On Tuesday, 56 books are brought back. How many books are there now?

28. Mixing Multiplication and Division: There is a group of 10 people who are ordering pizza. If each person gets 2 slices and each pizza has 4 slices, how many pizzas should they order?

29. Mixing Multiplication, Addition and Subtraction: Lana has 2 bags with 2 marbles in each bag. Markus has 2 bags with 3 marbles in each bag. How many more marbles does Markus have?

30. Mixing Division, Addition and Subtraction: Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?

Ordering and number sense word problems

31. Counting to Preview Multiplication: There are 2 chalkboards in your classroom. If each chalkboard needs 2 pieces of chalk, how many pieces do you need in total?

32. Counting to Preview Division: There are 3 chalkboards in your classroom. Each chalkboard has 2 pieces of chalk. This means there are 6 pieces of chalk in total. If you take 1 piece of chalk away from each chalkboard, how many will there be in total?

33. Composing Numbers: What number is 6 tens and 10 ones?

34. Guessing Numbers: I have a 7 in the tens place. I have an even number in the ones place. I am lower than 74. What number am I?

35. Finding the Order: In the hockey game, Mitchell scored more points than William but fewer points than Auston. Who scored the most points? Who scored the fewest points?

Fractions word problems

Best for: 3rd grade, 4th grade, 5th grade, 6th grade

36. Finding Fractions of a Group: Julia went to 10 houses on her street for Halloween. 5 of the houses gave her a chocolate bar. What fraction of houses on Julia’s street gave her a chocolate bar?

37. Finding Unit Fractions: Heather is painting a portrait of her best friend, Lisa. To make it easier, she divides the portrait into 6 equal parts. What fraction represents each part of the portrait?

38. Adding Fractions with Like Denominators: Noah walks ⅓ of a kilometre to school each day. He also walks ⅓ of a kilometre to get home after school. How many kilometres does he walk in total?

39. Subtracting Fractions with Like Denominators: Last week, Whitney counted the number of juice boxes she had for school lunches. She had ⅗ of a case. This week, it’s down to ⅕ of a case. How much of the case did Whitney drink?

40. Adding Whole Numbers and Fractions with Like Denominators: At lunchtime, an ice cream parlor served 6 ¼ scoops of chocolate ice cream, 5 ¾ scoops of vanilla and 2 ¾ scoops of strawberry. How many scoops of ice cream did the parlor serve in total?

41. Subtracting Whole Numbers and Fractions with Like Denominators: For a party, Jaime had 5 ⅓ bottles of cola for her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left?

42. Adding Fractions with Unlike Denominators: Kevin completed ½ of an assignment at school. When he was home that evening, he completed ⅚ of another assignment. How many assignments did Kevin complete?

43. Subtracting Fractions with Unlike Denominators: Packing school lunches for her kids, Patty used ⅞ of a package of ham. She also used ½ of a package of turkey. How much more ham than turkey did Patty use?

44. Multiplying Fractions: During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. How many kilometres did the students run on Thursday? Write your answer as a fraction.

45. Dividing Fractions: A clothing manufacturer uses ⅕ of a bottle of colour dye to make one pair of pants. The manufacturer used ⅘ of a bottle yesterday. How many pairs of pants did the manufacturer make?

46. Multiplying Fractions with Whole Numbers: Mark drank ⅚ of a carton of milk this week. Frank drank 7 times more milk than Mark. How many cartons of milk did Frank drink? Write your answer as a fraction, or as a whole or mixed number.

Decimals word problems

Best for: 4th grade, 5th grade

47. Adding Decimals: You have 2.6 grams of yogurt in your bowl and you add another spoonful of 1.3 grams. How much yogurt do you have in total?

48. Subtracting Decimals: Gemma had 25.75 grams of frosting to make a cake. She decided to use only 15.5 grams of the frosting. How much frosting does Gemma have left?

49. Multiplying Decimals with Whole Numbers: Marshall walks a total of 0.9 kilometres to and from school each day. After 4 days, how many kilometres will he have walked?

50. Dividing Decimals by Whole Numbers: To make the Leaning Tower of Pisa from spaghetti, Mrs. Robinson bought 2.5 kilograms of spaghetti. Her students were able to make 10 leaning towers in total. How many kilograms of spaghetti does it take to make 1 leaning tower?

51. Mixing Addition and Subtraction of Decimals: Rocco has 1.5 litres of orange soda and 2.25 litres of grape soda in his fridge. Antonio has 1.15 litres of orange soda and 0.62 litres of grape soda. How much more soda does Rocco have than Angelo?

52. Mixing Multiplication and Division of Decimals: 4 days a week, Laura practices martial arts for 1.5 hours. Considering a week is 7 days, what is her average practice time per day each week?

Comparing and sequencing word problems

Best for: Kindergarten, 1st grade, 2nd grade

53. Comparing 1-Digit Integers: You have 3 apples and your friend has 5 apples. Who has more?

54. Comparing 2-Digit Integers: You have 50 candies and your friend has 75 candies. Who has more?

55. Comparing Different Variables: There are 5 basketballs on the playground. There are 7 footballs on the playground. Are there more basketballs or footballs?

56. Sequencing 1-Digit Integers: Erik has 0 stickers. Every day he gets 1 more sticker. How many days until he gets 3 stickers?

57. Skip-Counting by Odd Numbers: Natalie began at 5. She skip-counted by fives. Could she have said the number 20?

58. Skip-Counting by Even Numbers: Natasha began at 0. She skip-counted by eights. Could she have said the number 36?

59. Sequencing 2-Digit Numbers: Each month, Jeremy adds the same number of cards to his baseball card collection. In January, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April?

Time word problems

66. Converting Hours into Minutes: Jeremy helped his mom for 1 hour. For how many minutes was he helping her?

69. Adding Time: If you wake up at 7:00 a.m. and it takes you 1 hour and 30 minutes to get ready and walk to school, at what time will you get to school?

70. Subtracting Time: If a train departs at 2:00 p.m. and arrives at 4:00 p.m., how long were passengers on the train for?

71. Finding Start and End Times: Rebecca left her dad’s store to go home at twenty to seven in the evening. Forty minutes later, she was home. What time was it when she arrived home?

Money word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade

60. Adding Money: Thomas and Matthew are saving up money to buy a video game together. Thomas has saved $30. Matthew has saved $35. How much money have they saved up together in total?

61. Subtracting Money: Thomas has $80 saved up. He uses his money to buy a video game. The video game costs $67. How much money does he have left?

62. Multiplying Money: Tim gets $5 for delivering the paper. How much money will he have after delivering the paper 3 times?

63. Dividing Money: Robert spent $184.59 to buy 3 hockey sticks. If each hockey stick was the same price, how much did 1 cost?

64. Adding Money with Decimals: You went to the store and bought gum for $1.25 and a sucker for $0.50. How much was your total?

65. Subtracting Money with Decimals: You went to the store with $5.50. You bought gum for $1.25, a chocolate bar for $1.15 and a sucker for $0.50. How much money do you have left?

67. Applying Proportional Relationships to Money: Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional.

68. Applying Percentages to Money: Retta put $100.00 in a bank account that gains 20% interest annually. How much interest will be accumulated in 1 year? And if she makes no withdrawals, how much money will be in the account after 1 year?

Physical measurement word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade

72. Comparing Measurements: Cassandra’s ruler is 22 centimetres long. April’s ruler is 30 centimetres long. How many centimetres longer is April’s ruler?

73. Contextualizing Measurements: Picture a school bus. Which unit of measurement would best describe the length of the bus? Centimetres, metres or kilometres?

74. Adding Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Last year, Micha’s dad used 100 litres of gas. This year, her dad used 90 litres of gas. How much gas did he use in total for the two years?

75. Subtracting Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Over the past two years, Micha’s dad used 200 litres of gas. This year, he used 100 litres of gas. How much gas did he use last year?

76. Multiplying Volume and Mass: Kiera wants to make sure she has strong bones, so she drinks 2 litres of milk every week. After 3 weeks, how many litres of milk will Kiera drink?

77. Dividing Volume and Mass: Lillian is doing some gardening, so she bought 1 kilogram of soil. She wants to spread the soil evenly between her 2 plants. How much will each plant get?

78. Converting Mass: Inger goes to the grocery store and buys 3 squashes that each weigh 500 grams. How many kilograms of squash did Inger buy?

79. Converting Volume: Shad has a lemonade stand and sold 20 cups of lemonade. Each cup was 500 millilitres. How many litres did Shad sell in total?

80. Converting Length: Stacy and Milda are comparing their heights. Stacy is 1.5 meters tall. Milda is 10 centimetres taller than Stacy. What is Milda’s height in centimetres?

81. Understanding Distance and Direction: A bus leaves the school to take students on a field trip. The bus travels 10 kilometres south, 10 kilometres west, another 5 kilometres south and 15 kilometres north. To return to the school, in which direction does the bus have to travel? How many kilometres must it travel in that direction?

Ratios and percentages word problems

Best for: 4th grade, 5th grade, 6th grade

82. Finding a Missing Number: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. Jenny has 28 trophies. How many does Meredith have?

83. Finding Missing Numbers: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. The difference between the numbers is 12. What are the numbers?

84. Comparing Ratios: The school’s junior band has 10 saxophone players and 20 trumpet players. The school’s senior band has 18 saxophone players and 29 trumpet players. Which band has the higher ratio of trumpet to saxophone players?

85. Determining Percentages: Mary surveyed students in her school to find out what their favourite sports were. Out of 1,200 students, 455 said hockey was their favourite sport. What percentage of students said hockey was their favourite sport?

86. Determining Percent of Change: A decade ago, Oakville’s population was 67,624 people. Now, it is 190% larger. What is Oakville’s current population?

87. Determining Percents of Numbers: At the ice skate rental stand, 60% of 120 skates are for boys. If the rest of the skates are for girls, how many are there?

88. Calculating Averages: For 4 weeks, William volunteered as a helper for swimming classes. The first week, he volunteered for 8 hours. He volunteered for 12 hours in the second week, and another 12 hours in the third week. The fourth week, he volunteered for 9 hours. For how many hours did he volunteer per week, on average?

Probability and data relationships word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade

89. Understanding the Premise of Probability: John wants to know his class’s favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?

90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?

91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.

92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?

93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Maurita’s 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felice’s 4 tests: 3, 3, 3. If Maurita’s mean for the 4 tests is 1 point higher than Felice’s, what’s the score of Felice’s 4th test?

94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for $6.00. Which store has the better deal?

95. Writing Equations for Proportional Relationships: Lionel loves soccer, but has trouble motivating himself to practice. So, he incentivizes himself through video games. There is a proportional relationship between the amount of drills Lionel completes, in x , and for how many hours he plays video games, in y . When Lionel completes 10 drills, he plays video games for 30 minutes. Write the equation for the relationship between x and y .

Geometry word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade

96. Introducing Perimeter:  The theatre has 4 chairs in a row. There are 5 rows. Using rows as your unit of measurement, what is the perimeter?

97. Introducing Area: The theatre has 4 chairs in a row. There are 5 rows. How many chairs are there in total?

98. Introducing Volume: Aaron wants to know how much candy his container can hold. The container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. What is the container’s volume?

99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?

100. Finding the Perimeter of 2D Shapes: Mitchell wrote his homework questions on a piece of square paper. Each side of the paper is 8 centimetres. What is the perimeter?

101. Determining the Area of 2D Shapes: A single trading card is 9 centimetres long by 6 centimetres wide. What is its area?

102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?

103. Determining the Surface Area of 3D Shapes: What is the surface area of a cube that has a width of 2cm, height of 2 cm and length of 2 cm?

104. Determining the Volume of 3D Shapes: Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?

105. Identifying Right-Angled Triangles: A triangle has the following side lengths: 3 cm, 4 cm and 5 cm. Is this triangle a right-angled triangle?

106. Identifying Equilateral Triangles: A triangle has the following side lengths: 4 cm, 4 cm and 4 cm. What kind of triangle is it?

107. Identifying Isosceles Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 5 cm. What kind of triangle is it?

108. Identifying Scalene Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 6 cm. What kind of triangle is it?

109. Finding the Perimeter of Triangles: Luigi built a tent in the shape of an equilateral triangle. The perimeter is 21 metres. What is the length of each of the tent’s sides?

110. Determining the Area of Triangles: What is the area of a triangle with a base of 2 units and a height of 3 units?

111. Applying Pythagorean Theorem: A right triangle has one non-hypotenuse side length of 3 inches and the hypotenuse measures 5 inches. What is the length of the other non-hypotenuse side?

112. Finding a Circle’s Diameter: Jasmin bought a new round backpack. Its area is 370 square centimetres. What is the round backpack’s diameter?

113. Finding a Circle's Area: Captain America’s circular shield has a diameter of 76.2 centimetres. What is the area of his shield?

114. Finding a Circle’s Radius: Skylar lives on a farm, where his dad keeps a circular corn maze. The corn maze has a diameter of 2 kilometres. What is the maze’s radius?

Variables word problems

Best for: 6th grade, 7th grade, 8th grade

115. Identifying Independent and Dependent Variables: Victoria is baking muffins for her class. The number of muffins she makes is based on how many classmates she has. For this equation, m is the number of muffins and c is the number of classmates. Which variable is independent and which variable is dependent?

116. Writing Variable Expressions for Addition: Last soccer season, Trish scored g goals. Alexa scored 4 more goals than Trish. Write an expression that shows how many goals Alexa scored.

117. Writing Variable Expressions for Subtraction: Elizabeth eats a healthy, balanced breakfast b times a week. Madison sometimes skips breakfast. In total, Madison eats 3 fewer breakfasts a week than Elizabeth. Write an expression that shows how many times a week Madison eats breakfast.

118. Writing Variable Expressions for Multiplication: Last hockey season, Jack scored g goals. Patrik scored twice as many goals than Jack. Write an expression that shows how many goals Patrik scored.

119. Writing Variable Expressions for Division: Amanda has c chocolate bars. She wants to distribute the chocolate bars evenly among 3 friends. Write an expression that shows how many chocolate bars 1 of her friends will receive.

120. Solving Two-Variable Equations: This equation shows how the amount Lucas earns from his after-school job depends on how many hours he works: e = 12h . The variable h represents how many hours he works. The variable e represents how much money he earns. How much money will Lucas earn after working for 6 hours?

How to easily make your own math word problems & word problems worksheets

Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:

• Link to Student Interests:  By framing your word problems with student interests, you’ll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
• Make Questions Topical:  Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
• Include Student Names:  Naming a question’s characters after your students is an easy way make subject matter relatable, helping them work through the problem.
• Be Explicit:  Repeating keywords distills the question, helping students focus on the core problem.
• Test Reading Comprehension:  Flowery word choice and long sentences can hide a question’s key elements. Instead, use concise phrasing and grade-level vocabulary.
• Focus on Similar Interests:  Framing too many questions with related interests -- such as football and basketball -- can alienate or disengage some students.
• Feature Red Herrings:  Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.

A key to differentiated instruction , word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.

Final thoughts about math word problems

You’ll likely get the most out of this resource by using the problems as templates, slightly modifying them by applying the above tips. In doing so, they’ll be more relevant to -- and engaging for -- your students.

Regardless, having 120 curriculum-aligned math word problems at your fingertips should help you deliver skill-building challenges and thought-provoking assessments.

The result?

A greater understanding of how your students process content and demonstrate understanding, informing your ongoing teaching approach.

[FREE] Fun Math Games & Activities Packs

Always on the lookout for fun math games and activities in the classroom? Try our ready-to-go printable packs for students to complete independently or with a partner!

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Solving equations

Here you will learn about solving equations, including linear and quadratic algebraic equations, and how to solve them.

Students will first learn about solving equations in grade 8 as a part of expressions and equations, and again in high school as a part of reasoning with equations and inequalities.

What is solving an equation?

Solving equations is a step-by-step process to find the value of the variable. A variable is the unknown part of an equation, either on the left or right side of the equals sign. Sometimes, you need to solve multi-step equations which contain algebraic expressions.

To do this, you must use the order of operations, which is a systematic approach to equation solving. When you use the order of operations, you first solve any part of an equation located within parentheses. An equation is a mathematical expression that contains an equals sign.

For example,

\begin{aligned}y+6&=11\\\\ 3(x-3)&=12\\\\ \cfrac{2x+2}{4}&=\cfrac{x-3}{3}\\\\ 2x^{2}+3&x-2=0\end{aligned}

There are two sides to an equation, with the left side being equal to the right side. Equations will often involve algebra and contain unknowns, or variables, which you often represent with letters such as x or y.

You can solve simple equations and more complicated equations to work out the value of these unknowns. They could involve fractions, decimals or integers.

Common Core State Standards

How does this relate to 8 th grade and high school math?

• Grade 8 – Expressions and Equations (8.EE.C.7) Solve linear equations in one variable.
• High school – Reasoning with Equations and Inequalities (HSA.REI.B.3) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

[FREE] Solving Equations Worksheet (Grade 6 to 8)

Use this worksheet to check your grade 6 to 8 students’ understanding of solving equations. 15 questions with answers to identify areas of strength and support!

How to solve equations

In order to solve equations, you need to work out the value of the unknown variable by adding, subtracting, multiplying or dividing both sides of the equation by the same value.

• Combine like terms .
• Simplify the equation by using the opposite operation to both sides.
• Isolate the variable on one side of the equation.

Solving equations examples

Example 1: solve equations involving like terms.

Solve for x.

Combine like terms.

Combine the q terms on the left side of the equation. To do this, subtract 4q from both sides.

The goal is to simplify the equation by combining like terms. Subtracting 4q from both sides helps achieve this.

After you combine like terms, you are left with q=9-4q.

2 Simplify the equation by using the opposite operation on both sides.

Add 4q to both sides to isolate q to one side of the equation.

The objective is to have all the q terms on one side. Adding 4q to both sides accomplishes this.

After you move the variable to one side of the equation, you are left with 5q=9.

3 Isolate the variable on one side of the equation.

Divide both sides of the equation by 5 to solve for q.

Dividing by 5 allows you to isolate q to one side of the equation in order to find the solution. After dividing both sides of the equation by 5, you are left with q=1 \cfrac{4}{5} \, .

Example 2: solve equations with variables on both sides

Combine the v terms on the same side of the equation. To do this, add 8v to both sides.

7v+8v=8-8v+8v

After combining like terms, you are left with the equation 15v=8.

Simplify the equation by using the opposite operation on both sides and isolate the variable to one side.

Divide both sides of the equation by 15 to solve for v. This step will isolate v to one side of the equation and allow you to solve.

15v \div 15=8 \div 15

The final solution to the equation 7v=8-8v is \cfrac{8}{15} \, .

Example 3: solve equations with the distributive property

Combine like terms by using the distributive property.

The 3 outside the parentheses needs to be multiplied by both terms inside the parentheses. This is called the distributive property.

\begin{aligned}& 3 \times c=3 c \\\\ & 3 \times(-5)=-15 \\\\ &3 c-15-4=2\end{aligned}

Once the 3 is distributed on the left side, rewrite the equation and combine like terms. In this case, the like terms are the constants on the left, –15 and –4. Subtract –4 from –15 to get –19.

Simplify the equation by using the opposite operation on both sides.

The goal is to isolate the variable, c, on one side of the equation. By adding 19 to both sides, you move the constant term to the other side.

\begin{aligned}& 3 c-19+19=2+19 \\\\ & 3 c=21\end{aligned}

Isolate the variable to one side of the equation.

To solve for c, you want to get c by itself.

Dividing both sides by 3 accomplishes this.

On the left side, \cfrac{3c}{3} simplifies to c, and on the right, \cfrac{21}{3} simplifies to 7.

The final solution is c=7.

As an additional step, you can plug 7 back into the original equation to check your work.

Example 4: solve linear equations

Combine like terms by simplifying.

Using steps to solve, you know that the goal is to isolate x to one side of the equation. In order to do this, you must begin by subtracting from both sides of the equation.

\begin{aligned} & 2x+5=15 \\\\ & 2x+5-5=15-5 \\\\ & 2x=10 \end{aligned}

Continue to simplify the equation by using the opposite operation on both sides.

Continuing with steps to solve, you must divide both sides of the equation by 2 to isolate x to one side.

\begin{aligned} & 2x \div 2=10 \div 2 \\\\ & x= 5 \end{aligned}

Isolate the variable to one side of the equation and check your work.

Plugging in 5 for x in the original equation and making sure both sides are equal is an easy way to check your work. If the equation is not equal, you must check your steps.

\begin{aligned}& 2(5)+5=15 \\\\ & 10+5=15 \\\\ & 15=15\end{aligned}

Example 5: solve equations by factoring

Solve the following equation by factoring.

Combine like terms by factoring the equation by grouping.

Multiply the coefficient of the quadratic term by the constant term.

2 x (-20) = -40

Look for two numbers that multiply to give you –40 and add up to the coefficient of 3. In this case, the numbers are 8 and –5 because 8 x -5=–40, and 8+–5=3.

Split the middle term using those two numbers, 8 and –5. Rewrite the middle term using the numbers 8 and –5.

2x^2+8x-5x-20=0

Group the terms in pairs and factor out the common factors.

2x^2+8x-5x-20=2x(x + 4)-5(x+4)=0

Now, you’ve factored the equation and are left with the following simpler equations 2x-5 and x+4.

This step relies on understanding the zero product property, which states that if two numbers multiply to give zero, then at least one of those numbers must equal zero.

Let’s relate this back to the factored equation (2x-5)(x+4)=0

Because of this property, either (2x-5)=0 or (x+4)=0

Isolate the variable for each equation and solve.

When solving these simpler equations, remember that you must apply each step to both sides of the equation to maintain balance.

\begin{aligned}& 2 x-5=0 \\\\ & 2 x-5+5=0+5 \\\\ & 2 x=5 \\\\ & 2 x \div 2=5 \div 2 \\\\ & x=\cfrac{5}{2} \end{aligned}

\begin{aligned}& x+4=0 \\\\ & x+4-4=0-4 \\\\ & x=-4\end{aligned}

The solution to this equation is x=\cfrac{5}{2} and x=-4.

Example 6: solve quadratic equations

Solve the following quadratic equation.

Combine like terms by factoring the quadratic equation when terms are isolated to one side.

To factorize a quadratic expression like this, you need to find two numbers that multiply to give -5 (the constant term) and add to give +2 (the coefficient of the x term).

The two numbers that satisfy this are -1 and +5.

So you can split the middle term 2x into -1x+5x: x^2-1x+5x-5-1x+5x

Now you can take out common factors x(x-1)+5(x-1).

And since you have a common factor of (x-1), you can simplify to (x+5)(x-1).

The numbers -1 and 5 allow you to split the middle term into two terms that give you common factors, allowing you to simplify into the form (x+5)(x-1).

Let’s relate this back to the factored equation (x+5)(x-1)=0.

Because of this property, either (x+5)=0 or (x-1)=0.

Now, you can solve the simple equations resulting from the zero product property.

\begin{aligned}& x+5=0 \\\\ & x+5-5=0-5 \\\\ & x=-5 \\\\\\ & x-1=0 \\\\ & x-1+1=0+1 \\\\ & x=1\end{aligned}

The solutions to this quadratic equation are x=1 and x=-5.

Teaching tips for solving equations

• Use physical manipulatives like balance scales as a visual aid. Show how you need to keep both sides of the equation balanced, like a scale. Add or subtract the same thing from both sides to keep it balanced when solving. Use this method to practice various types of equations.
• Emphasize the importance of undoing steps to isolate the variable. If you are solving for x and 3 is added to x, subtracting 3 undoes that step and isolates the variable x.
• Relate equations to real-world, relevant examples for students. For example, word problems about tickets for sports games, cell phone plans, pizza parties, etc. can make the concepts click better.
• Allow time for peer teaching and collaborative problem solving. Having students explain concepts to each other, work through examples on whiteboards, etc. reinforces the process and allows peers to ask clarifying questions. This type of scaffolding would be beneficial for all students, especially English-Language Learners. Provide supervision and feedback during the peer interactions.

Easy mistakes to make

• Forgetting to distribute or combine like terms One common mistake is neglecting to distribute a number across parentheses or combine like terms before isolating the variable. This error can lead to an incorrect simplified form of the equation.
• Misapplying the distributive property Incorrectly distributing a number across terms inside parentheses can result in errors. Students may forget to multiply each term within the parentheses by the distributing number, leading to an inaccurate equation.
• Failing to perform the same operation on both sides It’s crucial to perform the same operation on both sides of the equation to maintain balance. Forgetting this can result in an imbalanced equation and incorrect solutions.
• Making calculation errors Simple arithmetic mistakes, such as addition, subtraction, multiplication, or division errors, can occur during the solution process. Checking calculations is essential to avoid errors that may propagate through the steps.
• Ignoring fractions or misapplying operations When fractions are involved, students may forget to multiply or divide by the common denominator to eliminate them. Misapplying operations on fractions can lead to incorrect solutions or complications in the final answer.

Related math equations lessons

• Math equations
• Rearranging equations
• How to find the equation of a line
• Solve equations with fractions
• Linear equations
• Writing linear equations
• Substitution
• Identity math
• One step equation

Practice solving equations questions

1. Solve 4x-2=14.

Add 2 to both sides.

Divide both sides by 4.

2. Solve 3x-8=x+6.

Add 8 to both sides.

Subtract x from both sides.

Divide both sides by 2.

3. Solve 3(x+3)=2(x-2).

Expanding the parentheses.

Subtract 9 from both sides.

Subtract 2x from both sides.

4. Solve \cfrac{2 x+2}{3}=\cfrac{x-3}{2}.

Multiply by 6 (the lowest common denominator) and simplify.

Expand the parentheses.

Subtract 4 from both sides.

Subtract 3x from both sides.

5. Solve \cfrac{3 x^{2}}{2}=24.

Multiply both sides by 2.

Divide both sides by 3.

Square root both sides.

6. Solve by factoring:

Use factoring to find simpler equations.

Set each set of parentheses equal to zero and solve.

x=3 or x=10

Solving equations FAQs

The first step in solving a simple linear equation is to simplify both sides by combining like terms. This involves adding or subtracting terms to isolate the variable on one side of the equation.

Performing the same operation on both sides of the equation maintains the equality. This ensures that any change made to one side is also made to the other, keeping the equation balanced and preserving the solutions.

To handle variables on both sides of the equation, start by combining like terms on each side. Then, move all terms involving the variable to one side by adding or subtracting, and simplify to isolate the variable. Finally, perform any necessary operations to solve for the variable.

To deal with fractions in an equation, aim to eliminate them by multiplying both sides of the equation by the least common denominator. This helps simplify the equation and make it easier to isolate the variable. Afterward, proceed with the regular steps of solving the equation.

The next lessons are

• Inequalities
• Types of graph
• Coordinate plane

Still stuck?

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Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

Find out how we can help your students achieve success with our math tutoring programs .

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40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!

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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html