problem solving addition lesson 4.9

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

4.9: Strategies for Solving Applications and Equations

• Last updated
• Save as PDF
• Page ID 113697

Learning Objectives

By the end of this section, you will be able to:

• Use a problem solving strategy for word problems
• Solve number word problems
• Solve percent applications
• Solve simple interest applications

Before you get started, take this readiness quiz.

• Translate “six less than twice x ” into an algebraic expression. If you missed this problem, review [link] .
• Convert 4.5% to a decimal. If you missed this problem, review [link] .
• Convert 0.6 to a percent. If you missed this problem, review [link] .

Have you ever had any negative experiences in the past with word problems? When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. Realize that your negative experiences with word problems are in your past. To move forward you need to calm your fears and change your negative feelings.

Start with a fresh slate and begin to think positive thoughts. Repeating some of the following statements may be helpful to turn your thoughts positive. Thinking positive thoughts is a first step towards success.

• I think I can! I think I can!
• While word problems were hard in the past, I think I can try them now.
• I am better prepared now—I think I will begin to understand word problems.
• I am able to solve equations because I practiced many problems and I got help when I needed it—I can try that with word problems.
• It may take time, but I can begin to solve word problems.
• You are now well prepared and you are ready to succeed. If you take control and believe you can be successful, you will be able to master word problems.

Use a Problem Solving Strategy for Word Problems

Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.

EXAMPLE \(\PageIndex{1}\)

Normal yearly snowfall at the local ski resort is 12 inches more than twice the amount it received last season. The normal yearly snowfall is 62 inches. What was the snowfall last season at the ski resort?

EXAMPLE \(\PageIndex{2}\)

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was three more than twice the number of notebooks. He bought seven textbooks. How many notebooks did he buy?

He bought two notebooks

EXAMPLE \(\PageIndex{3}\)

Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?

He did seven crosswords puzzles

We summarize an effective strategy for problem solving.

PROBLEM SOLVING STRATEGY FOR WORD PROBLEMS

• Read the problem. Make sure all the words and ideas are understood.
• Identify what you are looking for.
• Name what you are looking for. Choose a variable to represent that quantity.
• Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
• Solve the equation using proper algebra techniques.
• Check the answer in the problem to make sure it makes sense.
• Answer the question with a complete sentence.

Solve Number Word Problems

We will now apply the problem solving strategy to “number word problems.” Number word problems give some clues about one or more numbers and we use these clues to write an equation. Number word problems provide good practice for using the Problem Solving Strategy.

EXAMPLE \(\PageIndex{4}\)

The sum of seven times a number and eight is thirty-six. Find the number.

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

EXAMPLE \(\PageIndex{5}\)

The sum of four times a number and two is fourteen. Find the number.

EXAMPLE \(\PageIndex{6}\)

The sum of three times a number and seven is twenty-five. Find the number.

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far, we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

EXAMPLE \(\PageIndex{7}\)

The sum of two numbers is negative fifteen. One number is nine less than the other. Find the numbers.

EXAMPLE \(\PageIndex{8}\)

The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

\(−15,−8\)

EXAMPLE \(\PageIndex{9}\)

The sum of two numbers is negative eighteen. One number is forty more than the other. Find the numbers.

\(−29,11\)

Consecutive Integers (optional)

Some number problems involve consecutive integers . Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:

\[\begin{array}{rrrr} 1, & 2, & 3, & 4 \\ −10, & −9, & −8, & −7\\ 150, & 151, & 152, & 153 \end{array}\]

Notice that each number is one more than the number preceding it. Therefore, if we define the first integer as n , the next consecutive integer is \(n+1\). The one after that is one more than \(n+1\), so it is \(n+1+1\), which is \(n+2\).

\[\begin{array}{ll} n & 1^{\text{st}} \text{integer} \\ n+1 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; & 2^{\text{nd}}\text{consecutive integer} \\ n+2 & 3^{\text{rd}}\text{consecutive integer} \;\;\;\;\;\;\;\; \text{etc.} \end{array}\]

We will use this notation to represent consecutive integers in the next example.

EXAMPLE \(\PageIndex{10}\)

Find three consecutive integers whose sum is \(−54\).

EXAMPLE \(\PageIndex{11}\)

Find three consecutive integers whose sum is \(−96\).

\(−33,−32,−31\)

EXAMPLE \(\PageIndex{12}\)

Find three consecutive integers whose sum is \(−36\).

\(−13,−12,−11\)

Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers . Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:

\[24, 26, 28\]

\[−12,−10,−8\]

Notice each integer is two more than the number preceding it. If we call the first one n , then the next one is \(n+2\). The one after that would be \(n+2+2\) or \(n+4\).

Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 63, 65, and 67.

\[63, 65, 67\]

\[n,n+2,n+4\]

Does it seem strange to have to add two (an even number) to get the next odd number? Do we get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you do not have to do anything different. The pattern is still the same—to get to the next odd or the next even integer, add two.

EXAMPLE \(\PageIndex{13}\)

Find three consecutive even integers whose sum is \(120\).

EXAMPLE \(\PageIndex{14}\)

Find three consecutive even integers whose sum is 102.

\(32, 34, 36\)

EXAMPLE \(\PageIndex{15}\)

Find three consecutive even integers whose sum is \(−24\).

\(−10,−8,−6\)

When a number problem is in a real life context, we still use the same strategies that we used for the previous examples.

EXAMPLE \(\PageIndex{16}\)

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,400. This was $1,600 less than six times the cost in 1975. What was the average cost of a car in 1975?

The average cost was $5,000.

EXAMPLE \(\PageIndex{18}\)

US Census data shows that the median price of new home in the U.S. in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

The median price was $19,300.

Solve Percent Applications

There are several methods to solve percent equations. In algebra, it is easiest if we just translate English sentences into algebraic equations and then solve the equations. Be sure to change the given percent to a decimal before you use it in the equation.

EXAMPLE \(\PageIndex{19}\)

Translate and solve:

• What number is 45% of 84?
• 8.5% of what amount is $4.76?
• 168 is what percent of 112?
• What number is 45% of 80?
• 7.5% of what amount is $1.95?
• 110 is what percent of 88?

ⓐ 36 ⓑ $26 ⓒ \(125 \% \)

EXAMPLE \(\PageIndex{21}\)

• What number is 55% of 60?
• 8.5% of what amount is $3.06?
• 126 is what percent of 72?

ⓐ 33 ⓑ $36 ⓐ \(175 \% \)

Now that we have a problem solving strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we will solve involve everyday situations, you can rely on your own experience.

EXAMPLE \(\PageIndex{22}\)

The label on Audrey’s yogurt said that one serving provided 12 grams of protein, which is 24% of the recommended daily amount. What is the total recommended daily amount of protein?

EXAMPLE \(\PageIndex{23}\)

One serving of wheat square cereal has 7 grams of fiber, which is 28% of the recommended daily amount. What is the total recommended daily amount of fiber?

EXAMPLE \(\PageIndex{24}\)

One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium?

Remember to put the answer in the form requested. In the next example we are looking for the percent.

EXAMPLE \(\PageIndex{25}\)

Veronica is planning to make muffins from a mix. The package says each muffin will be 240 calories and 60 calories will be from fat. What percent of the total calories is from fat?

EXAMPLE \(\PageIndex{26}\)

Mitzi received some gourmet brownies as a gift. The wrapper said each 28% brownie was 480 calories, and had 240 calories of fat. What percent of the total calories in each brownie comes from fat? Round the answer to the nearest whole percent.

EXAMPLE \(\PageIndex{27}\)

The mix Ricardo plans to use to make brownies says that each brownie will be 190 calories, and 76 calories are from fat. What percent of the total calories are from fat? Round the answer to the nearest whole percent.

It is often important in many fields—business, sciences, pop culture—to talk about how much an amount has increased or decreased over a certain period of time. This increase or decrease is generally expressed as a percent and called the percent change .

To find the percent change, first we find the amount of change, by finding the difference of the new amount and the original amount. Then we find what percent the amount of change is of the original amount.

FIND PERCENT CHANGE

\[\text{change}= \text{new amount}−\text{original amount}\]

change is what percent of the original amount?

EXAMPLE \(\PageIndex{28}\)

Recently, the California governor proposed raising community college fees from $36 a unit to $46 a unit. Find the percent change. (Round to the nearest tenth of a percent.)

EXAMPLE \(\PageIndex{29}\)

Find the percent change. (Round to the nearest tenth of a percent.) In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents.

\(8.8 \% \)

EXAMPLE \(\PageIndex{30}\)

Find the percent change. (Round to the nearest tenth of a percent.) In 1995, the standard bus fare in Chicago was $1.50. In 2008, the standard bus fare was 2.25.

Applications of discount and mark-up are very common in retail settings.

When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate , usually given as a percent, is used to determine the amount of the discount . To determine the amount of discount, we multiply the discount rate by the original price.

The price a retailer pays for an item is called the original cost . The retailer then adds a mark-up to the original cost to get the list price , the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.

\[ \begin{align} \text{amount of discount} &= \text{discount rate}· \text{original price} \\ \text{sale price} &= \text{original amount}– \text{discount price} \end{align}\]

The sale price should always be less than the original price.

\[\begin{align} \text{amount of mark-up} &= \text{mark-up rate}·\text{original price} \\ \text{list price} &= \text{original cost}–\text{mark-up} \end{align}\]

The list price should always be more than the original cost.

EXAMPLE \(\PageIndex{31}\)

Liam’s art gallery bought a painting at an original cost of $750. Liam marked the price up 40%. Find

• the amount of mark-up and
• the list price of the painting.

EXAMPLE \(\PageIndex{32}\)

Find ⓐ the amount of mark-up and ⓑ the list price: Jim’s music store bought a guitar at original cost $1,200. Jim marked the price up 50%.

ⓐ $600 ⓑ $1,800

EXAMPLE \(\PageIndex{33}\)

Find ⓐ the amount of mark-up and ⓑ the list price: The Auto Resale Store bought Pablo’s Toyota for $8,500. They marked the price up 35%.

ⓐ $2,975 ⓑ $11,475

Solve Simple Interest Applications

Interest is a part of our daily lives. From the interest earned on our savings to the interest we pay on a car loan or credit card debt, we all have some experience with interest in our lives.

The amount of money you initially deposit into a bank is called the principal , P , and the bank pays you interest, I. When you take out a loan, you pay interest on the amount you borrow, also called the principal.

In either case, the interest is computed as a certain percent of the principal, called the rate of interest , r . The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable t , (for time) represents the number of years the money is saved or borrowed.

Interest is calculated as simple interest or compound interest. Here we will use simple interest.

SIMPLE INTEREST

If an amount of money, P , called the principal, is invested or borrowed for a period of t years at an annual interest rate r , the amount of interest, I , earned or paid is

\[ \begin{array}{ll} I=Prt \; \; \; \; \; \; \; \; \; \; \; \; \text{where} & { \begin{align} I &= \text{interest} \\ P &= \text{principal} \\ r &= \text{rate} \\ t &= \text{time} \end{align}} \end{array}\]

Interest earned or paid according to this formula is called simple interest .

The formula we use to calculate interest is \(I=Prt\). To use the formula we substitute in the values for variables that are given, and then solve for the unknown variable. It may be helpful to organize the information in a chart.

EXAMPLE \(\PageIndex{34}\)

Areli invested a principal of $950 in her bank account that earned simple interest at an interest rate of 3%. How much interest did she earn in five years?

\( \begin{aligned} I & = \; ? \\ P & = \; \$ 950 \\ r & = \; 3 \% \\ t & = \; 5 \text{ years} \end{aligned}\)

\(\begin{array}{ll} \text{Identify what you are asked to find, and choose a} & \text{What is the simple interest?} \\ \text{variable to represent it.} & \text{Let } I= \text{interest.} \\ \text{Write the formula.} & I=Prt \\ \text{Substitute in the given information.} & I=(950)(0.03)(5) \\ \text{Simplify.} & I=142.5 \\ \text{Check.} \\ \text{Is } \$142.50 \text{ a reasonable amount of interest on } \$ \text{ 950?} \; \;\;\;\;\; \;\;\;\;\;\; \\ \text{Yes.} \\ \text{Write a complete sentence.} & \text{The interest is } \$ \text{142.50.} \end{array}\)

EXAMPLE \(\PageIndex{35}\)

Nathaly deposited $12,500 in her bank account where it will earn 4% simple interest. How much interest will Nathaly earn in five years?

He will earn $2,500.

EXAMPLE \(\PageIndex{36}\)

Susana invested a principal of $36,000 in her bank account that earned simple interest at an interest rate of 6.5%.6.5%. How much interest did she earn in three years?

She earned $7,020.

There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we do not know the rate.

EXAMPLE \(\PageIndex{37}\)

Hang borrowed $7,500 from her parents to pay her tuition. In five years, she paid them $1,500 interest in addition to the $7,500 she borrowed. What was the rate of simple interest?

\( \begin{aligned} I & = \; \$ 1500 \\ P & = \; \$ 7500 \\ r & = \; ? \\ t & = \; 5 \text{ years} \end{aligned}\)

Identify what you are asked to find, and choose What is the rate of simple interest? a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Change to percent form. Let r = rate of interest. I = P r t 1,500 = ( 7,500 ) r ( 5 ) 1,500 = 37,500 r 0.04 = r 4 % = r Check. I = P r t 1,500 = ? ( 7,500 ) ( 0.04 ) ( 5 ) 1,500 = 1,500 ✓ Write a complete sentence. The rate of interest was 4%. Identify what you are asked to find, and choose What is the rate of simple interest? a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Change to percent form. Let r = rate of interest. I = P r t 1 ,500 = ( 7,500 ) r ( 5 ) 1,500 = 37,500 r 0.04 = r 4 % = r Check. I = P r t 1 ,500 = ? ( 7,500 ) ( 0.04 ) ( 5 ) 1,500 = 1, 500 ✓ Write a complete sentence. The rate of interest was 4%.

EXAMPLE \(\PageIndex{38}\)

Jim lent his sister $5,000 to help her buy a house. In three years, she paid him the $5,000, plus $900 interest. What was the rate of simple interest?

The rate of simple interest was 6%.

EXAMPLE \(\PageIndex{39}\)

Loren lent his brother $3,000 to help him buy a car. In four years, his brother paid him back the $3,000 plus $660 in interest. What was the rate of simple interest?

The rate of simple interest was 5.5%.

In the next example, we are asked to find the principal—the amount borrowed.

EXAMPLE \(\PageIndex{40}\)

Sean’s new car loan statement said he would pay $4,866,25 in interest from a simple interest rate of 8.5% over five years. How much did he borrow to buy his new car?

\( \begin{aligned} I & = \; 4,866.25 \\ P & = \; ? \\ r & = \; 8.5 \% \\ t & = \; 5 \text{ years} \end{aligned}\)

Identify what you are asked to find, What is the amount borrowed (the principal)? and choose a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Let P = principal borrowed. I = P r t 4,866.25 = P ( 0.085 ) ( 5 ) 4,866.25 = 0.425 P 11,450 = P Check. I = P r t 4,866.25 = ? ( 11,450 ) ( 0.085 ) ( 5 ) 4,866.25 = 4,866.25 ✓ Write a complete sentence. The principal was $11,450. Identify what you are asked to find, What is the amount borrowed (the principal)? and choose a variable to represent it. Write the formula. Substitute in the given information. Multiply. Divide. Let P = principal borrowed. I = P r t 4 ,866.25 = P ( 0.085 ) ( 5 ) 4,866.25 = 0.425 P 11,450 = P Check. I = P r t 4 ,866.25 = ? ( 11,450 ) ( 0.085 ) ( 5 ) 4,866.25 = 4,866.25 ✓ Write a complete sentence. The principal was $11,450.

EXAMPLE \(\PageIndex{41}\)

Eduardo noticed that his new car loan papers stated that with a 7.5% simple interest rate, he would pay $6,596.25 in interest over five years. How much did he borrow to pay for his car?

He paid $17,590.

EXAMPLE \(\PageIndex{42}\)

In five years, Gloria’s bank account earned $2,400 interest at 5% simple interest. How much had she deposited in the account?

She deposited $9,600.

Access this online resource for additional instruction and practice with using a problem solving strategy.

• Begining Arithmetic Problems

Key Concepts

\(\text{change}=\text{new amount}−\text{original amount}\)

\(\text{change is what percent of the original amount?}\)

• \( \begin{align} \text{amount of discount} &= \text{discount rate}· \text{original price} \\ \text{sale price} &= \text{original amount}– \text{discount price} \end{align}\)
• \(\begin{align} \text{amount of mark-up} &= \text{mark-up rate}·\text{original price} \\ \text{list price} &= \text{original cost}–\text{mark-up} \end{align}\)
• If an amount of money, P , called the principal, is invested or borrowed for a period of t years at an annual interest rate r , the amount of interest, I , earned or paid is: \[\begin{aligned} &{} &{} &{I=interest} \nonumber\\ &{I=Prt} &{\text{where} \space} &{P=principal} \nonumber\\ &{} &{\space} &{r=rate} \nonumber\\ &{} &{\space} &{t=time} \nonumber \end{aligned}\]

Practice Makes Perfect

1. List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

Answers will vary.

2. List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

3. There are \(16\) girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.

4. There are \(18\) Cub Scouts in Troop 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.

5. Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is \(12\) less than three times the number of hardbacks. Huong had \(162\) paperbacks. How many hardback books were there?

58 hardback books

6. Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are \(42\) adult bicycles. How many children’s bicycles are there?

In the following exercises, solve each number word problem.

7. The difference of a number and \(12\) is three. Find the number.

8. The difference of a number and eight is four. Find the number.

9. The sum of three times a number and eight is \(23\). Find the number.

10. The sum of twice a number and six is \(14\). Find the number.

11 . The difference of twice a number and seven is \(17\). Find the number.

12. The difference of four times a number and seven is \(21\). Find the number.

13. Three times the sum of a number and nine is \(12\). Find the number.

14. Six times the sum of a number and eight is \(30\). Find the number.

15. One number is six more than the other. Their sum is \(42\). Find the numbers.

\(18, \;24\)

16. One number is five more than the other. Their sum is \(33\). Find the numbers.

17. The sum of two numbers is \(20\). One number is four less than the other. Find the numbers.

\(8, \;12\)

18 . The sum of two numbers is \(27\). One number is seven less than the other. Find the numbers.

19. One number is \(14\) less than another. If their sum is increased by seven, the result is \(85\). Find the numbers.

\(32,\; 46\)

20 . One number is \(11\) less than another. If their sum is increased by eight, the result is \(71\). Find the numbers.

21. The sum of two numbers is \(14\). One number is two less than three times the other. Find the numbers.

\(4,\; 10\)

22. The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

23. The sum of two consecutive integers is \(77\). Find the integers.

\(38,\; 39\)

24. The sum of two consecutive integers is \(89\). Find the integers.

25. The sum of three consecutive integers is \(78\). Find the integers.

\(25,\; 26,\; 27\)

26. The sum of three consecutive integers is \(60\). Find the integers.

27. Find three consecutive integers whose sum is \(−36\).

\(−11,\;−12,\;−13\)

28. Find three consecutive integers whose sum is \(−3\).

29. Find three consecutive even integers whose sum is \(258\).

\(84,\; 86,\; 88\)

30. Find three consecutive even integers whose sum is \(222\).

31. Find three consecutive odd integers whose sum is \(−213\).

\(−69,\;−71,\;−73\)

32. Find three consecutive odd integers whose sum is \(−267\).

33. Philip pays \($1,620\) in rent every month. This amount is \($120\) more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

34. Marc just bought an SUV for \($54,000\). This is \($7,400\) less than twice what his wife paid for her car last year. How much did his wife pay for her car?

35. Laurie has \($46,000\) invested in stocks and bonds. The amount invested in stocks is \($8,000\) less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

\($13,500\)

36. Erica earned a total of \($50,450\) last year from her two jobs. The amount she earned from her job at the store was \($1,250\) more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?

In the following exercises, translate and solve.

37. a. What number is 45% of 120? b. 81 is 75% of what number? c. What percent of 260 is 78?

a. 54 b. 108 c. 30%

38. a. What number is 65% of 100? b. 93 is 75% of what number? c. What percent of 215 is 86?

39. a. 250% of 65 is what number? b. 8.2% of what amount is $2.87? c. 30 is what percent of 20?

a. 162.5 b. $35 c. 150%

40. a. 150% of 90 is what number? b. 6.4% of what amount is $2.88? c. 50 is what percent of 40?

In the following exercises, solve.

41. Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16% of the total bill as a tip. How much should the tip be?

42. When Hiro and his co-workers had lunch at a restaurant near their work, the bill was $90.50. They want to leave 18% of the total bill as a tip. How much should the tip be?

43. One serving of oatmeal has 8 grams of fiber, which is 33% of the recommended daily amount. What is the total recommended daily amount of fiber?

44. One serving of trail mix has 67 grams of carbohydrates, which is 22% of the recommended daily amount. What is the total recommended daily amount of carbohydrates?

45. A bacon cheeseburger at a popular fast food restaurant contains 2070 milligrams (mg) of sodium, which is 86% of the recommended daily amount. What is the total recommended daily amount of sodium?

46. A grilled chicken salad at a popular fast food restaurant contains 650 milligrams (mg) of sodium, which is 27% of the recommended daily amount. What is the total recommended daily amount of sodium?

47. The nutrition fact sheet at a fast food restaurant says the fish sandwich has 380 calories, and 171 calories are from fat. What percent of the total calories is from fat?

48. The nutrition fact sheet at a fast food restaurant says a small portion of chicken nuggets has 190 calories, and 114 calories are from fat. What percent of the total calories is from fat?

49. Emma gets paid $3,000 per month. She pays $750 a month for rent. What percent of her monthly pay goes to rent?

50. Dimple gets paid $3,200 per month. She pays $960 a month for rent. What percent of her monthly pay goes to rent?

51. Tamanika received a raise in her hourly pay, from $15.50 to $17.36. Find the percent change.

52. Ayodele received a raise in her hourly pay, from $24.50 to $25.48. Find the percent change.

53. Annual student fees at the University of California rose from about $4,000 in 2000 to about $12,000 in 2010. Find the percent change.

54. The price of a share of one stock rose from $12.50 to $50. Find the percent change.

55. A grocery store reduced the price of a loaf of bread from $2.80 to $2.73. Find the percent change.

−2.5%

56. The price of a share of one stock fell from $8.75 to $8.54. Find the percent change.

57. Hernando’s salary was $49,500 last year. This year his salary was cut to $44,055. Find the percent change.

58. In ten years, the population of Detroit fell from 950,000 to about 712,500. Find the percent change.

In the following exercises, find a. the amount of discount and b. the sale price.

59. Janelle bought a beach chair on sale at 60% off. The original price was $44.95.

a. $26.97 b. $17.98

60. Errol bought a skateboard helmet on sale at 40% off. The original price was $49.95.

In the following exercises, find a. the amount of discount and b. the discount rate (Round to the nearest tenth of a percent if needed.)

61. Larry and Donna bought a sofa at the sale price of $1,344. The original price of the sofa was $1,920.

a. $576 b. 30%

62. Hiroshi bought a lawnmower at the sale price of $240. The original price of the lawnmower is $300.

In the following exercises, find a. the amount of the mark-up and b. the list price.

63. Daria bought a bracelet at original cost $16 to sell in her handicraft store. She marked the price up 45%. What was the list price of the bracelet?

a. $7.20 b. $23.20

64. Regina bought a handmade quilt at original cost $120 to sell in her quilt store. She marked the price up 55%. What was the list price of the quilt?

65. Tom paid $0.60 a pound for tomatoes to sell at his produce store. He added a 33% mark-up. What price did he charge his customers for the tomatoes?

a. $0.20 b. $0.80

66. Flora paid her supplier $0.74 a stem for roses to sell at her flower shop. She added an 85% mark-up. What price did she charge her customers for the roses?

67. Casey deposited $1,450 in a bank account that earned simple interest at an interest rate of 4%. How much interest was earned in two years?

68 . Terrence deposited $5,720 in a bank account that earned simple interest at an interest rate of 6%. How much interest was earned in four years?

69. Robin deposited $31,000 in a bank account that earned simple interest at an interest rate of 5.2%. How much interest was earned in three years?

70. Carleen deposited $16,400 in a bank account that earned simple interest at an interest rate of 3.9% How much interest was earned in eight years?

71. Hilaria borrowed $8,000 from her grandfather to pay for college. Five years later, she paid him back the $8,000, plus $1,200 interest. What was the rate of simple interest?

72. Kenneth lent his niece $1,200 to buy a computer. Two years later, she paid him back the $1,200, plus $96 interest. What was the rate of simple interest?

73. Lebron lent his daughter $20,000 to help her buy a condominium. When she sold the condominium four years later, she paid him the $20,000, plus $3,000 interest. What was the rate of simple interest?

74. Pablo borrowed $50,000 to start a business. Three years later, he repaid the $50,000, plus $9,375 interest. What was the rate of simple interest?

75. In 10 years, a bank account that paid 5.25% simple interest earned $18,375 interest. What was the principal of the account?

76. In 25 years, a bond that paid 4.75% simple interest earned $2,375 interest. What was the principal of the bond?

77. Joshua’s computer loan statement said he would pay $1,244.34 in simple interest for a three-year loan at 12.4%. How much did Joshua borrow to buy the computer?

78. Margaret’s car loan statement said she would pay $7,683.20 in simple interest for a five-year loan at 9.8%. How much did Margaret borrow to buy the car?

Everyday Math

79 . Tipping At the campus coffee cart, a medium coffee costs $1.65. MaryAnne brings $2.00 with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave?

80 . Tipping Four friends went out to lunch and the bill came to $53.75 They decided to add enough tip to make a total of $64, so that they could easily split the bill evenly among themselves. What percent tip did they leave?

Writing Exercises

81. What has been your past experience solving word problems? Where do you see yourself moving forward?

82. Without solving the problem “44 is 80% of what number” think about what the solution might be. Should it be a number that is greater than 44 or less than 44? Explain your reasoning.

83. After returning from vacation, Alex said he should have packed 50% fewer shorts and 200% more shirts. Explain what Alex meant.

84. Because of road construction in one city, commuters were advised to plan that their Monday morning commute would take 150% of their usual commuting time. Explain what this means.

a. After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.

b. After reviewing this checklist, what will you do to become confident for all objectives?

Curriculum  /  Math  /  4th Grade  /  Unit 1: Place Value, Rounding, Addition, and Subtraction  /  Lesson 17

Place Value, Rounding, Addition, and Subtraction

Lesson 17 of 19

Criteria for Success

Tips for teachers, anchor tasks.

Problem Set

Target Task

Additional practice.

Fluently subtract multi-digit whole numbers using the standard algorithm involving multiple decompositions. Solve one-step word problems involving subtraction.

Common Core Standards

Core standards.

The core standards covered in this lesson

Number and Operations in Base Ten

4.NBT.B.4 — Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Foundational Standards

The foundational standards covered in this lesson

3.NBT.A.2 — Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Subtract multi-digit numbers with multiple decompositions. 
• Solve one-step word problems involving subtraction with multiple decompositions, using a letter to represent the unknown (MP.4).

Suggestions for teachers to help them teach this lesson

Students have solved one-step word problems involving subtraction in previous grade levels (1.OA.1, 2.OA.1, 3.OA.8), just not yet with whole numbers of this magnitude. So, the intention of the word problems in this lesson is to have students solve contextual problems that involve computations expected of this grade level (4.NBT.4) as well as prepare students to solve multi-step word problems involving addition and subtraction later in the unit, and eventually multiplication (Unit 3) and division (Unit 4) (4.OA.3).

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

Place any digit, 1 through 9, in the boxes below to create the smallest possible difference. Each digit can only be used once.

Guiding Questions

Subtraction to Get the Smallest Difference by is made available on Open Middle under the CC BY-NC-SA 4.0  license. Accessed Sept. 13, 2017, 8:46 a.m..

Estimate. Then solve.

a.   253,421 −  $$\square$$  = 75,832

b.    $$\begin{align} 1,000,000 \\ {-\ 978,564}& \\ \hline \end{align}$$

Grade 4 Mathematics > Module 1 > Topic E > Lesson 15 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

Last year, there were 620,073 people in attendance at a local parade. There were 163,278 fewer people at the parade this year than there were last year. How many people were in attendance this year?

Unlock the answer keys for this lesson's problem set and extra practice problems to save time and support student learning.

Discussion of Problem Set

• Look at #1(d). If you had added 13 to both numbers, would it have resulted in the same difference? What is the advantage of adding 13 to both numbers? 
• Look at #1(e) and 1(f). Did anyone notice a pattern that could be used to solve these problems?
• Did anyone count up to solve #1(g)? Why did you decide to do that? What is the advantage of that strategy? 
• What was different about #1(i) than the previous problems? 
• What mistake did the student make in #4? How would you correct it? What is the correct difference? 
• Look at #5 and #6. How are these problems similar? How are they different? 
• How do you know when you are ready to subtract across the problem?
• How can you check your answer when subtracting? 
• Is there a number that you can subtract from 1,000,000 without decomposing across to the ones (other than 1,000,000)? 100,000? 10,000?
• How can decomposing multiple times be challenging? 
• How does the tape diagram help you determine which operation to use to find the answer?

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Solve. Show or explain your work.

a.      ________ = 32,010 – 2,546

b.   829,403 = 175,368 + _____________

Student Response

A company plans to give free stickers to 100,000 customers. On the first day of the giveaway, they give free stickers to 6,512 customers. How many stickers do they still have to give away? Show or explain your work. 

An example response to the Target Task at the level of detail expected of the students.

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Answer keys for Problem Sets and Extra Practice Problems are available with a Fishtank Plus subscription.

Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Topic A: Place Value of Multi-Digit Whole Numbers

Review place value understanding for numbers within 1,000.

4.NBT.A.1 4.NBT.A.2 4.NBT.A.3 4.NBT.B.4

Build numbers to 10,000 and write numbers to that place value in standard, unit, and expanded form. 

4.NBT.A.1 4.NBT.A.2

Model and write numbers to 10,000 with more than 9 of any unit in standard and unit form.

4.NBT.A.1 4.NBT.A.2 4.NBT.B.4

Build numbers to 1,000,000 and write numbers to that place value in standard and unit form.

Multiply and divide single units by 10. Recognize that a digit represents 10 times the value of what it represents in the place to its right.

Multiply and divide multiple units by 10. Apply the property of “ten times greater” in the context of problems.

Create a free account to access thousands of lesson plans.

Already have an account? Sign In

Topic B: Reading, Writing, and Comparing Multi-Digit Whole Numbers

Read and write multi-digit numbers using base-ten numerals, number names, and expanded form.

Locate multi-digit numbers on a number line and explain their placement.

4.NBT.A.2 4.NBT.A.3

Compare numbers based on the meanings of the digits using >, <, or = to record the comparison.

Topic C: Rounding Multi-Digit Whole Numbers

Round multi-digit numbers to their largest place.

Round multi-digit numbers to any place.

Round multi-digit numbers to any place in more complex cases, including those involving real-world contexts and/or assessing the reasonableness of that estimate. 

Topic D: Multi-Digit Whole-Number Addition and Subtraction

Fluently add multi-digit whole numbers using the standard algorithm involving up to two compositions. Solve one-step word problems involving addition.

Fluently add multi-digit whole numbers using the standard algorithm involving multiple compositions. Solve one-step word problems involving addition.

Solve multi-step word problems involving addition, using rounding to assess the reasonableness of answers.

4.NBT.B.4 4.OA.A.3

Fluently subtract multi-digit whole numbers using the standard algorithm involving up to two decompositions. Solve one-step word problems involving subtraction.

Solve multi-step word problems involving subtraction, using rounding to assess the reasonableness of answers.

Solve multi-step word problems involving addition and subtraction, using rounding to assess the reasonableness of answers.

Request a Demo

See all of the features of Fishtank in action and begin the conversation about adoption.

Learn more about Fishtank Learning School Adoption.

Contact Information

School information, what courses are you interested in, are you interested in onboarding professional learning for your teachers and instructional leaders, any other information you would like to provide about your school.

Effective Instruction Made Easy

Access rigorous, relevant, and adaptable math lesson plans for free

addition and subtraction problem solving worksheets

All Formats

Resource types, all resource types.

• Rating Count
• Price (Ascending)
• Price (Descending)
• Most Recent

Addition and subtraction problem solving worksheets

Grade 2 Addition and Subtraction with Length Problem Solving Worksheets

Fractions Worksheets Freebie Add , Subtract and Solve Word Problems

• Easel Activity

2nd Grade Addition & Subtract Math Word Problem Solving Intervention Worksheets

Step by Step Word Problems Worksheets - Addition & Subtraction Problem Solving

Add and Subtract to Solve Word Problems Seatwork Worksheet Independent- 4 Pages!

1st Grade Word Problems Math Worksheets Addition & Subtraction within 20

3rd Grade Math Review - Spiral Review Worksheets | Print and Digital

• Google Apps™

Line Plots with Fractions Worksheets Activity 4th Grade Interpret Data & Graphs

Addition and Subtraction Kindergarten Worksheets and Activities - Within 10

4th Grade Math Review | Spiral Review Worksheets

Counting and Recognizing Coins Worksheets 1st Grade Math Money Word Problems

Addition and Subtraction Word Problems Worksheets Second Grade Math 2.OA.1

Picture Word Problems Printable Worksheets - Addition & Subtraction Kindergarten

Skip Counting by 2, 5 and 10 - number cards & worksheets Bundle

Addition and Subtraction Word Problems Worksheets First Grade Math 1.OA.1

2 Digit Addition & Subtraction Word Problems within 100 Math Worksheets

Adding and Subtracting Fractions with Unlike Denominators Coloring Worksheet

Using Part-Part-Whole Models to Solve Addition and Subtraction Word Problems

Addition & Subtraction Unit - Worksheets & Word Problems - Grade 1 Math Ontario

Addition and Subtraction Fact Families Centers Activities Worksheets and Games

No Prep Triple Digit Addition and Subtraction Printables

First Grade Math Topic 1, Solve Addition and Subtraction Problems to 10

Addition & Subtraction Word Problems within 20 1st Grade Worksheets Anchor Chart

Adding and Subtracting Fractions Unlike Denominators Worksheets 5.NF.1 5.NF.2

• We're hiring
• Help & FAQ
• Privacy policy
• Student privacy
• Terms of service
• Tell us what you think

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

UP Class 6th

Course: up class 6th   >   unit 2.

• Adding and subtracting on number line

Addition on a number line

• Commutative law of addition
• Associative law of addition
• Applying properties of whole number addition
• (Choice A)   A
• (Choice B)   B
• (Choice C)   C

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&