problem solving estimation questions

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Course: 3rd grade   >   Unit 8

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Estimation Textbook Exercise

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estimating, approximation, approximating

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Here is everything you need to know about estimation for. You’ll learn the definition of estimation and how to use various estimation strategies to make calculations quicker and easier.

Students will first learn about estimation in 3rd grade math as part of their work with operations and algebraic thinking. They will expand upon that knowledge as they progress through elementary math.

What is estimation?

Estimation is when you use approximate values in a calculation to give an approximate answer rather than an exact answer.

Estimation helps to make calculations quicker and easier.

Let’s look at some estimation strategies.

Common Core State Standards

How does this relate to 4th grade math and 5th grade math?

• Grade 3 – Operations and Algebraic Thinking (3.OA.D.8) Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
• Grade 4 – Number and Operations – Base Ten (4.NBT.A.3) Use place value understanding to round multi-digit whole numbers to any place.
• Grade 5 – Number and Operations – Base Ten (5.NBT.A.4) Use place value understanding to round decimals to any place

How to estimate calculations

In order to estimate using compatible numbers.

Round each number so that they are compatible.

Create the new calculation using the compatible numbers.

Do the calculation using the compatible numbers to give an approximate answer, using the ‘approximately equal to’ symbol \textbf{≈}

In order to estimate using rounding.

Round each number in the calculation the nearest whole number, nearest ten, nearest hundred, or nearest thousand.

Create the calculation using the rounded numbers.

Do the calculation using the rounded numbers to give an approximate answer, using the ‘approximately equal to’ symbol \textbf{≈}

In order to estimate a  real world scenario using rounding or compatible numbers.

Read the problem and decide which operation(s) to use.

Create an equation.

Use rounding or compatible numbers to write a new equation.

Do the calculation(s) using the rounded numbers of the compatible numbers to give an approximate answer, using the ‘approximately equal to’ symbol \textbf{≈}

Label the answer.

[FREE] Rounding Numbers Check for Understanding (Grade 3 to 5)

Use this quiz to check your grade 3 to 5 students’ understanding of rounding numbers. 10+ questions with answers covering a range of 3rd, 4th, and 5th grade rounding numbers topics to identify areas of strength and support!

Estimation examples

Example 1: estimate the product using compatible numbers.

Estimate the value of 228 \times 19 by using compatible numbers.

228 is close to 220.

19 is close to 20

2 Create the new calculation using the compatible numbers.

220 \times 20 =

This calculation is much easier to do than 228\times 19.

3 Do the calculation(s) using the rounded numbers of the compatible numbers to give an approximate answer, using the ‘approximately equal to’ symbol \textbf{≈}

220\times20=4400

228\times19=4,332

228\times19\approx4,400

Example 2: estimate quotients using compatible numbers

Estimate the value of 2774\div73

2774 is close to 2800.

73 is close to 70.

2800\div70=40

2774\div73=38

2774\div73\approx40

Example 3: estimate the product of decimal numbers by rounding to the nearest whole number

Estimate the value by rounding to the nearest whole number.

6.9\times 9.6

6.9 rounded to the nearest whole number is 7. \;\; 9 > 5 so round up by adding 1 to the rounded digit. 6 + 1 = 7

9.6 rounded to the nearest whole number is 10. \;\; 6 > 5, so round up by adding 1 to the rounded digit. 9 + 1 = 10

This calculation is much easier to do than 6.9\times9.6

7\times10=70

6.9\times9.6=66.24

6.9\times9.6\approx70

Example 4: estimate the quotient of decimal numbers using rounding

783.12\div2.4

783.12 round to the nearest ten. 3 < 5, so round down by leaving the rounded digit alone. The number will round to 780.

2.4 round to the nearest whole number. 4 < 5, so round down by leaving the rounded digit alone. The number will round to 2.

780\div2=390

783.12\div 2.4 = 326.3

783.12\div2.4\approx390

Example 5: word problem using estimation

Benny has saved \$127.54. Sandy has saved 2.5 times as much money as Benny.

Estimate how much money Sandy has saved.

Multiplication will be used to solve this problem.

127.54\times2.5= \; ?

Rounding strategy will be used.

127.54 round to the nearest ten. 7 > 5, so round up by adding one to the rounding digit. 2 + 1 = 3, so it will round to 130.

2.5 rounded to the nearest whole number is 3. \; 5 = 5, so round up by adding one to the rounded digit.

130\times3=?

130\times3=390

127.54\times 2.5 = 318.85

127.54\times2.5\approx390

Sandy saved approximately \$390.

Example 6: word problem using estimation

Janna has \$21.16. She wants to buy the large candy bars for her party. The large candy bars cost \$1.84. Estimate how many large candy bars she can buy.

Division is the operation that will be used.

21.16\div 1.84=

21.16 round to the nearest ten. 1 < 5, so round down by leaving the rounded digit alone. The number will round to be 20.

1.84 round to the nearest whole number. 8 > 5, so round up by adding 1 to the rounded digit. 1 + 1 = 2. The number will round to be 2.

20\div 2 = 10

21.16\div1.84 = 11.5

21.16\div1.84\approx 10

Janna can buy approximately 10 large candy bars.

Teaching tips for Estimation

• Reinforcing number sense is essential to estimation strategies. Use number lines and manipulatives to help students build a deep understanding of numbers and how they relate to one another.
• Have students check their answers for reasonableness by estimating products, estimating quotients, estimating sums and estimating differences.
• Estimating strategies are used in real-life constantly. Use real-life learning activities in class so that students can see the importance of estimating skills.
• When teaching the strategy of compatible numbers, emphasize to students that they estimated pair of numbers must work well together. Rounding strategies such as rounding to the nearest whole number might not give a pair of compatible numbers.
• Demonstrate to students that the closer the rounded numbers are to the actual number the better estimate they’ll get.
• Although worksheets with estimation math problems help students practice the strategies, having students do real-life problems is more meaningful.

Our favorite mistakes

• It is common to find the actual calculation rather than use estimation Estimation strategies are used to make the calculation quick and easily done mentally. 5.8\times2.3 is not easy to do quickly. So using the estimation strategy of rounding 5.8 to 6 and 2.3 to 2 will make the calculation quick and easy. 6\times 2 = 12. The actual answer is 13.34.
• Students have difficulty understanding estimation or applying an estimation strategy. Students have trouble estimating “how many” because they have weak number sense. Number sense helps you to understand how numbers relate to one another. Reinforcing number lines and visual models helps students to make sense of the numbers.
• Students think they have to use one particular estimation strategy. Estimation strategies exist to help students make calculations quick and easy. There is NOT one right strategy to estimate. Students should use the strategy they are most comfortable with.

Related lessons on rounding numbers

• Rounding numbers
• Rounding decimals
• Place value
• Significant figures

Practice Estimation questions

1. Use compatible numbers to estimate the value of 22\times 19?

22 is close to 20

19 is close to 20 20\times20 = 400 22\times 19\approx 400

The actual answer is 418 which is close to 400.

2. What are the appropriate compatible numbers to estimate the value of 219\div74

Compatible numbers are a pair of numbers that work well together.

In this case, estimating 219 to be 210 and estimating 74 to be 70 will make a pair of compatible numbers. 210\div30 is easy and quick to calculate.

The compatible numbers are 210 and 70.

3. Estimate the value of 7.2 \times 98 using rounding.

7.2 rounded to the nearest whole number is 7. \;\; 2 < 5 , so round down by leaving the rounding digit alone.

98 rounded to the nearest hundred is 100. \;\; 8 > 5, so round up by adding 1 to the rounded digit.

4. Estimate the value of 389\div81 using rounding.

389 round to the nearest hundred. 8 > 5, so round up by adding one to the rounded digit. 3 + 1 = 4, so 400.

81 round to the nearest ten. 1 < 5, so round down by leaving the rounding digit alone.

5. Jenny has saved \$114.65 in her savings account. Brandon has saved 1.5 times the amount of money. Estimate the amount of money Brandon has saved.

You will estimate the value of 114.65\times 1.5.

Round \$114.65 to the nearest ten. 4 < 5 so leave the rounded digit alone, so it will round to be \$110. \; 1.5 round to the nearest whole number. 5 = 5, so round up by adding one to the rounded digit. 1 + 1 = 2.

6. There are 678 boxes in 11 storage spaces. Estimate how many boxes there are in 1 storage space by rounding the values to the nearest ten.

\approx{57} boxes

\approx{68} boxes

\approx{64} boxes

\approx{58} boxes

You will estimate the value of 678\div11.

Round 678 to the nearest ten. 8 > 5, round up by adding 1 to the rounded digit. 7 + 1 = 8, so the rounded number is 680.

Round 11 to the nearest ten. 1 < 5, leave the rounded digit alone, so the rounded number is 10.

Estimation FAQs

Rounding is taking a number and rounding it to a specific place, for example rounding to the nearest tens place, nearest hundreds place, nearest hundredths place, etc.. Compatible numbers are numbers that work well with each other.

There is no right or wrong way to estimate. In many cases it depends on the actual problem.

Round to the place that will make the calculations the easiest to do.

The closer your estimated answer is to the exact answer, the better the estimate.

The next lessons are

• Factors and multiples
• Fractions operations

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General Problem Solving

Problem solving and estimating, learning outcomes.

• Identify and apply a solution pathway for multi-step problems

Problem solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backwards, asking “what information and procedures will I need to find this?” Very few interesting questions can be answered in one mathematical step; often times you will need to chain together a solution pathway , a series of steps that will allow you to answer the question.

Problem Solving Process

• Identify the question you’re trying to answer.
• Work backwards, identifying the information you will need and the relationships you will use to answer that question.
• Continue working backwards, creating a solution pathway.
• If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
• Solve the problem, following your solution pathway.

In most problems we work, we will be approximating a solution, because we will not have perfect information. We will begin with a few examples where we will be able to approximate the solution using basic knowledge from our lives.

Recall: operations on Fractions

When simplifying algebraic expressions, we may sometimes need to add, subtract, simplify, multiply, or divide fractions. It is important to be able to do these operations on the fractions without converting them to decimals.

•  [latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex]
•  [latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}[/latex]
•  [latex]\dfrac{24}{32}=\dfrac{2\cdot2\cdot2\cdot3}{2\cdot2\cdot2\cdot2\cdot2}=\dfrac{3}{2\cdot2}=\dfrac{3}{4}[/latex]
•  [latex]\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad \pm bc}{bd}[/latex]

In the first example, we will need to think about time scales, we are asked to find how many times a heart beats in a year, but usually we measure heart rate in beats per minute.

How many times does your heart beat in a year?

This question is asking for the rate of heart beats per year. Since a year is a long time to measure heart beats for, if we knew the rate of heart beats per minute, we could scale that quantity up to a year. So the information we need to answer this question is heart beats per minute. This is something you can easily measure by counting your pulse while watching a clock for a minute.

Suppose you count 80 beats in a minute. To convert this to beats per year:

[latex]\displaystyle\frac{80\text{ beats}}{1\text{ minute}}\cdot\frac{60\text{ minutes}}{1\text{ hour}}\cdot\frac{24\text{ hours}}{1\text{ day}}\cdot\frac{365\text{ days}}{1\text{ year}}=42,048,000\text{ beats per year}[/latex]

The technique that helped us solve the last problem was to get the number of heartbeats in a minute translated into the number of heartbeats in a year. Converting units from one to another, like minutes to years is a common tool for solving problems.

In the next example, we show how to infer the thickness of something too small to measure with every-day tools. Before precision instruments were widely available, scientists and engineers had to get creative with ways to measure either very small or very large things. Imagine how early astronomers inferred the distance to stars, or the circumference of the earth.

How thick is a single sheet of paper? How much does it weigh?

While you might have a sheet of paper handy, trying to measure it would be tricky. Instead we might imagine a stack of paper, and then scale the thickness and weight to a single sheet. If you’ve ever bought paper for a printer or copier, you probably bought a ream, which contains 500 sheets. We could estimate that a ream of paper is about 2 inches thick and weighs about 5 pounds. Scaling these down,

[latex]\displaystyle\frac{2\text{ inches}}{\text{ream}}\cdot\frac{1\text{ ream}}{500\text{ pages}}=0.004\text{ inches per sheet}[/latex]

[latex]\displaystyle\frac{5\text{ pounds}}{\text{ream}}\cdot\frac{1\text{ ream}}{500\text{ pages}}=0.01\text{ pounds per sheet, or }=0.16\text{ ounces per sheet.}[/latex]

The first two example questions in this set are examined in more detail here.

We can infer a measurement by using scaling.  If 500 sheets of paper is two inches thick, then we could use proportional reasoning to infer the thickness of one sheet of paper.

In the next example, we use proportional reasoning to determine how many calories are in a mini muffin when you are given the amount of calories for a regular sized muffin.

A recipe for zucchini muffins states that it yields 12 muffins, with 250 calories per muffin. You instead decide to make mini-muffins, and the recipe yields 20 muffins. If you eat 4, how many calories will you consume?

There are several possible solution pathways to answer this question. We will explore one.

To answer the question of how many calories 4 mini-muffins will contain, we would want to know the number of calories in each mini-muffin. To find the calories in each mini-muffin, we could first find the total calories for the entire recipe, then divide it by the number of mini-muffins produced. To find the total calories for the recipe, we could multiply the calories per standard muffin by the number per muffin. Notice that this produces a multi-step solution pathway. It is often easier to solve a problem in small steps, rather than trying to find a way to jump directly from the given information to the solution.

We can now execute our plan:

[latex]\displaystyle{12}\text{ muffins}\cdot\frac{250\text{ calories}}{\text{muffin}}=3000\text{ calories for the whole recipe}[/latex]

[latex]\displaystyle\frac{3000\text{ calories}}{20\text{ mini-muffins}}=\text{ gives }150\text{ calories per mini-muffin}[/latex]

[latex]\displaystyle4\text{ mini-muffins}\cdot\frac{150\text{ calories}}{\text{mini-muffin}}=\text{totals }600\text{ calories consumed.}[/latex]

View the following video for more about the zucchini muffin problem.

We have found that ratios are very helpful when we know some information but it is not in the right units, or parts to answer our question. Making comparisons mathematically often involves using ratios and proportions. In the next examples we will

You need to replace the boards on your deck. About how much will the materials cost?

There are two approaches we could take to this problem: 1) estimate the number of boards we will need and find the cost per board, or 2) estimate the area of the deck and find the approximate cost per square foot for deck boards. We will take the latter approach.

For this solution pathway, we will be able to answer the question if we know the cost per square foot for decking boards and the square footage of the deck. To find the cost per square foot for decking boards, we could compute the area of a single board, and divide it into the cost for that board. We can compute the square footage of the deck using geometric formulas. So first we need information: the dimensions of the deck, and the cost and dimensions of a single deck board.

Suppose that a rectangular deck measures 16 ft by 24 ft, for a total area of 384 ft 2 .

From a visit to the local home store, you find that an 8 foot by 4 inch cedar deck board costs about $7.50. The area of this board, doing the necessary conversion from inches to feet, is:

[latex]\displaystyle{8}\text{ feet}\cdot4\text{ inches}\cdot\frac{1\text{ foot}}{12\text{ inches}}=2.667\text{ft}^2{.}[/latex] The cost per square foot is then [latex]\displaystyle\frac{\$7.50}{2.667\text{ft}^2}=\$2.8125\text{ per ft}^2{.}[/latex]

This will allow us to estimate the material cost for the whole 384 ft 2 deck

[latex]\displaystyle\$384\text{ft}^2\cdot\frac{\$2.8125}{\text{ft}^2}=\$1080\text{ total cost.}[/latex]

Of course, this cost estimate assumes that there is no waste, which is rarely the case. It is common to add at least 10% to the cost estimate to account for waste.

This example is worked through in the following video.

Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata?

To make this decision, we must first decide what our basis for comparison will be. For the purposes of this example, we’ll focus on fuel and purchase costs, but environmental impacts and maintenance costs are other factors a buyer might consider.

It might be interesting to compare the cost of gas to run both cars for a year. To determine this, we will need to know the miles per gallon both cars get, as well as the number of miles we expect to drive in a year. From that information, we can find the number of gallons required from a year. Using the price of gas per gallon, we can find the running cost.

From Hyundai’s website, the 2013 Sonata will get 24 miles per gallon (mpg) in the city, and 35 mpg on the highway. The hybrid will get 35 mpg in the city, and 40 mpg on the highway.

An average driver drives about 12,000 miles a year. Suppose that you expect to drive about 75% of that in the city, so 9,000 city miles a year, and 3,000 highway miles a year.

We can then find the number of gallons each car would require for the year.

Sonata: [latex]\displaystyle{9000}\text{ city miles}\cdot\frac{1\text{ gallon}}{24\text{ city miles}}+3000\text{ highway miles}\cdot\frac{1\text{ gallon}}{35\text{ highway miles}}=460.7\text{ gallons}[/latex]

Hybrid:  [latex]\displaystyle{9000}\text{ city miles}\cdot\frac{1\text{ gallon}}{35\text{ city miles}}+3000\text{ highway miles}\cdot\frac{1\text{ gallon}}{40\text{ highway miles}}=332.1\text{ gallons}[/latex]

If gas in your area averages about $3.50 per gallon, we can use that to find the running cost:

Sonata:  [latex]\displaystyle{460.7}\text{ gallons}\cdot\frac{\$3.50}{\text{gallon}}=\$1612.45[/latex]

Hybrid: [latex]\displaystyle{332.1}\text{ gallons}\cdot\frac{\$3.50}{\text{gallon}}=\$1162.35[/latex]

The hybrid will save $450.10 a year. The gas costs for the hybrid are about [latex]\displaystyle\frac{\$450.10}{\$1612.45}[/latex] = 0.279 = 27.9% lower than the costs for the standard Sonata.

While both the absolute and relative comparisons are useful here, they still make it hard to answer the original question, since “is it worth it” implies there is some tradeoff for the gas savings. Indeed, the hybrid Sonata costs about $25,850, compared to the base model for the regular Sonata, at $20,895.

To better answer the “is it worth it” question, we might explore how long it will take the gas savings to make up for the additional initial cost. The hybrid costs $4965 more. With gas savings of $451.10 a year, it will take about 11 years for the gas savings to make up for the higher initial costs.

We can conclude that if you expect to own the car 11 years, the hybrid is indeed worth it. If you plan to own the car for less than 11 years, it may still be worth it, since the resale value of the hybrid may be higher, or for other non-monetary reasons. This is a case where math can help guide your decision, but it can’t make it for you.

This question pulls together all the skills discussed previously on this page, as the video demonstration illustrates.

• Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
• Problem Solving. Authored by : David Lippman. Located at : http://www.opentextbookstore.com/mathinsociety/ . Project : Math in Society. License : CC BY-SA: Attribution-ShareAlike
• Estimating with imperfect information. Authored by : OCLPhase2's channel. Located at : https://youtu.be/xF5BNEr0gjo . License : CC BY: Attribution
• Multistep proportions / problem solving process. Authored by : OCLPhase2's channel. Located at : https://youtu.be/NVCwFO-w2z4 . License : CC BY: Attribution
• Estimating the cost of a deck. Authored by : OCLPhase2's channel. Located at : https://youtu.be/adPGfeTy-Pc . License : CC BY: Attribution
• Guiding decision using math: Sonata vs Hybrid. Authored by : OCLPhase2's channel. Located at : https://youtu.be/HXmc-EkOYJE . License : CC BY: Attribution

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Estimating worded problems, problem solving and reasoning

Subject: Mathematics

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7 October 2018

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How to Tackle Estimation Interview Questions

Jul 4, 2017 | Career

“How would you estimate the number of people living in Iowa?”

“How would you estimate expenses if you were planning the next Olympics?”

“How many people could you fit on a football field?”

“If a lottery ticket costs $3 and pays $12 if you win, what is the probability of winning?”

These are questions I received while interviewing for actuarial internships. I never received more than 1 of these questions per interview, but your response can make or break the interview.

The most important thing to remember about estimation questions is: employers aren’t looking for a “correct” answer. They are looking for a few key qualities:

• Problem-solving skills – can you logically think through a difficult, open-ended question to develop a solution?
• Reasonable assumption-setting – given your problem-solving approach, can you set reasonable assumptions and justify them?
• Communication skills – can you clearly explain your thought process?

Unlike behavioral questions, you can’t anticipate which estimation problems you’ll receive. That doesn’t mean you can’t prepare for them.

You can follow a few general principles to think through any of these estimation questions and impress the interviewer:

1. Don’t focus on getting the correct answer

I mentioned this above, but it’s worth repeating.

Especially as a student, you are trained to think in terms of black-and-white with math problems – your answer is either right or wrong.

Not only does this miss the point of these estimation questions, but it puts unnecessary pressure on you. I bombed my first estimation interview question because of this pressure. My internal dialogue went something like:

“I don’t know how much it costs to plan the Olympics.”

“Should I be thinking in terms of millions or billions?”

“There is no way I’m going to come close to the correct answer.”

And I proceeded to stumble my way through a response. Not surprisingly, I didn’t get a follow-up interview.

Notice that my internal dialogue was entirely focused on the final answer. I focused on what I didn’t know, instead of what I did. I needed to flip the script, leading us to the next principle…

2. Deconstruct the question

The first step to solve any challenging problem is to break it into manageable chunks. Let’s say we’re asked for the number of Starbucks stores in the United States.

Most of us don’t have this information memorized. However, what if we broke this question down into 3 smaller questions:

• How many Starbucks stores are in your town?
• How many people live in your town?
• How many people live in the United States?

These questions are easier to answer. My hometown has 1 Starbucks store, and it has a population of about 25,000. Knowing that there are around 300 million people in the United States, I would estimate:

300,000,000 people / 25,000 people per store = 12,000 stores

 After doing a quick online search, I found that there are around 13,000 Starbucks in the United States as of 2016 (total company-operated + licensed stores). The estimate ended up being close, although that’s not the point.

The interviewer is paying more attention to the thought process – how did I break the problem into smaller chunks? Did I do so in a logical manner? What assumptions did I use and were they reasonable?

3. Explain the assumptions & limitations

When you’re answering a difficult estimation question without the help of outside resources, you’ll end up making simplifying assumptions. Again, the point is not to be correct in your assumptions, but to be thoughtful about how you set them.

In the Starbucks question above, I broke the question into three unknown quantities: the US population, my hometown population, and the number of Starbucks in my hometown.

I took a sampling approach to answer the question. My method implied that my hometown was a representative sample of Starbucks per capita across the US. I’m sure that assumption is inaccurate, but that’s okay as long as I address this limitation.

There are also sources of error in the quantities I used for my variables. I’m not using exact population numbers, and I may not remember the correct number of Starbucks stores in the town. These estimates greatly affect my answer.

When you’re explaining your estimation process, it’s important to not only explain your assumptions but also address their limitations and sources of error .

Companies like to ask open-ended estimation questions to see how you approach challenging and unexpected problems.

You can’t predict what they will ask, but you can practice the general framework of deconstructing the question, setting assumptions, and communicating the limitations of your process.

The key is to remain composed while responding. The interviewers know they are asking a tough question, and they will be especially impressed if you piece together a logical response in a confident, well-spoken manner.

I recommend searching for “estimation interview questions” and practicing your response to several questions. You’ll be much more comfortable in a real interview if you already have experience responding to difficult questions of this nature.

The goal is not to prepare for any specific question, but instead, become more comfortable with difficult interview questions.

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PM estimation interview questions (how-to guide with examples)

Estimation questions are common in product manager interviews at companies like Google, Facebook, and Amazon. 

Here are a few examples of PM estimation interview questions:

• What is the market size for driverless cars in 2025?
• What is the storage space required to host all images on Google Street View?
• What is the required Internet bandwidth for an average college campus?
• How many kindergarten teachers are there in the US?
• What is the weight of a school bus?

Note that the above examples are real interview questions that were reported by Google PM candidates on Glassdoor .

Estimation questions can be really intimidating because they often cover huge topics with a lot of variables.

The good news is that they're fairly easy to answer if you know how to approach them. So let's walk through our 4-step process for solving them, as well as a few examples and shortcuts.

• How to answer estimation questions
• Example with answer: YouTube's daily revenue
• Cheat sheet
• List of practice questions
• How to practice estimation questions

Click here to practice 1-on-1 with FAANG ex-interviewers

1. how to answer product manager estimation questions ↑.

We recommend using a four-step approach to answer product estimation questions. Let's discuss each step one by one.

• Ask clarification questions
• Map out your calculations
• Round numbers and calculate
• Sense-check your results

Step one: Ask clarification questions

The first thing you should do is ask a few clarification questions to make sure that you know exactly what number you need to calculate.

For instance, let's imagine your interviewer asks: "How much money is spent on gas in the US every year?" Here are a few important clarification questions you would need to ask:

• Should we focus on gas consumption from road transportation only? Or also include home heating, air transportation, etc.?
• Should we focus on personal cars only? Or include consumption from commercial vehicles, too?

These questions are useful to make sure that you calculate exactly what your interviewer wants. In addition, they are also a great way to buy yourself some time. While your conscious brain asks these questions, your subconscious brain can start working on step two of the approach.

Step two: Map out your calculations

Once you know exactly what number you want to calculate, you then need to map the calculation steps to get to that number. Let's imagine we want to calculate the quantity of gas consumed by personal cars in the US.

Here is a step-by-step approach that you could use to get to that number:

• Calculate the number of personal cars in the US
• Calculate the consumption of an average personal car
• Calculate the total consumption in gallons and then dollars

If you struggle to find a starting point, an alternative approach is to begin from the number you want to calculate and draw an issue tree from top to bottom, as we have below. In addition, the advantage of this approach is that it will force you to be mutually exclusive and collectively exhaustive (MECE) .

Both approaches are equally acceptable. But in both cases you should VALIDATE your approach with your interviewer by telling them the different calculations steps you are going to take BEFORE starting to calculate numbers (step three). This is important because it gives your interviewer a chance to rectify your course of action if they had another plan in mind.

Skipping this validation step is extremely risky because your interviewer might realize too late in the exercise that you are approaching the problem in a different way to what they wanted. In our experience it's almost impossible to recover from that situation.

Step three: Round numbers and calculate

Once you've agreed on an approach, it's time to start calculating. You will need various data points (e.g. number of miles driven by an average US driver each year) to get to your final estimate. In most cases your interviewer will ask you to make your own assumptions to get to the final number. But in some other cases they might share data with you.

When making assumptions it is vital that you pick simple numbers. For instance, you should use 350 days in a year instead of 365. You should assume that the average cost per gallon of gas is $3.00 instead of $2.50. Etc. Rounding numbers will make your calculations easier and decrease your likelihood of making a mistake.

In addition, you should talk out loud when doing calculations so that your interviewer can follow your thought process. Your interviewer is interested in what's going on in your head - not the final result.

Note: if you are looking for ways to become faster at mental math calculations we recommend reading our interview maths guide .

Step four: Sense-check your results

Most candidates stop talking at the end of step three. They look up and expect their interviewer to tell them if they got to the right result or not. This is a mistake. The best candidates sense-check their results and try to spot their own mistakes before telling their interviewer they are done.

Mental calculation errors happen frequently. If your interviewer spots a mistake in your calculations this will definitely play against you. But if you spot your own mistake you still have a chance.

Now that you know what technique to use to answer product estimation questions let's apply it to an example.

2. Example - Estimate YouTube's daily revenue ↑

Try answering the question below accurately and as fast as possible.

Try this question:

How much revenue does YouTube make per day?

1. Ask clarification questions

Here are some clarification questions that immediately come to mind for this estimation question:

• Do we want to calculate all of YouTube's revenues? Or just advertising which is likely to be the main source?
• Does YouTube charge advertisers per 1,000 views or per click? Are we interested in YouTube's revenues globally or just the US?

Let's assume here that we want to calculate the advertising revenues ($) made by YouTube in the US only . And let's also assume that YouTube mainly charges advertisers per 1,000 views for simplicity.

2. Map out your calculations

We could estimate YouTube's daily revenues in the US by taking a three-step approach.

• Calculate the number of daily US YouTube users
• Calculate the number of advertising videos they view
• Calculate the corresponding revenue

In a real interview, you would need to lay out that approach to your interviewer and check that it works for them before starting your calculations.

3. Round numbers and calculate

Let's step through our approach:

1. Calculate the number of daily US YouTube users

• There are ~300m people in the US. Let's assume that the average life expectancy is 80 years old and that the population is equally distributed across each age.
• 0 to 10 y.o.: does not watch
• 10 to 20 y.o.: every day
• 20 to 40 y.o.: every day
• 40 to 60 y.o.: 75% of days
• 60 to 80 y.o.: 50% of days
• 0 to 10 y.o.: 0
• 10 to 20 y.o.: every day = 100% x 300m / 8 = ~40m
• 20 to 40 y.o.: every day = 100% x 300m / 4  = 75m
• 40 to 60 y.o.: 75% of days = 75% x 75m = ~55m
• 60 to 80 y.o.: 50% of days = 50% x 75m = ~40m
• Total = 40 + 75 + 55 + 40 = ~210m
• The number of daily YouTube users is therefore approximately 200m in the US

In an actual interview it's a great idea to justify the assumptions you make by relating them to your personal experience. For instance here you could say, "I'm going to assume that the 40 to 60 y.o. group watches YouTube 75% of days as this is roughly how often my parents watch it. And I'll assume 50% for the 60 to 80 y.o. group, as this is how often my grandparents watch it."

2. Calculate the total number of advertising videos viewed

Let's now estimate the total number of ad views per day on YouTube.

• The number of videos viewed by users probably varies quite dramatically. Users who stream music on YouTube probably view 20 or more videos on an average day. Other users might just watch a single video. So, let's assume the average is ten videos per day. We've got 200m users, so that's 2bn views per day.
• YouTube does not systematically play an ad at the beginning of a video. Based on our experience, let's assume only 50% of videos viewed start with an ad. So, that's 1bn advertising video views per day.

3. Calculate the corresponding revenue

• Assuming YouTube charges advertisers $10 per 1,000 views, the total revenues it generates in a day is therefore: 1bn x $10 / 1,000 = $10m

We therefore estimate that YouTube generates about $10m per day in ad revenues in the US only.

4. Sense-check your results

As mentioned above, it is crucial that you spend some time sense-checking your results. In practice, you should check your intermediate results as you progress through the estimation and your final result at the end. But here we are checking all the numbers in one place so it's easier for you to keep track of what we are doing.

• Intermediate result: If 200m people watch YouTube every day this means about 2/3rds of the country watches at least a video on any given day. That sounds reasonable.
• Final result: Similarly, if YouTube generates $10m per day, that means it generates $3.65bn per year. Again, this is in the right order of magnitude given Google made about $120bn in ad revenue across all geographies in 2018.

2.1 Example - Estimate time stopped at stop lights

Now you've learned how to use a framework, watch the mock interview below to see it in action. We recommend pausing the video at regular intervals so you can come up with your own answers.

Note: it's a good idea to learn some numbers for estimation questions such as the cost per thousand views in advertising ($10) or Google's ad revenues in 2018 ($120bn). We've put together a cheat sheet later in this article to get you started with this.

3. Estimation questions cheat sheet ↑

At this stage, we feel it will also be useful for you to memorise some common assumptions, such as population and life expectancy, as they will often come up in estimation questions.

We've created a cheat sheet with socio-economic data, advertising data, and some company information below. You can download the slides by simply clicking this link: estimation questions cheat sheet

4. List of practice questions ↑

Here is a list product estimation questions that were asked in PM interviews at Google according to data from Glassdoor.com. These are great examples of the questions you may encounter in your interviews. 

If you'd like to learn about the other types of questions you may face, you can also visit our  ultimate guide to product manager interview questions .

• What is the market size for toilet paper in the US?
• How many restaurant reviews are written on Google Reviews every month?
• How many millennials own homes in the US?
• How much ad revenue does GMail make every year?
• How many computers does Google own?
• How many dentists are there in New York?
• How many bicycles do you need to start a bike sharing service in New York?
• How many passengers are in the air on a plane at any given time in the US?
• What is the weight of the Empire State building?
• You are opening a new Walmart store. How many cash registers do you need?
• How much money is spent on gas in the US every year?

Now that you have a list of sample questions to work with, it’s important to consider how you will practice with these questions.

5. How to practice estimation questions ↑

With a lot to cover, it’s best to take a systematic approach to make the most of your practice time. 

Below you’ll find links to free resources and four introductory steps that you can take to prepare for estimation questions.

5.1 Study the company you're applying to

Get acquainted with the company you’ve applied to. In many cases, the product questions you’ll be presented with will be based on real-life cases the company is facing. If you’re applying to a specific team, study up on their products, the user, etc.

Take the time to find out which products you’ll most likely be working with, based on the job description, and research them. Look up relevant press releases, product descriptions, product reviews, and other resources in order to discuss what’s most important to the role: the company’s product.

If you'd like to learn more about a specific company's PM interviews, then we'd encourage you to check out our guide for that company below :

• Google product manager interview guide
• Facebook product manager interview guide
• Amazon product manager interview guide
• Microsoft product manager interview guide
• LinkedIn product manager interview guide
• Uber product manager interview guide
• Stripe product manager interview guide
• Lyft product manager interview guide
• Apple product manager interview guide
• TikTok product manager interview guide
• Coinbase product manager interview guide
• Airbnb product manager interview guide
• DoorDash product manager interview guide

5.2 Learn a consistent method for answering estimation questions

In this article, we’ve outlined a step-by-step method you can use to solve estimation questions. We’d encourage you to first memorize the basic steps, and then try solving a couple of the sample questions on paper.

This will help you to understand the structure of a good answer.  This is a good first step, BUT just knowing the method is not enough, you also need to be able to apply the steps in interview conditions. 

5.3 Practice by yourself or with peers

In our experience, practicing by yourself is a great way to prepare for PM interviews. You can ask and answer questions out loud, to help you get a feel for the different types of PM interview questions. Practicing by yourself will help you perfect your step-by-step approach for each question type. It also gives you time to correct your early mistakes.

If you have friends or peers who can do mock interviews with you, that's a great option too. This can be especially helpful if your friend has experience with PM interviews, or is at least familiar with the process.

5.4 Practice with experienced PM interviewers

Finally, you should also try to practice product manager mock interviews with expert ex-interviewers, as they’ll be able to give you much more accurate feedback than friends and peers. 

If you know a Product Manager who can help you, that's fantastic! But for most of us, it's tough to find the right connections to make this happen. And it might also be difficult to practice multiple hours with that person unless you know them really well.

Here's the good news. We've already made the connections for you. W e’ve created a coaching service where you can practice 1-on-1 with ex-interviewers from Google, Amazon, and other leading tech companies. Learn more and start scheduling sessions today .

Keep reading: product manager interview articles

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1.5: Problem Solving and Estimating

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Finally, we will bring together the mathematical tools we’ve reviewed, and use them to approach more complex problems. In many problems, it is tempting to take the given information, plug it into whatever formulas you have handy, and hope that the result is what you were supposed to find. Chances are, this approach has served you well in other math classes.

This approach does not work well with real life problems. Instead, problem solving is best approached by first starting at the end: identifying exactly what you are looking for. From there, you then work backwards, asking “what information and procedures will I need to find this?” Very few interesting questions can be answered in one mathematical step; often times you will need to chain together a solution pathway, a series of steps that will allow you to answer the question.

Problem Solving Process

• Identify the question you’re trying to answer.
• Work backwards, identifying the information you will need and the relationships you will use to answer that question.
• Continue working backwards, creating a solution pathway.
• If you are missing necessary information, look it up or estimate it. If you have unnecessary information, ignore it.
• Solve the problem, following your solution pathway.

In most problems we work, we will be approximating a solution, because we will not have perfect information. We will begin with a few examples where we will be able to approximate the solution using basic knowledge from our lives.

How many times does your heart beat in a year?

This question is asking for the rate of heart beats per year. Since a year is a long time to measure heart beats for, if we knew the rate of heart beats per minute, we could scale that quantity up to a year. So the information we need to answer this question is heart beats per minute. This is something you can easily measure by counting your pulse while watching a clock for a minute.

Suppose you count 80 beats in a minute. To convert this beats per year:

\(\frac{80 \text { beats }}{1 \text { minute }} \cdot \frac{60 \text { minutes }}{1 \text { hour }} \cdot \frac{24 \text { hours }}{1 \text { day }} \cdot \frac{365 \text { days }}{1 \text { year }}=42,048,000\) beats per year

How thick is a single sheet of paper? How much does it weigh?

While you might have a sheet of paper handy, trying to measure it would be tricky. Instead we might imagine a stack of paper, and then scale the thickness and weight to a single sheet. If you’ve ever bought paper for a printer or copier, you probably bought a ream, which contains 500 sheets. We could estimate that a ream of paper is about 2 inches thick and weighs about 5 pounds. Scaling these down,

\(\frac{2 \text { inches }}{\text { ream }} \cdot \frac{1 \text { ream }}{500 \text { pages }}=0.004\) inches per sheet

\(\frac{5 \text { pounds }}{\text { ream }} \cdot \frac{1 \text { ream }}{500 \text { pages }}=0.01\) pounds per sheet, or 0.16 ounces per sheet.

A recipe for zucchini muffins states that it yields 12 muffins, with 250 calories per muffin. You instead decide to make mini-muffins, and the recipe yields 20 muffins. If you eat 4, how many calories will you consume?

There are several possible solution pathways to answer this question. We will explore one.

To answer the question of how many calories 4 mini-muffins will contain, we would want to know the number of calories in each mini-muffin. To find the calories in each mini-muffin, we could first find the total calories for the entire recipe, then divide it by the number of mini-muffins produced. To find the total calories for the recipe, we could multiply the calories per standard muffin by the number per muffin. Notice that this produces a multi-step solution pathway. It is often easier to solve a problem in small steps, rather than trying to find a way to jump directly from the given information to the solution.

We can now execute our plan:

\(12 \text{muffins} $\cdot \frac{250 \text { calories }}{\text { muffin }}=3000$\) calories for the whole recipe

\(\frac{3000 \text { calories }}{20 \text { mini }-\text { muffins }}\) gives 150 calories per mini-muffin

\(4\text{ mini muffins } \cdot \frac{150 \text { calories }}{\text { mini - muffin }}\) totals 600 calories consumed.

You need to replace the boards on your deck. About how much will the materials cost?

There are two approaches we could take to this problem: 1) estimate the number of boards we will need and find the cost per board, or 2) estimate the area of the deck and find the approximate cost per square foot for deck boards. We will take the latter approach.

For this solution pathway, we will be able to answer the question if we know the cost per square foot for decking boards and the square footage of the deck. To find the cost per square foot for decking boards, we could compute the area of a single board, and divide it into the cost for that board. We can compute the square footage of the deck using geometric formulas. So first we need information: the dimensions of the deck, and the cost and dimensions of a single deck board.

Suppose that measuring the deck, it is rectangular, measuring 16 ft by 24 ft, for a total area of \(384 \mathrm{ft}^{2}\).

From a visit to the local home store, you find that an 8 foot by 4 inch cedar deck board costs about $7.50. The area of this board, doing the necessary conversion from inches to feet, is:

\(8 \text { feet } \cdot 4 \text { inches } \cdot \frac{1 \text { foot }}{12 \text { inches }}=2.667 \mathrm{ft}^{2}\). The cost per square foot is then

\(\frac{\$ 7.50}{2.667 \mathrm{ft}^{2}}=\$ 2.8125 \text { per } \mathrm{ft}^{2}\).

This will allow us to estimate the material cost for the whole \(384 \mathrm{ft}^{2}\) deck

\(\$ 384 \mathrm{ft}^{2} \cdot \frac{\$ 2.8125}{\mathrm{ft}^{2}}=\$ 1080\) total cost.

Of course, this cost estimate assumes that there is no waste, which is rarely the case. It is common to add at least 10% to the cost estimate to account for waste.

Is it worth buying a Hyundai Sonata hybrid instead the regular Hyundai Sonata?

To make this decision, we must first decide what our basis for comparison will be. For the purposes of this example, we’ll focus on fuel and purchase costs, but environmental impacts and maintenance costs are other factors a buyer might consider.

It might be interesting to compare the cost of gas to run both cars for a year. To determine this, we will need to know the miles per gallon both cars get, as well as the number of miles we expect to drive in a year. From that information, we can find the number of gallons required from a year. Using the price of gas per gallon, we can find the running cost.

From Hyundai’s website, the 2013 Sonata will get 24 miles per gallon (mpg) in the city, and 35 mpg on the highway. The hybrid will get 35 mpg in the city, and 40 mpg on the highway.

An average driver drives about 12,000 miles a year. Suppose that you expect to drive about 75% of that in the city, so 9,000 city miles a year, and 3,000 highway miles a year.

We can then find the number of gallons each car would require for the year.

\(9000\text{ city miles } \cdot \frac{1 \text { gallon }}{24 \text { city miles }}+3000\text{ hightway miles}. \frac{1 \text { gallon }}{35 \text { highway miles }}=460.7\text{ gallons}\)

\(9000\text{ city miles }\cdot \frac{1 \text { gallon }}{35 \text { city miles }}+3000\text{ hightway miles}. \frac{1 \text { gallon }}{40 \text { highway miles }}=332.1\text{ gallons}\)

If gas in your area averages about $3.50 per gallon, we can use that to find the running cost:

Sonata: \(460.7 \text { gallons } \cdot \frac{\$ 3.50}{\text { gallon }}=\$ 1612.45\)

Hybrid: \(\text { 332.1 gallons } \cdot \frac{\$ 3.50}{\text { gallon }}=\$ 1162.35\)

The hybrid will save $450.10 a year. The gas costs for the hybrid are about \(\frac{\$ 450.10}{\$ 1612.45} = 0.279 = 27.9\%\) lower than the costs for the standard Sonata.

While both the absolute and relative comparisons are useful here, they still make it hard to answer the original question, since “is it worth it” implies there is some tradeoff for the gas savings. Indeed, the hybrid Sonata costs about $25,850, compared to the base model for the regular Sonata, at $20,895.

To better answer the “is it worth it” question, we might explore how long it will take the gas savings to make up for the additional initial cost. The hybrid costs $4965 more. With gas savings of $451.10 a year, it will take about 11 years for the gas savings to make up for the higher initial costs.

We can conclude that if you expect to own the car 11 years, the hybrid is indeed worth it. If you plan to own the car for less than 11 years, it may still be worth it, since the resale value of the hybrid may be higher, or for other non-monetary reasons. This is a case where math can help guide your decision, but it can’t make it for you.

Try it Now 6

If traveling from Seattle, WA to Spokane WA for a three-day conference, does it make more sense to drive or fly?

There is not enough information provided to answer the question, so we will have to make some assumptions, and look up some values.

Assumptions:

a) We own a car. Suppose it gets 24 miles to the gallon. We will only consider gas cost.

b) We will not need to rent a car in Spokane, but will need to get a taxi from the airport to the conference hotel downtown and back.

c) We can get someone to drop us off at the airport, so we don’t need to consider airport parking.

d) We will not consider whether we will lose money by having to take time off work to drive.

Values looked up (your values may be different)

a) Flight cost: \(\$184\)

b) Taxi cost: \(\$25\) each way (estimate, according to hotel website)

c) Driving distance: \(280\) miles each way

d) Gas cost: \(\$3.79\) a gallon

Cost for flying: \(\$184\text{ flight cost }+ \$50\text{ in taxi fares }= \$234\).

Cost for driving: \(560\) miles round trip will require 23.3 gallons of gas, costing \(\$88.31\).

Based on these assumptions, driving is cheaper. However, our assumption that we only include gas cost may not be a good one. Tax law allows you deduct \(\$0.55\) (in 2012) for each mile driven, a value that accounts for gas as well as a portion of the car cost, insurance, maintenance, etc. Based on this number, the cost of driving would be \(\$319\).