how to solve percent equations word problems

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Course: 7th grade   >   Unit 2

Solving percent problems.

• Equivalent expressions with percent problems
• Percent word problem: magic club
• Percent problems
• Percent word problems: tax and discount
• Tax and tip word problems
• Percent word problem: guavas
• Discount, markup, and commission word problems
• Multi-step ratio and percent problems

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5.2.1: Solving Percent Problems

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Learning Objectives

• Identify the amount, the base, and the percent in a percent problem.
• Find the unknown in a percent problem.

Introduction

Percents are a ratio of a number and 100, so they are easier to compare than fractions, as they always have the same denominator, 100. A store may have a 10% off sale. The amount saved is always the same portion or fraction of the price, but a higher price means more money is taken off. Interest rates on a saving account work in the same way. The more money you put in your account, the more money you get in interest. It’s helpful to understand how these percents are calculated.

Parts of a Percent Problem

Jeff has a coupon at the Guitar Store for 15% off any purchase of $100 or more. He wants to buy a used guitar that has a price tag of $220 on it. Jeff wonders how much money the coupon will take off the original $220 price.

Problems involving percents have any three quantities to work with: the percent , the amount , and the base .

• The percent has the percent symbol (%) or the word “percent.” In the problem above, 15% is the percent off the purchase price.
• The base is the whole amount. In the problem above, the whole price of the guitar is $220, which is the base.
• The amount is the number that relates to the percent. It is always part of the whole. In the problem above, the amount is unknown. Since the percent is the percent off , the amount will be the amount off of the price.

You will return to this problem a bit later. The following examples show how to identify the three parts: the percent, the base, and the amount.

Identify the percent, amount, and base in this problem.

30 is 20% of what number?

Percent: The percent is the number with the % symbol: 20%.

Base : The base is the whole amount, which in this case is unknown.

Amount: The amount based on the percent is 30.

Percent=20%

Base=unknown

The previous problem states that 30 is a portion of another number. That means 30 is the amount. Note that this problem could be rewritten: 20% of what number is 30?

Identify the percent, base, and amount in this problem:

What percent of 30 is 3?

The percent is unknown, because the problem states " What percent?" The base is the whole in the situation, so the base is 30. The amount is the portion of the whole, which is 3 in this case.

Solving with Equations

Percent problems can be solved by writing equations. An equation uses an equal sign (=) to show that two mathematical expressions have the same value.

Percents are fractions, and just like fractions, when finding a percent (or fraction, or portion) of another amount, you multiply.

The percent of the base is the amount.

Percent of the Base is the Amount.

\[\ \text { Percent } {\color{red}\cdot}\text { Base }{\color{blue}=}\text { Amount } \nonumber \]

In the examples below, the unknown is represented by the letter \(\ n\). The unknown can be represented by any letter or a box \(\ \square\) or even a question mark.

Write an equation that represents the following problem.

\(\ 20 \% \cdot n=30\)

Once you have an equation, you can solve it and find the unknown value. To do this, think about the relationship between multiplication and division. Look at the pairs of multiplication and division facts below, and look for a pattern in each row.

Multiplication and division are inverse operations. What one does to a number, the other “undoes.”

When you have an equation such as \(\ 20 \% \cdot n=30\), you can divide 30 by 20% to find the unknown: \(\ n=30 \div 20 \%\).

You can solve this by writing the percent as a decimal or fraction and then dividing.

\(\ n=30 \div 20 \%=30 \div 0.20=150\)

What percent of 72 is 9?

\(\ 12.5 \% \text { of } 72 \text { is } 9\).

You can estimate to see if the answer is reasonable. Use 10% and 20%, numbers close to 12.5%, to see if they get you close to the answer.

\(\ 10 \% \text { of } 72=0.1 \cdot 72=7.2\)

\(\ 20 \% \text { of } 72=0.2 \cdot 72=14.4\)

Notice that 9 is between 7.2 and 14.4, so 12.5% is reasonable since it is between 10% and 20%.

What is 110% of 24?

\(\ 26.4 \text { is } 110 \% \text { of } 24\).

This problem is a little easier to estimate. 100% of 24 is 24. And 110% is a little bit more than 24. So, 26.4 is a reasonable answer.

18 is what percent of 48?

• \(\ 0.375 \%\)
• \(\ 8.64 \%\)
• \(\ 37.5 \%\)
• \(\ 864 \%\)

Incorrect. You may have calculated properly, but you forgot to move the decimal point when you rewrote your answer as a percent. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Incorrect. You may have used \(\ 18\) or \(\ 48\) as the percent, rather than the amount or base. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Correct. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives \(\ 37.5 \%\).

Incorrect. You probably used 18 or 48 as the percent, rather than the amount or base, and also forgot to rewrite the percent as a decimal before multiplying. The equation for this problem is \(\ n \cdot 48=18\). The corresponding division is \(\ 18 \div 48\), so \(\ n=0.375\). Rewriting this decimal as a percent gives the correct answer, \(\ 37.5 \%\).

Using Proportions to Solve Percent Problems

Percent problems can also be solved by writing a proportion. A proportion is an equation that sets two ratios or fractions equal to each other. With percent problems, one of the ratios is the percent, written as \(\ \frac{n}{100}\). The other ratio is the amount to the base.

\(\ \text { Percent }=\frac{\text { amount }}{\text { base }}\)

Write a proportion to find the answer to the following question.

30 is 20% of 150.

18 is 125% of what number?

• \(\ 0.144\)
• \(\ 694 \frac{4}{9}\) (or about \(\ 694.4\))

Incorrect. You probably didn’t write a proportion and just divided 18 by 125. Or, you incorrectly set up one fraction as \(\ \frac{18}{125}\) and set this equal to the base, \(\ n\). The percent in this case is 125%, so one fraction in the proportion should be \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Correct. The percent in this case is 125%, so one fraction in the proportion should be \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Incorrect. You probably put the amount (18) over 100 in the proportion, rather than the percent (125). Perhaps you thought 18 was the percent and 125 was the base. The correct percent fraction for the proportion is \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Incorrect. You probably confused the amount (18) with the percent (125) when you set up the proportion. The correct percent fraction for the proportion is \(\ \frac{125}{100}\). The base is unknown and the amount is 18, so the other fraction is \(\ \frac{18}{n}\). Solving the proportion \(\ \frac{125}{100}=\frac{18}{n}\) gives \(\ n=14.4\).

Let’s go back to the problem that was posed at the beginning. You can now solve this problem as shown in the following example.

Jeff has a coupon at the Guitar Store for 15% off any purchase of $100 or more. He wants to buy a used guitar that has a price tag of $220 on it. Jeff wonders how much money the coupon will take off of the $220 original price .

The coupon will take $33 off the original price.

You can estimate to see if the answer is reasonable. Since 15% is half way between 10% and 20%, find these numbers.

\(\ \begin{array}{l} 10 \% \text { of } 220=0.1 \cdot 220=22 \\ 20 \% \text { of } 220=0.2 \cdot 220=44 \end{array}\)

The answer, 33, is between 22 and 44. So $33 seems reasonable.

There are many other situations that involve percents. Below are just a few.

Evelyn bought some books at the local bookstore. Her total bill was $31.50, which included 5% tax. How much did the books cost before tax?

The books cost $30 before tax.

Susana worked 20 hours at her job last week. This week, she worked 35 hours. In terms of a percent, how much more did she work this week than last week?

Since 35 is 175% of 20, Susana worked 75% more this week than she did last week. (You can think of this as, “Susana worked 100% of the hours she worked last week, as well as 75% more.”)

Percent problems have three parts: the percent, the base (or whole), and the amount. Any of those parts may be the unknown value to be found. To solve percent problems, you can use the equation, \(\ \text { Percent } \cdot \text { Base }=\text { Amount }\), and solve for the unknown numbers. Or, you can set up the proportion, \(\ \text { Percent }=\frac{\text { amount }}{\text { base }}\), where the percent is a ratio of a number to 100. You can then use cross multiplication to solve the proportion.

Percentages in Word Problems

Hi, and welcome to this video lesson on percentages in word problems.

I know word problems are most people’s worst nightmare, but never fear, we’re going to learn how to turn a big, scary, word percentage problem into a 3-step breeze!

Okay, let’s look at our problem:

The bill for dinner is $62.00. The diners decide to leave their server a 20% tip. Determine the total cost of dining at the restaurant, including tip.

Okay, so what is our goal? We always want to understand the goal in a word problem. Our goal here is: “Determine the total cost of dining at the restaurant, including tip.” That means finding the cost of the meal and finding the cost of the tip so we can add them together. We already know the bill for dinner, so we’re halfway home. Let’s solve the rest of this problem in three easy steps.

STEP 1: Change the percentage to a decimal. Remove the % sign from the 20% and drop a period in front of the 20 so we have .20. We are allowed to do this because when we are finding percents, we are really multiplying a decimal number against another number. This is because 20 percent of a number can be written as a ratio of a part per hundred: \(20\% = \frac{20}{100}=.20\)

STEP 2: Multiply the bill by 0.20 to find the amount of the tip: \($62.00(0.20)=$12.40\)

STEP 3: Add the tip and bill to find the total. The total cost of dining will be the sum of the bill for dinner and the tip: \($62.00+$12.40=$74.40\)

The total cost is $74.40.

I hope that helps. Thanks for watching this video lesson, and, until next time, happy studying.

Percent Word Problems

  Lauren went to her favorite taco truck for lunch. Her bill was $24.80, and she wants to leave a 20% tip. Help Lauren determine what her tip should be.

The correct answer is Tip $4.96. In order to calculate Lauren’s tip, we need to determine what 20% of $24.80 is. Let’s convert 20% to a decimal, which would be 0.20. Now we can simply multiply \($24.80×0.20\) in order to determine the tip. \($24.80×0.20=$4.96\)

  Michael wants to mow lawns in order to make some extra money this summer, but he needs to find a lawn mower to use. Michael’s brother tells him that he will loan Michael his lawn mower if he gives him 4% of the money he makes on each lawn. If Michael agrees, and he earns 40 dollars on his first lawn mowed, how much money does he own his brother?

The correct answer is $1.60. In order to calculate 4% of 40, we need to convert 4% to a decimal. 4% is 0.04 as a decimal. Now we can multiply 0.04 and $40 in order to determine what Michael owes his brother. \(0.04×$40=1.6=$1.60\)

  In a study of 250 high school students, 90% of students have taken the driver’s education course. How many students have not taken the course?

15 students

20 students

25 students

30 students

The correct answer is 25 students. 90% of the students have taken the driver’s education course, and there are 250 students total. Let’s start by determining how many students have taken the course. To do this we can multiply \(0.9×250\) which equals 225. This means that 225 students have taken the course. If 225 students have taken the course, and there are 250 students total, we can find the difference between 225 and 250 in order to determine how many students have not taken the course. \(250-225=25\) students have not taken the course.

  Julian scored 90% on his math test. The test had 60 questions. How many questions did he answer correctly?

The correct answer is 54. If Julian answered 90% of the questions correctly, and there were 60 questions total, we can calculate 90% of 60 in order to determine how many questions he answered correctly. Let’s convert 90% to a decimal (0.9), and then multiply this by 60. \(0.9×60=54\) questions answered correctly

  A video game costs $45 before tax. If the sales tax is 5%, what will the total cost of the game be including tax?

The correct answer is $47.25. Let’s first calculate the tax. If the game costs $45 and the tax is 5%, we can multiply \(45×0.05\) in order to determine the tax. \(45×0.05 = 2.25\), which means there will be a $2.25 tax on the purchase. Now let’s add this tax to the price of the game in order to calculate the total cost of the game plus the tax. \($45+$2.25=$47.25\)

by Mometrix Test Preparation | This Page Last Updated: December 28, 2023

Percent Word Problems

In these lessons we look at some examples of percent word problems. The videos will illustrate how to use the block diagrams (Singapore Math) method to solve word problems.

Related Pages More Math Word Problems Algebra Word Problems More Singapore Math Word Problems

How to solve percent problems with bar models? Examples:

• Marilyn saves 30% of the money she earns each month. She earns $1350 each month. How much does she save?
• At the Natural History Museum, 40% of the visitors are children. There are 36 children at the museum. How many visitors altogether are at the museum?
• Bill bought cards to celebrate Pi day. He sent 60% of his cards to his friends. He sent 42 cards to his friends. How many cards did he buy altogether?
• Bruce cooked 80% of the pancakes at the pancake breakfast last weekend. They made 1120 pancakes. How many pancakes did Bruce cook?

Sales Tax and Discount An example of finding total price with sales tax and an example of finding cost after discount.

• Alejandro bought a TV for $900 and paid a sales tax of 8%. How much did he pay for the TV?
• Alice saved for a new bike. The bike was on sale for a discount of 35%. The original cost of the bike was $270. How much did she pay for the bike?

Percent Word Problems Example: There are 600 children on a field. 30% of them were boys. After 5 teams of boys join the children on the field, the percentage of children who were boys increased to 40%. How many boys were there in the 5 teams altogether?

Problem Solving - Choosing a strategy to solve percent word problems An explanation of how to solve multi-step percentage problems using bar models or choosing an operation. Example: The $59.99 dress is on sale for 15% off. How much is the price of the dress?

How to solve percent problems using a tape diagram or bar diagram? Example: An investor offers $200,000 for a 20% stake in a new company. What amount does the investor believe the toatl value of the business is worth at this time? How to use a tape diagram or bar diagram to find the answer?

• First draw a bar that represents the company’s whole value.
• Divide into 5 equal parts because 100%/20% = 5.
• Label one side with the percentages.
• Label the other side $200,000 across from 20% because that was given.
• Finish labeling the money side.
• Find solution.

Solve Percent Problems Using a Tape Diagram (Bar Diagram) Example: a) If $300 is increased by 25% what is the new amount? b) What is 19% of 120? c) Joe went to an athletic store to purchase new running shoes. To his surprise, the store was having a 20% off athletic shoes sale. He purchased a new pair of shoes that were regularly priced $60. How much did Joe pay for his shoes?

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Solving Word Questions

With LOTS of examples!

In Algebra we often have word questions like:

Example: Sam and Alex play tennis.

On the weekend Sam played 4 more games than Alex did, and together they played 12 games.

How many games did Alex play?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

• Read the whole thing first
• Do a sketch if possible
• Assign letters for the values
• Find or work out formulas

You should also write down what is actually being asked for , so you know where you are going and when you have arrived!

Also look for key words:

Thinking Clearly

Some wording can be tricky, making it hard to think "the right way around", such as:

Example: Sam has 2 dollars less than Alex. How do we write this as an equation?

• Let S = dollars Sam has
• Let A = dollars Alex has

Now ... is that: S − 2 = A

or should it be: S = A − 2

or should it be: S = 2 − A

The correct answer is S = A − 2

( S − 2 = A is a common mistake, as the question is written "Sam ... 2 less ... Alex")

Example: on our street there are twice as many dogs as cats. How do we write this as an equation?

• Let D = number of dogs
• Let C = number of cats

Now ... is that: 2D = C

or should it be: D = 2C

Think carefully now!

The correct answer is D = 2C

( 2D = C is a common mistake, as the question is written "twice ... dogs ... cats")

Let's start with a really simple example so we see how it's done:

Example: A rectangular garden is 12m by 5m, what is its area ?

Turn the English into Algebra:

• Use w for width of rectangle: w = 12m
• Use h for height of rectangle: h = 5m

Formula for Area of a Rectangle : A = w × h

We are being asked for the Area.

A = w × h = 12 × 5 = 60 m 2

The area is 60 square meters .

Now let's try the example from the top of the page:

Example: Sam and Alex play Tennis. On the weekend Sam played 4 more games than Alex did, and together they played 12 games. How many games did Alex play?

• Use S for how many games Sam played
• Use A for how many games Alex played

We know that Sam played 4 more games than Alex, so: S = A + 4

And we know that together they played 12 games: S + A = 12

We are being asked for how many games Alex played: A

Which means that Alex played 4 games of tennis.

Check: Sam played 4 more games than Alex, so Sam played 8 games. Together they played 8 + 4 = 12 games. Yes!

A slightly harder example:

Example: Alex and Sam also build tables. Together they make 10 tables in 12 days. Alex working alone can make 10 in 30 days. How long would it take Sam working alone to make 10 tables?

• Use a for Alex's work rate
• Use s for Sam's work rate

12 days of Alex and Sam is 10 tables, so: 12a + 12s = 10

30 days of Alex alone is also 10 tables: 30a = 10

We are being asked how long it would take Sam to make 10 tables.

30a = 10 , so Alex's rate (tables per day) is: a = 10/30 = 1/3

Which means that Sam's rate is half a table a day (faster than Alex!)

So 10 tables would take Sam just 20 days.

Should Sam be paid more I wonder?

And another "substitution" example:

Example: Jenna is training hard to qualify for the National Games. She has a regular weekly routine, training for five hours a day on some days and 3 hours a day on the other days. She trains altogether 27 hours in a seven day week. On how many days does she train for five hours?

• The number of "5 hour" days: d
• The number of "3 hour" days: e

We know there are seven days in the week, so: d + e = 7

And she trains 27 hours in a week, with d 5 hour days and e 3 hour days: 5d + 3e = 27

We are being asked for how many days she trains for 5 hours: d

The number of "5 hour" days is 3

Check : She trains for 5 hours on 3 days a week, so she must train for 3 hours a day on the other 4 days of the week.

3 × 5 hours = 15 hours, plus 4 × 3 hours = 12 hours gives a total of 27 hours

Some examples from Geometry:

Example: A circle has an area of 12 mm 2 , what is its radius?

• Use A for Area: A = 12 mm 2
• Use r for radius

And the formula for Area is: A = π r 2

We are being asked for the radius.

We need to rearrange the formula to find the area

Example: A cube has a volume of 125 mm 3 , what is its surface area?

Make a quick sketch:

• Use V for Volume
• Use A for Area
• Use s for side length of cube
• Volume of a cube: V = s 3
• Surface area of a cube: A = 6s 2

We are being asked for the surface area.

First work out s using the volume formula:

Now we can calculate surface area:

An example about Money:

Example: Joel works at the local pizza parlor. When he works overtime he earns 1¼ times the normal rate. One week Joel worked for 40 hours at the normal rate of pay and also worked 12 hours overtime. If Joel earned $660 altogether in that week, what is his normal rate of pay?

• Joel's normal rate of pay: $N per hour
• Joel works for 40 hours at $N per hour = $40N
• When Joel does overtime he earns 1¼ times the normal rate = $1.25N per hour
• Joel works for 12 hours at $1.25N per hour = $(12 × 1¼N) = $15N
• And together he earned $660, so:

$40N + $(12 × 1¼N) = $660

We are being asked for Joel's normal rate of pay $N.

So Joel’s normal rate of pay is $12 per hour

Joel’s normal rate of pay is $12 per hour, so his overtime rate is 1¼ × $12 per hour = $15 per hour. So his normal pay of 40 × $12 = $480, plus his overtime pay of 12 × $15 = $180 gives us a total of $660

More about Money, with these two examples involving Compound Interest

Example: Alex puts $2000 in the bank at an annual compound interest of 11%. How much will it be worth in 3 years?

This is the compound interest formula:

So we will use these letters:

• Present Value PV = $2,000
• Interest Rate (as a decimal): r = 0.11
• Number of Periods: n = 3
• Future Value (the value we want): FV

We are being asked for the Future Value: FV

Example: Roger deposited $1,000 into a savings account. The money earned interest compounded annually at the same rate. After nine years Roger's deposit has grown to $1,551.33 What was the annual rate of interest for the savings account?

The compound interest formula:

• Present Value PV = $1,000
• Interest Rate (the value we want): r
• Number of Periods: n = 9
• Future Value: FV = $1,551.33

We are being asked for the Interest Rate: r

So the annual rate of interest is 5%

Check : $1,000 × (1.05) 9 = $1,000 × 1.55133 = $1,551.33

And an example of a Ratio question:

Example: At the start of the year the ratio of boys to girls in a class is 2 : 1 But now, half a year later, four boys have left the class and there are two new girls. The ratio of boys to girls is now 4 : 3 How many students are there altogether now?

• Number of boys now: b
• Number of girls now: g

The current ratio is 4 : 3

Which can be rearranged to 3b = 4g

At the start of the year there was (b + 4) boys and (g − 2) girls, and the ratio was 2 : 1

b + 4 g − 2 = 2 1

Which can be rearranged to b + 4 = 2(g − 2)

We are being asked for how many students there are altogether now: b + g

There are 12 girls !

And 3b = 4g , so b = 4g/3 = 4 × 12 / 3 = 16 , so there are 16 boys

So there are now 12 girls and 16 boys in the class, making 28 students altogether .

There are now 16 boys and 12 girls, so the ratio of boys to girls is 16 : 12 = 4 : 3 At the start of the year there were 20 boys and 10 girls, so the ratio was 20 : 10 = 2 : 1

And now for some Quadratic Equations :

Example: The product of two consecutive even integers is 168. What are the integers?

Consecutive means one after the other. And they are even , so they could be 2 and 4, or 4 and 6, etc.

We will call the smaller integer n , and so the larger integer must be n+2

And we are told the product (what we get after multiplying) is 168, so we know:

n(n + 2) = 168

We are being asked for the integers

That is a Quadratic Equation , and there are many ways to solve it. Using the Quadratic Equation Solver we get −14 and 12.

Check −14: −14(−14 + 2) = (−14)×(−12) = 168 YES

Check 12: 12(12 + 2) = 12×14 = 168 YES

So there are two solutions: −14 and −12 is one, 12 and 14 is the other.

Note: we could have also tried "guess and check":

• We could try, say, n=10: 10(12) = 120 NO (too small)
• Next we could try n=12: 12(14) = 168 YES

But unless we remember that multiplying two negatives make a positive we might overlook the other solution of (−14)×(−12).

Example: You are an Architect. Your client wants a room twice as long as it is wide. They also want a 3m wide veranda along the long side. Your client has 56 square meters of beautiful marble tiles to cover the whole area. What should the length of the room be?

Let's first make a sketch so we get things right!:

• the length of the room: L
• the width of the room: W
• the total Area including veranda: A
• the width of the room is half its length: W = ½L
• the total area is the (room width + 3) times the length: A = (W+3) × L = 56

We are being asked for the length of the room: L

This is a quadratic equation , there are many ways to solve it, this time let's use factoring :

And so L = 8 or −14

There are two solutions to the quadratic equation, but only one of them is possible since the length of the room cannot be negative!

So the length of the room is 8 m

L = 8, so W = ½L = 4

So the area of the rectangle = (W+3) × L = 7 × 8 = 56

There we are ...

... I hope these examples will help you get the idea of how to handle word questions. Now how about some practice?

• Pre-algebra lessons
• Pre-algebra word problems
• Algebra lessons
• Algebra word problems
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Percentage word problems

Before you take a look at the percentage word problems in this lesson and their solutions, it may help to review the lesson about  formula for percentage or you can use the different techniques that I use here.

Different types of percentage word problems

There are three different types of percentage word problems. We will show how to solve them using proportions. 

• What is 80% of 20? ( example #1 )
• 50 is 25% of what number? ( example #2 )
• 18 is what percent of 24? What percent of 2000 is 3500? ( example #3 and example #4 )

Solving percentage word problems using proportions

You can solve problems involving percents using the proportion you see in the figure above:   ( n% / 100% = Part / Whole )

First, study the figure carefully! Then, we will show how to use the proportion to solve percentage word problems by creating diagrams to visualize relationships.

Example #1: A test has 20 questions. If peter gets 80% correct, how many questions did peter miss?

First, you need to find the number of correct answers by looking for 80% of 20.

When the problem involves looking for the part or the problem says something like, "Find 80% of 20" or "Find 30% of 50," just change the percent to a decimal and multiply.

80% of 20 = (80 / 100) × 20 = 0.80 × 20 = 16

Since the test has 20 questions and he got 16 correct answers, the number of questions Peter missed is 20 − 16 = 4

Recall that 16 is called the percentage. It is the answer you get when you take the percent of a number.

Percentage  =   Part

Example #2: In a school, 25% of the teachers teach basic math. If there are 50 basic math teachers, how many teachers are there in the school?

Once again set the problem up as shown in the figure below. Notice that the question is, " How many teachers are in the school?"

Therefore, the whole is missing this time!

Method #2 I shall help you reason the problem out!

When we say that 25% of the teachers teach basic math, we mean 25% of all teachers in the school equals number of teachers teaching basic math.

Since we don't know how many teachers there are in the school, we replace this with x or a blank. However, we know that the number of teachers teaching basic math is equal to the percentage = part =  50 Putting it all together, we get the following equation: 25% of ____ = 50 or 25% × ____ = 50 or 0.25 × ____ = 50 Thus, the question is 0.25 times what gives me 50? A simple division of 50 by 0.25 will get you the answer 50 / 0.25 = 200 Therefore, we have 200 teachers in the school In fact, 0.25 × 200 = 50

More percentage word problems

Example #3: 24 students in a class took an algebra test. If 18 students passed the test, what percent do not pass?

Solution First, find out how many student did not pass. Number of students who did not pass is 24 − 18 = 6

Then, write down the following equation: x% of 24 = 6 or x% × 24 = 6

To get x%, just divide 6 by 24 6 / 24 = 0.25 = 25 / 100 = 25% Therefore, 25% of students did not pass.

Example #4: A fundraising company would like to raise $2000 for a cause. The fundraiser was so successful that they ended up raising $3500. What percent of their goal did they raise?

Notice that the whole is 2000 since this is the whole money they expect to raise. The part is the amount that the fundraiser ended with and it usually lower than the amount they expect to raise. However, in this particular case, the part ended up being bigger than the whole. Keeping this in mind, here is how to set it up and solve it!

The fundraising company was able to raise 175% of the expected amount.

Example #5:

A department has a total of 22,000 units of stock. 25% of the garments are black and 10% of the garments are size 14.

a) How many black garments are there? 
b) How many size 14 garments are there? 
c) If 10% of the black garments are size 14,how many garments are black and size 14?

Note that the solution we show below for example #5 use a completely different approach or technique. Read it carefully and try to learn it as well!

25% = 25 per 100 = 250 per 1000 For 22,000 just multiply 250 by 22 250    ×    22   =  250 × (10 + 10 + 2)

                       =  2500 + 2500 + 500                        =  5000 + 500                        =  5500

So, there are 5500 black garments.

10% = 10 per 100 = 100 per 1000 For 22,000 just multiply 100 by 22 100 × 22 = 2200 So, 2200 of the garments are size 14.

If 10% or 10 per 100 of the black garments are size 14, then 100 per 1000 of the black garments are size 14.

500 per 5000 are size 14. However, you need to find it for 5500 black garments.

Then, what is 10% of 500? 10% = 10 per 100, so 50 per 500. So 550 of the black garments are size 14.

If you really understand the percentage word problems above, you can solve any other similar percentage word problems. If you still do not understand them, I strongly encourage you to study them again and again until you get it. The end result will be very rewarding!

Finding a percentage

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How to Solve Percent Problems? (+FREE Worksheet!)

Learn how to calculate and solve percent problems using the percent formula.

Related Topics

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Step by step guide to solve percent problems

• In each percent problem, we are looking for the base, or part or the percent.
• Use the following equations to find each missing section. Base \(= \color{black}{Part} \ ÷ \ \color{blue}{Percent}\) \(\color{ black }{Part} = \color{blue}{Percent} \ ×\) Base \(\color{blue}{Percent} = \color{ black }{Part} \ ÷\) Base

Percent Problems – Example 1:

\(2.5\) is what percent of \(20\)?

In this problem, we are looking for the percent. Use the following equation: \(\color{blue}{Percent} = \color{ black }{Part} \ ÷\) Base \(→\) Percent \(=2.5 \ ÷ \ 20=0.125=12.5\%\)

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Percent problems – example 2:.

\(40\) is \(10\%\) of what number?

Use the following formula: Base \(= \color{ black }{Part} \ ÷ \ \color{blue}{Percent}\) \(→\) Base \(=40 \ ÷ \ 0.10=400\) \(40\) is \(10\%\) of \(400\).

Percent Problems – Example 3:

\(1.2\) is what percent of \(24\)?

In this problem, we are looking for the percent. Use the following equation: \(\color{blue}{Percent} = \color{ black }{Part} \ ÷\) Base \(→\) Percent \(=1.2÷24=0.05=5\%\)

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Percent problems – example 4:.

\(20\) is \(5\%\) of what number?

Use the following formula: Base \(= \color{black}{Part} \ ÷ \ \color{blue}{Percent}\) \(→\) Base \(=20÷0.05=400\) \( 20\) is \(5\%\) of \(400\).

Exercises for Calculating Percent Problems

Solve each problem..

• \(51\) is \(340\%\) of what?
• \(93\%\) of what number is \(97\)?
• \(27\%\) of \(142\) is what number?
• What percent of \(125\) is \(29.3\)?
• \(60\) is what percent of \(126\)?
• \(67\) is \(67\%\) of what?

Download Percent Problems Worksheet

• \(\color{blue}{15}\)
• \(\color{blue}{104.3}\)
• \(\color{blue}{38.34}\)
• \(\color{blue}{23.44\%}\)
• \(\color{blue}{47.6\%}\)
• \(\color{blue}{100}\)

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by: Effortless Math Team about 4 years ago (category: Articles , Free Math Worksheets )

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Tricks to Solving Percentage Word Problems

How to Convert Percent to Decimal

Word problems test both your math skills and your reading comprehension skills. In order to answer them correctly, you'll need to examine the questions carefully. Always make sure you know what is being asked, what operations are necessary and what units, if any, you need to include in your answer.

Eliminate Extraneous Data

Sometimes, word problems include extraneous data that is not necessary to solve the problem. For example:

Kim won 80 percent of her games in June and 90 percent of her games in July. If she won 4 games in June and played 10 games in July, how many games did Kim win in July?

The simplest way to eliminate extraneous data is to identify the question; in this case, "How many games did Kim win in July?" In the example above, any information that doesn't deal with the month of July is unnecessary to answer the question. You are left with 90 percent of 10 games, allowing you to do a simple calculation:

0.9*10=9 games

Calculate Additional Data

Read the question portion twice to make sure you know what data you need to answer the question:

On a test with 80 questions, Abel got 4 answers wrong. What percentage of questions did he get right?

The word problem only gives you two numbers, so it would be easy to assume that the questions involves those two numbers. However, in this case, the question requires that you calculate another answer first: the number of questions Abel got right. You'll need to subtract 4 from 80, then calculate the percentage of the difference:

80-4=78, and 78/80*100=97.5 percent

Rephrase Difficult Problems

Remember that you can often rearrange problems to make them simpler. This is especially useful if you don't have a calculator available:

Gina needs to score at least 92 percent on her final exam to get an A for the semester. If there are 200 questions on the exam, how many questions does Gina need to get right in order to earn an A?

The standard approach would be to multiply 200 by 0.92: 200*.92=184. While this is a simple process, you can make the process even simpler. Instead of finding 92 percent of 200, find 200 percent of 92 by doubling it:

This method is particularly useful when you are dealing with numbers with known ratios. If, for example, the word problem asked you to find 77 percent of 50, you could simply find 50 percent of 77:

50*.77=38.5, or 77/2=38.5

Account for Units

Convert your answers into appropriate units:

Cassie works from 7 a.m. to 4 p.m. each weekday. If Cassie worked 82 percent of her shift on Wednesday and worked 100 percent of her other shifts, what percent of the week did she miss? How much time did she work in total?

First, calculate how many hours Cassie works per day, taking noon into account, then per week:

4+(12-7)=9 9*5=45

Next, calculate 82 percent of 9 hours:

0.82*9=7.38

Subtract the product from 9 for the total hours missed:

9-7.38=1.62

Calculate what percentage of the week she missed:

1.62/45*100=3.6 percent

The second question asks for an amount of time, which means you'll need to convert the decimal into time increments. Add the product to the other four work days:

7.38+(9*4)=43.38

Convert the decimal into minutes:

0.38*60=22.8

Convert the remaining decimal into seconds:

So Cassie missed 3.6 percent of her week, and worked 43 hours, 22 minutes and 48 seconds total.

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"Mixture" Word Problems

Explanation Examples

What are "mixture" word problems?

Mixture word problems are exercises which involve creating a mixture from two or more different things, and then determining some quantity (such as percentage, price, number of liters, etc) of the resulting mixture. There will always be a "rate" of some sort, such as miles per hour or cost per pound.

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Mixture Word Problems

Here is an example:

Your school is holding a family-friendly event this weekend. Students have been pre-selling tickets to the event; adult tickets are $5.00, and child tickets (for kids six years old and under) are $2.50 . From past experience, you expect about 13,000 people to attend the event.

You consult with your student ticket-sellers, and discover that they have not been keeping track of how many child tickets they have sold. The tickets are identical, until the ticket-seller punches a hole in the ticket, indicating that it is a child ticket.

But they don't remember how many holes they've punched. They only know that they've sold 548 tickets for $2460 . How much revenue from each of child and adult tickets can you expect?

To solve this, we need to figure out the ratio of tickets that have already been sold. If we work methodically, we can find the answer.

Let A stand for the number of adult tickets pre-sold. Since a total of 548 tickets have been sold, then the number of children tickets pre-sold thus far must be 548 − A .

this construction is important! When you've got a variable that stands for part of whatever it is that you're working with, then the amount that is left for the other part of whatever you're working with is found by subtracting the variable from the total. That is, (the total amount) less (the amount that is being represented by the variable) is (the amount that is left for ther other amount). This "how much is left" construction is something you will need to understand and use.

Since each adult ticket cost $5.00 , then 5A stands for the revenue brought in from the adult tickets pre-sold; likewise, 2.5(548 − A) stands for the revenue brought in from the child tickets. (Note: The per-ticket cost is the "rate" for this exercise.)

Organizing this information in a grid, we get:

From the last column, we get (total $ from the adult tickets) plus (total $ from the child tickets) is (the total $ so far), or, as an equation:

5A + 2.5((548 − A) = 2,460

5A + 1,370 − 2.5A = 2,460

1,370 + 2.5A = 2,460

2.5A = 1,090

A = (1,090)/(2.5) = 436

So 436 adult tickets were pre-sold, so the number of child tickets pre-sold is:

C = 548 − 436 = 112

So 112 child tickets were pre-sold.

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We expect about 13,000 people in total to attend this event. We have a ratio of adult pre-sold tickets to total pre-sold tickets. Assuming that the pre-sold rate (or ratio) is representative of the total number of adult tickets, we can set up a proportion (of pre-sold adult tickets to total pre-sold tickets), using a variable for the unknown total number of adult tickets expected to be sold:

436/548 = x /13,000

[(436)(13,000)]/548 = x

10,343.0656934... = x

This works out to about 10,343 adult tickets. The remaining 2,657 tickets, of the expected total of 13,000 , will be child tickets. Then the expected total ticket revenue is given by:

5(10,343) + 2.5(2,657) = 58,357.5

So the expected total revenues from ticket sales is $58,357.50 .

Let's try another one. This time, suppose you work in a lab. You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix 10% solution with 30% solution, to make your own 15% solution. You need 10 liters of the 15% acid solution. How many liters of 10% solution and 30% solution should you use?

Let w stand for the number of liters of the weaker 10% solution. Since the total number of liters is going to be 10 , then the number of liters left, after the first w liters have been poured, will be 10 −  w liters needed of the 30% solution.

(The clear labeling of the variable is important. While I picked w to stand for the w eaker acid, I might not remember this by the end of the exercise. If I don't label, I might not be able correctly to interpret my answer in the end.)

For mixture problems, it is often very helpful to do a grid, so let's do that here:

When the problem is set up like this, we can usually use the last column to write our equation. In this case, the liters of acid within the 10% solution, plus the liters of acid within the 30% solution, must add up to the liters of acid within the 15% solution. So, adding down the right-hand column, setting the inputs equal to the mixed output, we get:

0.10 w + 0.30(10 − w ) = 1.5

0.10 w + 3 − 0.30 w = 1.5

3 − 0.2 w = 1.5

1.5 = 0.2 w

(1.5)/(0.2) = w = 7.5

Looking back and confirming that the variable stands for the number of liters of the weaker acid, we see that we would need 7.5 liters of 10% acid . This means that we would need another 10 − 7.5 = 2.5 liters of the stronger 30% acid .

(If you think about it, this makes sense. Fifteen percent is closer to 10% than it is to 30% , so we ought to need more 10% solution in our mix.)

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• \mathrm{Lauren's\:age\:is\:half\:of\:Joe's\:age.\:Emma\:is\:four\:years\:older\:than\:Joe.\:The\:sum\:of\:Lauren,\:Emma,\:and\:Joe's\:age\:is\:54.\:How\:old\:is\:Joe?}
• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
• Is there a calculator that can solve word problems?
• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

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• High School Math Solutions – Inequalities Calculator, Exponential Inequalities Last post, we talked about how to solve logarithmic inequalities. This post, we will learn how to solve exponential...

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How does the percentage word problems calculator work, what 1 formula is used for the percentage word problems calculator, what 4 concepts are covered in the percentage word problems calculator.

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