how to solve percent proportion word problems

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Course: 7th grade   >   Unit 2

Solving percent problems.

• Equivalent expressions with percent problems
• Percent word problem: magic club
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Video transcript

Appendix A: Applications

Writing and solving percent proportions, learning outcomes.

• Translate a statement to a proportion
• Solve a percent proportion

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.

For example, [latex]\text{60%}={\Large\frac{60}{100}}[/latex] and we can simplify [latex]{\Large\frac{60}{100}}={\Large\frac{3}{5}}[/latex]. Since the equation [latex]{\Large\frac{60}{100}}={\Large\frac{3}{5}}[/latex] shows a percent equal to an equivalent ratio, we call it a percent proportion.

Using the vocabulary we used earlier:

[latex]{\Large\frac{\text{amount}}{\text{base}}}={\Large\frac{\text{percent}}{100}}[/latex] [latex]{\Large\frac{3}{5}}={\Large\frac{60}{100}}[/latex]

Percent Proportion

The amount is to the base as the percent is to [latex]100[/latex].

[latex]{\Large\frac{\text{amount}}{\text{base}}}={\Large\frac{\text{percent}}{100}}[/latex]

If we restate the problem in the words of a proportion, it may be easier to set up the proportion:

The amount is to the base as the percent is to one hundred.

We could also say:

The amount out of the base is the same as the percent out of one hundred.

First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

Translate to a proportion. What number is [latex]\text{75%}[/latex] of [latex]90[/latex]?

Solution If you look for the word “of”, it may help you identify the base.

Translate to a proportion. [latex]19[/latex] is [latex]\text{25%}[/latex] of what number?

Translate to a proportion. What percent of [latex]27[/latex] is [latex]9[/latex]?

Now that we have written percent equations as proportions, we are ready to solve the equations.

Translate and solve using proportions: What number is [latex]\text{45%}[/latex] of [latex]80[/latex]?

The following video shows a similar example of how to solve a percent proportion.

In the next example, the percent is more than [latex]100[/latex], which is more than one whole. So the unknown number will be more than the base.

Translate and solve using proportions: [latex]\text{125%}[/latex] of [latex]25[/latex] is what number?

Percents with decimals and money are also used in proportions.

Translate and solve: [latex]\text{6.5%}[/latex] of what number is [latex]\text{\$1.56}[/latex]?

In the following video we show a similar problem, note the different wording that results in the same equation.

Translate and solve using proportions: What percent of [latex]72[/latex] is [latex]9?[/latex]

Watch the following video to see a similar problem.

• Question ID 146843, 146840, 146839, 146828. Authored by : Lumen Learning. License : CC BY: Attribution
• Example 2: Solve a Percent Problem Using a Percent Proportion. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/wsBhmrmumJo . License : CC BY: Attribution
• Example 3: Determine What Percent One Number is of Another Using a Percent Proportion. Authored by : James Sousa (Mathispower4u.com). Located at : https://youtu.be/1GTPRROi1tE . License : CC BY: Attribution
• Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

Easy Peasy All-in-One Homeschool

A complete, free online christian homeschool curriculum for your family and mine, percent word problems.

In this lesson, we will be working with percents. If you have worked with fractions and know what a percent is, you already have a good base for this lesson. Think about percents and what they mean as a fraction. What does 25% mean? The word “percent” means “per 100.” If you remember it means 25 per 100 or 25 out of 100 or 25/100. A percent is a ratio.

All percent problems fit the proportion:

One could say that 40% of 60 is 24 . Placing these numbers into the proportion gives us

Remember, percent means  out of 100 , so the denominator of the percent ratio will always be 100. If one of the parts of the proportion is missing, we use a letter to represent the unknown. Then we can cross multiply to find our unknown portion.

Let’s take a look at a problem where we need to find some information. We use percents in our everyday life such as a sale on clothes at department stores, the percent of games won, and surveys.

At the department store, we find a pair of shoes on sale. We want to make sure we have enough to buy them. According to the sign the price of the shoes is 80% of the full price of $56. What is the sale price?

Set up of the proportion:

Use the cross product property to solve for x!!  In other words, cross multiply.

100x = 4480

“A percent of a number is another number.”   This phrase can be used to solve percent word problems.   The word “of” in math means to multiply.   The word “is” in math means equal. So you should multiply the percent (written as a decimal) times the given number.   The answer is the product.

Example 2  

Sales tax is 5% in Maryland.   Jim bought a pair of basketball shoes advertised for $62.   How much sales tax did he pay?

Convert your percent to a decimal.

5% = 0.05 (move the decimal two places to the left)

Multiply the decimal by the price.

0.05 times $62 equals the Maryland sales tax

0.05 (62) = ?

The sales tax was $3.10 in Maryland.

1. You surveyed 10 people. Six of them prefer lemonade to water. Write the number of people who prefer lemonade as a decimal.

2. Write 145% as a decimal.

3. Write .25 as both a fraction and a percent.

4. Write 220% as a fraction. Leave your answer in lowest term and as a mixed number. (Hint: Write as a decimal first.)

5. At Leslie’s Pizza Place, 85% of the pizzas sold were pepperoni. At Adriana’s Pizza, 41 out of 48 pizzas sold were pepperoni. Which pizza parlor sold a greater percentage of pepperoni pizza?

6. Last week, 1/5 of the mice in Adriana’s lab found the cheese at the end of a maze. In Leslie’s lab, 23% of the mice found the cheese at the end of a maze. Which lab had a greater percentage of mice who found the cheese?

Word Problems on Percentage

Word problems on percentage will help us to solve various types of problems related to percentage. Follow the procedure to solve similar type of percent problems.

Word problems on percentage:

1.  In an exam Ashley secured 332 marks. If she secured 83 % makes, find the maximum marks.

Let the maximum marks be m.

Ashley’s marks = 83% of m

Ashley secured 332 marks

Therefore, 83% of m = 332

⇒ 83/100 × m = 332

⇒ m = (332 × 100)/83

⇒ m =33200/83

Therefore, Ashley got 332 marks out of 400 marks.

2. An alloy contains 26 % of copper. What quantity of alloy is required to get 260 g of copper?

Let the quantity of alloy required = m g

Then 26 % of m =260 g

⇒ 26/100 × m = 260 g

⇒ m = (260 × 100)/26 g

⇒ m = 26000/26 g

⇒ m = 1000 g

3. There are 50 students in a class. If 14% are absent on a particular day, find the number of students present in the class.

Solution:             

Number of students absent on a particular day = 14 % of 50

                                          i.e., 14/100 × 50 = 7

Therefore, the number of students present = 50 - 7 = 43 students.

4. In a basket of apples, 12% of them are rotten and 66 are in good condition. Find the total number of apples in the basket.

Solution:             

Let the total number of apples in the basket be m

12 % of the apples are rotten, and apples in good condition are 66

Therefore, according to the question,

88% of m = 66

⟹ 88/100 × m = 66

⟹ m = (66 × 100)/88

⟹ m = 3 × 25

Therefore, total number of apples in the basket is 75.

5. In an examination, 300 students appeared. Out of these students; 28 % got first division, 54 % got second division and the remaining just passed. Assuming that no student failed; find the number of students who just passed.

The number of students with first division = 28 % of 300

                                                             = 28/100 × 300

                                                             = 8400/100

                                                             = 84

And, the number of students with second division = 54 % of 300

                                                                        = 54/100 × 300

                                                                        =16200/100

                                                                        = 162

Therefore, the number of students who just passed = 300 – (84 + 162)

                                                                           = 54

Questions and Answers on Word Problems on Percentage:

1. In a class 60% of the students are girls. If the total number of students is 30, what is the number of boys?

2. Emma scores 72 marks out of 80 in her English exam. Convert her marks into percent.

Answer: 90%

3. Mason was able to sell 35% of his vegetables before noon. If Mason had 200 kg of vegetables in the morning, how many grams of vegetables was he able to see by noon?

Answer: 70 kg

4. Alexander was able to cover 25% of 150 km journey in the morning. What percent of journey is still left to be covered?

Answer:  112.5 km

5. A cow gives 24 l milk each day. If the milkman sells 75% of the milk, how many liters of milk is left with him?

Answer: 6 l

6.  While shopping Grace spent 90% of the money she had. If she had $ 4500 on shopping, what was the amount of money she spent?

Answer:  $ 4050

Fraction into Percentage

Percentage into Fraction

Percentage into Ratio

Ratio into Percentage

Percentage into Decimal

Decimal into Percentage

Percentage of the given Quantity

How much Percentage One Quantity is of Another?

Percentage of a Number

Increase Percentage

Decrease Percentage

Basic Problems on Percentage

Solved Examples on Percentage

Problems on Percentage

Real Life Problems on Percentage

Application of Percentage

8th Grade Math Practice From Word Problems on Percentage to HOME PAGE

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Basic "Percent of" Word Problems

Basic Set-Up Markup / Markdown Increase / Decrease

When you learned how to translate simple English statements into mathematical expressions, you learned that "of" can indicate "times". This frequently comes up when using percentages.

Suppose you need to find 16% of 1400 . You would first convert the percentage " 16% " to its decimal form; namely, the number " 0.16 ".

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Percent Word Problems

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Why does the percentage have to be converted to decimal form?

When you are doing actual math, you need to use actual numbers. Percents, being the values with a "percent" sign tacked on, are not technically numbers. This is similar to your grade-point average ( gpa ), versus your grades. You can get an A in a class, but the letter "A" is not a numerical grade which can be averaged. Instead, you convert the "A" to the equivalent "4.0", and use this numerical value for finding your gpa .

When you're doing computations with percentages, remember always to convert the percent expressions to their equivalent decimal forms.

Once you've done this conversion of the percentage to decimal form, you note that "sixteen percent OF fourteen hundred" is telling you to multiply the 0.16 and the 1400 . The numerical result you get is (0.16)(1400) = 224 . This value tells you that 224 is sixteen percent of 1400 .

How do you turn "percent of" word problems into equations to solve?

Percentage problems usually work off of some version of the sentence "(this) is (some percentage) of (that)", which translates to "(this) = (some decimal) × (that)". You will be given two of the values — or at least enough information that you can figure out what two of the values must be — and then you'll need to pick a variable for the value you don't have, write an equation, and solve the equation for that variable.

What is an example of solving a "percent of" word problem?

• What percent of 20 is 30 ?

We have the original number 20 and the comparative number 30 . The unknown in this problem is the rate or percentage. Since the statement is "(thirty) is (some percentage) of (twenty)", then the variable stands for the percentage, and the equation is:

30 = ( x )(20)

30 ÷ 20 = x = 1.5

Since x stands for a percentage, I need to remember to convert this decimal back into a percentage:

Thirty is 150% of 20 .

What is the difference between "percent" and "percentage"?

"Percent" means "out of a hundred", its expression contains a specific number, and the "percent" sign can be used interchangeably with the word (such as " 24% " and "twenty-four percent"); "percentage" is used in less specific ways, to refer to some amount of some total (such as "a large percentage of the population"). ( Source )

In real life, though, including in math classes, we tend to be fairly sloppy in using these terms. So there's probably no need for you to worry overmuch about this technicallity.

• What is 35% of 80 ?

Here we have the rate (35%) and the original number (80) ; the unknown is the comparative number which constitutes 35% of 80 . Since the exercise statement is "(some number) is (thirty-five percent) of (eighty)", then the variable stands for a number and the equation is:

x = (0.35)(80)

Twenty-eight is 35% of 80 .

• 45% of what is 9 ?

Here we have the rate (45%) and the comparative number (9) ; the unknown is the original number that 9 is 45% of. The statement is "(nine) is (forty-five percent) of (some number)", so the variable stands for a number, and the equation is:

9 = (0.45)( x )

9 ÷ 0.45 = x = 20

Nine is 45% of 20 .

The format displayed above, "(this number) is (some percent) of (that number)", always holds true for percents. In any given problem, you plug your known values into this equation, and then you solve for whatever is left.

• Suppose you bought something that was priced at $6.95 , and the total bill was $7.61 . What is the sales tax rate in this city? (Round answer to one decimal place.)

The sales tax is a certain percentage of the price, so I first have to figure what the actual numerical amount of the tax was. The tax was:

7.61 – 6.95 = 0.66

Then (the sales tax) is (some percentage) of (the price), or, in mathematical terms:

0.66 = ( x )(6.95)

Solving for x , I get:

0.66 ÷ 6.95 = x      = 0.094964028... = 9.4964028...%

The sales tax rate is 9.5% .

In the above example, I first had to figure out what the actual tax was, before I could then find the answer to the exercise. Many percentage problems are really "two-part-ers" like this: they involve some kind of increase or decrease relative to some original value.

Note : Always figure the percentage of change of increase or decrease relative to the original value.

• Suppose a certain item used to sell for seventy-five cents a pound, you see that it's been marked up to eighty-one cents a pound. What is the percent increase?

First, I have to find the absolute (that is, the actual numerical value of the) increase:

81 – 75 = 6

The price has gone up six cents. Now I can find the percentage increase over the original price.

Note this language, "increase/decrease over the original", and use it to your advantage: it will remind you to put the increase or decrease over the original value, and then divide.

This percentage increase is the relative change:

6 / 75 = 0.08

...or an 8% increase in price per pound.

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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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