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3.6: Percent Problems

Created by: CK-12

Learning Objectives

  • Find a percent of a number.
  • Use the percent equation.
  • Find the percent of change.

Introduction

A percent is simply a ratio with a base unit of 100. When we write a ratio as a fraction, the base unit is the denominator. Whatever percentage we want to represent is the number on the numerator. For example, the following ratios and percents are equivalent.

Fraction Percent Fraction Percent
 \left (\frac{50} {100}\right ) 50%  \left (\frac{50} {1000}\right ) =  \left (\frac{0.5} {100}\right ) 0.5%
 \left (\frac{10} {100}\right ) 10%  \left (\frac{1} {25}\right ) = \left (\frac{4} {100}\right ) 4%
 \left (\frac{99} {100}\right ) 99%  \left (\frac{3} {5}\right ) = \left (\frac{60} {100}\right ) 60%
 \left (\frac{125} {100}\right ) 12.5%  \left (\frac{1} {10,000}\right ) = \left (\frac{0.01} {100}\right ) 0.01%

Fractions are easily converted to decimals, just as fractions with denominators of 10, 100, 1000, 10000 are converted to decimals. When we wish to convert a percent to a decimal, we divide by 100, or simply move the decimal point two units to the left.

Percent Decimal Percent Decimal Percent Decimal
10% 0.1 0.05% .0005 0% 0
99% 0.99 0.25% .0025 100% 1

Find a Percent of a Number

One thing we need to do before we work with percents is to practice converting between fractions, decimals and percentages. We will start by converting decimals to percents.

Example 1

Express 0.2 as a percent.

The word percent means “for every hundred”. Therefore, to find the percent, we want to change the decimal to a fraction with a denominator of 100. For the decimal 0.2 we know the following is true:

0.2 & = 0.2 \times 100 \times \left (\frac{1} {100}\right ) & & \text{Since } \ 100 \times \left (\frac{1} {100}\right ) = 1\\0.2 & = 20 \times \left (\frac{1} {100}\right ) \\0.2 & = \left (\frac{20} {100}\right ) = 20\%

Solution

0.2 = 20\%

We can take any number and multiply it by 100 \times \frac{1}{100} without changing that number. This is the key to converting numbers to percents.

Example 2

Express 0.07 as a percent.

0.07 & = 0.07 \times 100 \times \left (\frac{1} {100}\right )\\0.07 & = 7 \times \left (\frac{1} {100}\right )\\0.07 & = \left (\frac{7} {100}\right ) = 7\%

Solution

0.07 = 7\%

It is a simple process to convert percentages to decimals. Just remember that a percent is a ratio with a base (or denominator) of 100.

Example 3

Express 35% as a decimal.

 35\% = \left (\frac{35} {100}\right ) = 0.35

Example 4

Express 0.5% as a decimal.

 0.5\% = \left (\frac{0.5} {100}\right ) = \left (\frac{5} {1000}\right ) = 0.005

In practice, it is often easier to convert a percent to a decimal by moving the decimal point two spaces to the left.

The same trick works when converting a decimal to a percentage, just shift the decimal point two spaces to the right instead.

When converting fractions to percents, we can substitute \frac{x}{100} for x\%, where x is the unknown percentage we can solve for.

Example 5

Express  \frac{3} {5} as a percent.

We start by representing the unknown as x\% or \frac{x}{100}.

\left (\frac{3} {5}\right ) & = \frac{x} {100}& & \text{Cross multiply}.\\5x & = 100 \cdot 3 & & \text{Divide both sides by} \ 5 \ \text{to solve for} \ x.\\5x & = 300\\x & = \frac{300}{5} = 60

Solution

 \left (\frac{3} {5}\right ) = 60\%

Example 6

Express  \frac{13} {40} as a percent.

Again, represent the unknown percent as \frac{x}{100}, cross-multiply, and solve for x.

 \frac{13} {40} & = \frac{x} {100}\\40x & = 1300\\x & = \frac{1300} {40} = 32.5

Solution

 \left (\frac{13} {40}\right ) = 32.5\%

Converting percentages to simplified fractions is a case of writing the percentage ratio with all numbers written as prime factors:

Example 7

Express 28% as a simplified fraction.

First write as a ratio, and convert numbers to prime factors.

 28\% \left (\frac{28} {100}\right ) = \left (\frac{2\cdot 2\cdot 7} {5\cdot 5\cdot 2\cdot 2\cdot}\right )

Now cancel factors that appear on both numerator and denominator.

\left (\frac{\cancel{2}\cdot \cancel{2}\cdot 7}{\cancel{2}\cdot \cancel{2}\cdot 5\cdot 5}\right ) = \frac{7}{25}

Solution

 28\% = \left (\frac{7} {25}\right )

Multimedia Link The following video shows several more examples of finding percents and might be useful for reinforcing the procedure of finding the percent of a number. Khan Academy Taking Percentages (9:55)

Use the Percent Equation

The percent equation is often used to solve problems.

Percent Equation: Rate \times Total = Part or “R% of Total is Part”

Rate is the ratio that the percent represents (R% in the second version).

Total is often called the base unit.

Part is the amount we are comparing with the base unit.

Example 8

Find 25% of $80

Use the percent equation. We are looking for the part. The total is $80. ‘of’ means multiply. R% is 25% so the rate is \frac{25}{100} or 0.25.

0.25  \cdot \$80 = \$20

Solution

25% of $80 is $20.

Remember, to convert a percent to a decimal, you just need to move the decimal point two places to the left!

Example 9

Find 17% of $93

Use the percent equation. We are looking for the part. The total is $93. R% is 17% so the rate is 0.17.

0.17 \cdot 93 = 15.81

Solution

17% of $93 is $15.81.

Example 10

Express $90 as a percentage of $160.

Use the percent equation. This time we are looking for the rate. We are given the part ($90) and the total ($160). We will substitute in the given values.

\text{Rate}\times 160 & = 90 \qquad \qquad \text{Divide both sides by} \ 160\\\text{Rate} & = \left (\frac{90} {160}\right ) = 0.5625 = 0.5625 \left (\frac{100} {100}\right ) = \frac{56.25} {100}

Solution

$90 is 56.25% of 160.

Example 11

$50 is 15% of what total sum?

Use the percent equation. This time we are looking for the total. We are given the part ($50) and the rate (15% or 0.15). The total is our unknown in dollars, or x. We will substitute in these given values.

 0.15x & = 50 & & \text{Solve for} \ x \ \text{by dividing both sides by} \ 0.15.\\x & = \frac{50} {0.15} \approx 333.33

Solution

$50 is 15% of $333.33.

Find Percent of Change

A useful way to express changes in quantities is through percents. You have probably seen signs such as “20% extra free”, or “save 35% today.” When we use percents to represent a change, we generally use the formula.

\text{Percent change} &= \left (\frac{\text{final amount} - \text{original amount}} {\text{original amount}}\right ) \times 100\%\\& \text{Or}\\\frac{\text{percent change}} {100} &= \left (\frac{\text{actual change}} {\text{original amount}}\right )

A positive percent change would thus be an increase, while a negative change would be a decrease.

Example 12

A school of 500 students is expecting a 20% increase in students next year. How many students will the school have?

The percent change is +20. It is positive because it is an increase. The original amount is 500. We will show the calculation using both versions of the above equation. First we will substitute into the first formula.

\text{Percent change} &= \left (\frac{\text{final amount} - \text{original amount}} {\text{original\ amount}}\right ) \times 100\%\\20\% & = \left (\frac{\text{final amount} - 500} {500}\right ) \times 100\% & & \text{Divide both sides by} \ 100\%.\\& & & \text{Let} \ x = \text{final amount}.\\0.2 & = \frac{x - 500} {500} & & \text{Multiply both sides by} \ 500.\\100 & = x - 500 & & \text{Add} \ 500 \ \text{to both sides}.\\600 & = x

Solution

The school will have 600 students next year.

Example 13

A $150 mp3 player is on sale for 30% off. What is the price of the player?

The percent change is given, as is the original amount. We will substitute in these values to find the final amount in dollars (our unknown x). Note that a decrease means the change is negative. We will use the first equation.

\text{Percent change} &= \left (\frac{\text{final amount} - \text{original amount}} {\text{original amount}}\right ) \times 100\%\\\left (\frac{x - 150} {150}\right ) \cdot 100\% & = - 30\% & & \text{Divide both sides by} \ 100\%.\\\left (\frac{x - 150} {150}\right ) = \frac{30\%} {100\%} & = - 0.3\% & &  \text{Multiply both sides by} \ 150.\\x - 150 = 150 (-0.3) & = -45 & &  \text{Add} \ 150 \ \text{to both sides}.\\x & = -45 + 150

Solution

The mp3 player is on sale for $105.

We can also substitute straight into the second equation and solve for the change y.

 \frac{\text{percent change}} {100} &= \left (\frac{\text{actual change}} {\text{original amount}}\right)\\\frac{-30} {100} & = \frac{y} {150} & & \text{Multiply both sides by} \ 150.\\150(-0.3) & = y \\y & =-45

Solution

Since the actual change is -45($), the final price is \$150 - \$45 = \$105.

A mark-up is an increase from the price a store pays for an item from its supplier to the retail price it charges to the public. For example, a 100% mark-up (commonly known in business as keystone) means that the price is doubled. Half of the retail price covers the cost of the item from the supplier, half is profit.

Example 14 – Mark-up

A furniture store places a 30% mark-up on everything it sells. It offers its employees a 20% discount from the sales price. The employees are demanding a 25% discount, saying that the store would still make a profit. The manager says that at a 25% discount from the sales price would cause the store to lose money. Who is right?

We will consider this problem two ways. First, let us consider an item that the store buys from its supplier for $1000.

& \text{Item price} \ \$1000\\& \text{Mark-up} \ \$300 && (30\% \ \text{of} \ 1000  = 0.30 \cdot 1000 = 300 )\\& \text{Final retail price} \ \$1300

So a $1000 item would retail for $1300. $300 is the profit available to the store. Now, let us consider discounts.

& \text{Retail Price} & & \$1300\\& 20\% \ \text{discount} & & 0.20 \times \$1300 = \$260\\& 25\% \ \text{discount} & &  0.25 \times \$1300 = \$325

So with a 20% discount, employees pay \$1300 - \$260 = \$1040

With a 25% discount, employees pay \$1300 - \$325 = \$975

With a 20% discount, employees pay $40 more than the cost of the item.

At a 25% discount they pay $975, which is $25 less than the cost.

Finally, we will work algebraically. Consider an item whose wholesale price is x.

\text{Mark-up} & = 0.3x\\\text{Final retail price} & = 1.3x\\\text{Price at} \ 20\% \ \text{discount} & = 0.80 \times 1.3x = 1.04x\\\text{Price at} \ 25\% \ \text{discount} & = 0.75 \times 1.3x = 0.975x

Solution

The manager is right. A 20% discount from retail means the store makes around 4% profit. At a 25% discount, the store has a 2.5% loss.

Solve Real-World Problems Using Percents

Example 15

In 2004 the US Department of Agriculture had 112071 employees, of which 87846 were Caucasian. Of the remaining minorities, African-American and Hispanic employees had the two largest demographic groups, with 11754 and 6899 employees respectively.*

a) Calculate the total percentage of minority (non-Caucasian) employees at the USDA.

b) Calculate the percentage of African-American employees at the USDA.

c) Calculate the percentage of minority employees who were neither African-American nor Hispanic.

a) Use the percent equation \text{Rate} \times \text{Total} = \text{Part}.

The total number of employees is 112071. We know that the number of Caucasian employees is 87846, which means that there must be (112071 - 87846) = 24225 non-Caucasian employees. This is the part.

\text{Rate} \times 112071 & = 24225 & &  \text{Divide both sides by} \ 112071.\\\text{Rate} & = 0.216 & &  \text{Multiply by} \ 100 \ \text{to obtain percent}:\\\text{Rate}& =21.6\%

Solution

21.6% of USDA employees in 2004 were from minority groups.

b) \text{Total} = 112071\text{   Part} = 11754

\text{Rate} \times 112071 & = 11754 & & \text{Divide both sides by} \ 112071.\\\text{Rate} & = 0.105 & & \text{Multiply by} \ 100 \ \text{to obtain percent}:\\\text{Rate} & =10.5\%

Solution

10.5% of USDA employees in 2004 were African-American.

c) We now know there are 24225 non-Caucasian employees. This is now our total. That means there must be (24225 - 11754 - 6899) = 5572 minority employees who are neither African-American nor Hispanic. The part is 5572.

\text{Rate} \times 24225 & = 5572 & &  \text{Divide both sides by} \ 24225 \\\text{Rate} & = 0.230 & & \text{Multiply by} \ 100 \ \text{to obtain percent}.\\\text{Rate}& =23\%

Solution

23% of USDA minority employees in 2004 were neither African-American nor Hispanic.

Example 16

In 1995 New York had 18136000 residents. There were 827025 reported crimes, of which 152683 were violent. By 2005 the population was 19254630 and there were 85839 violent crimes out of a total of 491829 reported crimes. Calculate the percentage change from 1995 to 2005 in:  ^\dagger

a) Population of New York

b) Total reported crimes

c) violent crimes

This is a percentage change problem. Remember the formula for percentage change.

 \text{Percent change} = \left (\frac{\text{final amount} - \text{original amount}} {\text{original amount}}\right ) \times 100\%

In these cases, the final amount is the 2005 statistic. The initial amount is the 1995 statistic.

a) Population:

\text{Percent change} & = \left(\frac{19,254,630 - 18,136,000} {18,136,000}\right ) \times 100\%\\\text{Percent change} & = \left (\frac{1,118,630} {18,136,000}\right ) \times 100\%\\\text{Percent change} & = 0.0617 \times 100 \\\text{Percent change} & = 6.17\%

Solution

The population grew by 6.17%.

b) Total reported crimes

\text{Percent change} & = \left (\frac{491,829 - 827,025} {827,025}\right ) \times 100\%\\\text{Percent change} & = \left (\frac{-335,196} {827,025}\right ) \times 100\%\\\text{Percent change} & = - 0.4053 \times 100\\\text{Percent change} & = -40.53\%

Solution

The total number of reported crimes fell by 40.53%.

c) Violent crimes

\text{Percent change} & = \left (\frac{85,839 - 152,683} {152,683}\right ) \times 100\%\\\text{Percent change} & = \left (\frac{-66,844} {152,683}\right ) \times 100\%\\\text{Percent change} & = -0.4377 \times 100\\\text{Percent change} & = - 43.77\%

Solution

The total number of reported crimes fell by 43.77%.  ^\dagger Source: New York Law Enforcement Agency Uniform Crime Reports

Lesson Summary

  • A percent is simply a ratio with a base unit of 100, i.e. 13\% = \frac{13}{100}.
  • The percent equation is: Rate \times Total = Part or “R% of Total is Part”.
  • \text{Percent change}=\frac{\text{final amount}-\text{original amount}}{\text{original amount}} \times 100. A positive percent change means the value increased, while a negative percent change means the value decreased.

Review Questions

  1. Express the following decimals as a percent.
    1. 0.011
    2. 0.001
    3. 0.91
    4. 1.75
    5. 20
  2. Express the following fractions as a percent (round to two decimal places when necessary).
    1.  \frac{1} {6}
    2.  \frac{5} {24}
    3.  \frac{6} {7}
    4.  \frac{11} {7}
    5.  \frac{13} {97}
  3. Express the following percentages as a reduced fraction.
    1. 11%
    2. 65%
    3. 16%
    4. 12.5%
    5. 87.5%
  4. Find the following.
    1. 30% of 90
    2. 16.7% of 199
    3. 11.5% of 10.01
    4. y% of 3x
  5. A TV is advertised on sale. It is 35% off and has a new price of $195. What was the pre-sale price?
  6. An employee at a store is currently paid $9.50 per hour. If she works a full year she gets a 12% pay rise. What will her new hourly rate be after the raise?
  7. Store A and Store B both sell bikes, and both buy bikes from the same supplier at the same prices. Store A has a 40% mark-up for their prices, while store B has a 250% mark-up. Store B has a permanent sale and will always sell at 60% off those prices. Which store offers the better deal?

Review Answers

    1. 1.1%
    2. 0.1%
    3. 91%
    4. 175%
    5. 2000%
    1. 16.67%
    2. 20.83%
    3. 85.71%
    4. 157.14%
    5. -13.40%
    1. \frac{11}{100}
    2. \frac{13}{20}
    3. \frac{4}{25}
    4. \frac{1}{8}
    5. \frac{7}{8}
    1. 27
    2. 33.233
    3. -1.15115
    4. \frac{3xy}{100}
  1. $300
  2. $10.64
  3. Both stores’ final sale prices are identical.

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