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3.6: Percent Problems

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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 percentage we want to represent is the numerator, and the denominator is 100. For example, 43% is another way of writing \frac{43}{100}. \frac{43}{1000}, on the other hand, is equal to \frac{4.3}{100}, so it would be equivalent to 4.3%. \frac{2}{5} is equal to \frac{40}{100}, or 40%. To convert any fraction to a percent, just convert it to an equivalent fraction with a denominator of 100, and then take the numerator as your percent value.

To convert a percent to a decimal, just move the decimal point two spaces to the right:

 67\% &= 0.67\\0.2\% &= 0.002\\150\% &= 1.5

And to convert a decimal to a percent, just move the decimal point two spaces to the left:

 2.3 &= 230\%\\0.97 &= 97\%\\0.00002 &= 0.002\%

Finding and Converting Percentages

Before we work with percentages, we need to know how to convert between percentages, decimals and fractions.

Converting percentages to fractions is the easiest. The word “percent” simply means “per 100”—so, for example, 55% means 55 per 100, or \frac{55}{100}. This fraction can then be simplified to \frac{11}{20}.

Example 1

Convert 32.5% to a fraction.

Solution

32.5% is equal to 32.5 per 100, or \frac{32.5}{100}. To reduce this fraction, we first need to multiply it by \frac{10}{10} to get rid of the decimal point. \frac{325}{1000} then reduces to \frac{13}{40}.

Converting fractions to percentages can be a little harder. To convert a fraction directly to a percentage, you need to express it as an equivalent fraction with a denominator of 100.

Example 2

Convert \frac{7}{8} to a percent.

Solution

To get the denominator of this fraction equal to 100, we have to multiply it by 12.5. Multiplying the numerator by 12.5 also, we get \frac{87.5}{100}, which is equivalent to 87.5%.

But what about a fraction like \frac{1}{6}, where there’s no convenient number to multiply the denominator by to get 100? In a case like this, it’s easier to do the division problem suggested by the fraction in order to convert the fraction to a decimal, and then convert the decimal to a percent. 1 divided by 6 works out to 0.166666.... Moving the decimal two spaces to the right tells us that this is equivalent to about 16.7%.

Why can we convert from decimals to percents just by moving the decimal point? Because of what decimal places represent. 0.1 is another way of representing one tenth, and 0.01 is equal to one hundredth—and one hundredth is one percent. By the same logic, 0.02 is 2 percent, 0.35 is 35 percent, and so on.

Example 3

Convert 2.64 to a percent.

Solution

To convert to a percent, simply move the decimal two places to the right. 2.64 = 264\%.

Does a percentage greater than 100 even make sense? Sure it does—percentages greater than 100 come up in real life all the time. For example, a business that made 10 million dollars last year and 13 million dollars this year would have made 130% as much money this year as it did last year.

The only situation where a percentage greater than 100 doesn’t make sense is when you’re talking about dividing up something that you only have a fixed amount of—for example, if you took a survey and found that 56% of the respondents gave one answer and 72% gave another answer (for a total of 128%), you’d know something went wrong with your math somewhere, because there’s no way you could have gotten answers from more than 100% of the people you surveyed.

Converting percentages to decimals is just as easy as converting decimals to percentages—simply move the decimal to the left instead of to the right.

Example 4

Convert 58% to a decimal.

Solution

The decimal point here is invisible—it’s right after the 8. So moving it to the left two places gives us 0.58.

It can be hard to remember which way to move the decimal point when converting from decimals to percents or vice versa. One way to check if you’re moving it the right way is to check whether your answer is a bigger or smaller number than you started out with. If you’re converting from percents to decimals, you should end up with a smaller number—just think of how a number like 50 percent, where 50 is greater than 1, represents a fraction like \frac{1}{2} (or 0.50 in decimal form), where \frac{1}{2} is less than 1. Conversely, if you’re converting from decimals to percents, you should end up with a bigger number.

One way you might remember this is by remembering that a percent sign is bigger than a decimal point—so percents should be bigger numbers than decimals.

Example 5

Convert 3.4 to a percent.

Solution

If you move the decimal point to the left, you get 0.034%. That’s a smaller number than you started out with, but you’re moving from decimals to percents, so you want the number to get bigger, not smaller. Move it to the right instead to get 340%.

Now let’s try another fraction.

Example 6

Convert \frac{2}{7} to a percent.

Solution

\frac{2}{7} doesn’t convert easily unless you change it to a decimal first. 2 divided by 7 is approximately 0.285714..., and moving the decimal and rounding gives us 28.6%.

The following Khan Academy video shows several more examples of finding percents and might be useful for further practice: http://www.youtube.com/watch?v=_SpE4hQ8D_o.

Use the Percent Equation

The percent equation is often used to solve problems. It goes like this:

& \text{Rate} \times \text{Total} = \text{Part}\\& \qquad \qquad \text{or}\\ & R\% \ \text{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 7

Find 25% of $80.

Solution

We are looking for the part. The total is $80. ‘of’ means multiply. R\% is 25%, so we can use the second form of the equation: 25% of $80 is Part, or 0.25 \times 80 = \text{Part}.

0.25 \times 80 = 20, so the Part we are looking for is $20.

Example 8

Express $90 as a percentage of $160.

Solution

This time we are looking for the rate. We are given the part ($90) and the total ($160). Using the rate equation, we get \text{Rate} \times 160 = 90. Dividing both sides by 160 tells us that the rate is 0.5625, or 56.25%.

Example 9

$50 is 15% of what total sum?

This time we are looking for the total. We are given the part ($50) and the rate (15%, or 0.15). Using the rate equation, we get 0.15 \times \text{Total} = \$50. Dividing both sides by 0.15, we get \text{Total} = \frac{50}{0.15} \approx 333.33. So $50 is 15% of $333.33.

Find Percent of Change

A useful way to express changes in quantities is through percents. You’ve 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} = \frac{\text{final amount - original amount}}{\text{original amount}} \times 100\%

or

\frac{\text{percent change}}{100} = \frac{\text{actual change}}{\text{original amount}}

This means that a positive percent change is an increase, while a negative change is a decrease.

Example 10

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

Solution

First let’s solve this using the first formula. Since the 20% change is an increase, we represent it in the formula as 20 (if it were a decrease, it would be -20.) Plugging in all the numbers, we get

20\% = \frac{\text{final amount} - 500}{500} \times 100\%

Dividing both sides by 100%, we get 0.2 = \frac{\text{final amount} - 500}{500}.

Multiplying both sides by 500 gives us 100 = \text{final amount} - 500.

Then adding 500 to both sides gives us 600 as the final number of students.

How about if we use the second formula? Then we get \frac{20}{100} = \frac{\text{actual change}}{500}. (Reducing the first fraction to \frac{1}{5} will make the problem easier, so let’s rewrite the equation as \frac{1}{5} = \frac{\text{actual change}}{500}

Cross multiplying is our next step; that gives us 500 = 5 \times (\text{actual change}). Dividing by 5 tells us the change is equal to 100. We were told this was an increase, so if we start out with 500 students, after an increase of 100 we know there will be a total of 600.

Markup

A markup 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 11

A furniture store places a 30% markup 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?

Solution

We’ll consider this problem two ways. First, let’s consider an item that the store buys from its supplier for a certain price, say $1000. The markup would be 30% of 1000, or $300, so the item would sell for $1300 and the store would make a $300 profit.

And what if an employee buys the product? With a discount of 20%, the employee would pay 80% of the $1300 retail price, or 0.8 \times \$1300 = \$1040.

But with a 25% discount, the employee would pay 75% of the retail price, or 0.75 \times \$1300 = \$975.

So with a 20% employee discount, the store still makes a $40 profit on the item they bought for $1000—but with a 25% employee discount, the store loses $25 on the item.

Now let’s use algebra to see how this works for an item of any price. If x is the price of an item, then the store’s markup is 30% of x, or 0.3x, and the retail price of the item is x + 0.3x, or 1.3x. An employee buying the item at a 20% discount would pay 0.8 \times 1.3x = 1.04x, while an employee buying it at a 25% discount would pay 0.75 \times 1.3x = 0.975x.

So the manager is right: a 20% employee discount still allows the store to make a profit, while a 25% employee discount would cause the store to lose money.

It may not seem to make sense that the store would lose money after applying a 30% markup and only a 25% discount. The reason it does work out that way is that the discount is bigger in absolute dollars after the markup is factored in. That is, an employee getting 25% off an item is getting 25% off the original price plus 25% off the 30% markup, and those two numbers together add up to more than 30% of the original price.

Solve Real-World Problems Using Percents

Example 12

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.

Solution

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 - 87646 = 24225 non-Caucasian employees. This is the part. Plugging in the total and the part, we get \text{Rate} \times 112071 = 24225.

Divide both sides by 112071 to get \text{Rate} = \frac{24225}{112071} \approx 0.216. Multiply by 100 to get this as a percent: 21.6%.

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

b) Here, the total is still 112071 and the part is 11754, so we have \text{Rate} \times 112071 = 11754. Dividing, we get \text{Rate} = \frac{11754}{112071} \approx 0.105, or 10.5%.

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

c) Here, our total is just the number of non-Caucasian employees, which we found out is 24225. Subtracting the African-American and Hispanic employees leaves 24225 - 11754 - 6899 = 5572 employees in the group we’re looking at.

So with 24225 for the whole and 5572 for the part, our equation is \text{Rate} \times 24225 = 5572, or \text{Rate} = \frac{5572}{24225} \approx 0.230, or 23%.

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

Example 13

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. (Source: New York Law Enforcement Agency Uniform Crime Reports.) Calculate the percentage change from 1995 to 2005 in:

a) Population of New York

b) Total reported crimes

c) violent crimes

Solution

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

\text{Percent change} = \frac{\text{final amount - original amount}}{\text{original amount}} \times 100\%

In these problems, the final amount is the 2005 statistic, and the initial amount is the 1995 statistic.

a) Population:

\text{Percent change} &= \frac{19254630 - 18136000}{18136000} \times 100\%\\&= \frac{1118630}{18136000} \times 100\%\\&\approx 0.0617 \times 100\%\\&= 6.17\%

The population grew by 6.17%.

b) Total reported crimes:

\text{Percent change} &= \frac{491829 - 827025}{827025} \times 100\%\\&= \frac{-335196}{827025} \times 100\%\\&\approx -0.4053 \times 100\%\\&= -40.53\%

The total number of reported crimes fell by 40.53%.

c) Violent crimes:

\text{Percent change} &= \frac{85839 - 152683}{152683} \times 100\%\\&= \frac{-66844}{152683} \times 100\%\\&\approx -0.4377 \times 100\%\\&= -43.77\%

The total number of violent crimes fell by 43.77%.

Lesson Summary

  • A percent is simply a ratio with a base unit of 100—for example, 13\% = \frac{13}{100}.
  • The percent equation is \text{Rate} \times \text{Total} = \text{Part}, or R% of Total is Part.
  • The percent change equation is \text{Percent change} = \frac{\text{final amount - 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 percentages in decimal form.
    1. 15%
    2. 0.08%
    3. 222%
    4. 3.5%
    5. 341.9%
  3. 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}
  4. Express the following percentages as a reduced fraction.
    1. 11%
    2. 65%
    3. 16%
    4. 12.5%
    5. 87.5%
  5. Find the following.
    1. 30% of 90
    2. 16.7% of 199
    3. 11.5% of 10.01
    4. y\% of 3x
  6. A TV is advertised on sale. It is 35% off and now costs $195. What was the pre-sale price?
  7. An employee at a store is currently paid $9.50 per hour. If she works a full year she gets a 12% pay raise. What will her new hourly rate be after the raise?
  8. 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 the marked-up prices. Which store offers the better deal?

Texas Instruments Resources

In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9613.

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