Percents

Section 1.1 Percents

Suppose an administrator wants to compare the performance of two class sections. In one section, 23 out of 32 students passed; in the other, 18 out of 25 students passed. One way to compare their performance is by converting the fractions $\frac{23}{32}$ and $\frac{18}{25}$ to equivalent fractions with a common denominator. However, a more practical approach is to calculate and compare the percentage of students who passed each class. Using methods from this section, we find that 71.875% of the students in the first class passed, while 72% of the students in the second class passed. This shows that the second class performed slightly better than the first.

Objectives

Students will be able to:

Convert between fraction, decimal, and percent form.
Calculate percents, parts, and ending amounts.
Calculate absolute and relative change.
Interpret absolute and relative change in context.

Subsection 1.1.1 Percents

Percent, denoted by %, means “per 100,” or “parts per hundred.” The word percent comes from the Latin phrase per centum which means of a hundred. When we write 30% off, that means you save 30 out of every 100 dollars. A $100 item would be $30 off, a $200 item would be $60 off, and so on. We can write this percentage as the fraction $\frac{30}{100}\text{.}$ When we divide 30 by 100, we get

\begin{equation*} \frac{30}{100}=0.30 \end{equation*}

We can also say you would save 30 cents out of every dollar. Many equivalent fractions can be written that equal 30%. Notice that 60 out of 200 and 15 out of 50 are also 30%, since

\begin{equation*} \frac{60}{200}=\frac{15}{50}=\frac{30}{100}=0.30 \end{equation*}

Example 1.1.1.

As an example, let’s compare two jackets on sale and add some prices:

Jacket 1:

Original Price: $80
Sale: $20 off

Jacket 2:

Original Price: $100
Sale: 30% off

What is the sale price and sale percentage for each jacket? Which is the better buy?

Solution.

Jacket 1:

For this jacket we have the dollar amount of the sale so we will subtract it from the original price.

\begin{equation*} \$80-\$20 = \$60 \end{equation*}

Now we will divide to find the percentage of the sale:

\begin{equation*} \frac{\$20}{\$80} = 0.25 \end{equation*}

This jacket is 25% off and the sale price is $60.

Jacket 2:

For this jacket we know the percentage off so we need to find 30% of $100. We used a base of 100 so $30 off per $100 is $30.

$30\% \text{ off of } \$100 \text{ is } \$30$

Then we subtract the sale amount from the original price to get:

\begin{equation*} \$100-30 = \$70 \end{equation*}

This jacket is 30% off and the sale price is $70.

Which do you think is the better buy? You would save more with jacket 2, but jacket 1 is cheaper. If the jackets are similar in quality and desirability, jacket 1 would be a good choice. If jacket 2 is warmer, looks better on you, or offers better rain protection, you might consider whether it is worth spending an additional $10 to have a better jacket.

Subsection 1.1.2 Converting Between Percents and Decimals

To convert a percent to decimal form, we use the definition of "per 100." For example, $82\%$ means “82 per 100” so we divide by 100. Notice this is the same as moving the decimal two places to the left and removing the % symbol.

\begin{equation*} 82\% = \frac{82}{100}=0.82 \end{equation*}

To go the other way, if we have $0.1\text{,}$ we can multiply it by 100% or move the decimal two places to the right and add the % symbol.

\begin{equation*} 0.1 = 0.1(100\%)=10\% \end{equation*}

Example 1.1.2.

Change each percent to decimal form.

65%
200%
5.2%
0.1%

Solution.

$65\%=\frac{65}{100}=0.65\text{.}$ Notice this is the same as moving the decimal to the left two places.
$200\%=\frac{200}{100}=2.0$ or just $2\text{.}$
$5.2\%=\frac{5.2}{100}=0.052\text{.}$ If we use the method of moving the decimal two places to the left we use a zero to hold the place value and we get 0.052.
$0.1\%=\frac{0.1}{100}=0.001\text{.}$

Example 1.1.3.

Change each fraction or decimal into percent form.

1.25
0.06
0.003
$\displaystyle \frac{4}{5}$

Solution.

Multiplying $1.25(100\%)$ or moving the decimal two places to the right gives 125%.
Multiplying $0.06(100\%)$ or moving the decimal two places to the right gives 6%.
Multiplying $0.003(100\%)$ or moving the decimal two places to the right gives 0.3%. Note this is less than one percent.
First we divide $4$ by $5$ to get $0.80$ and then multiply by 100%. $0.80(100\%) = 80\%\text{.}$

Watch for Common Errors.

Be careful with fractions and decimals. Banks sometimes use fractions when expressing interest rates of loans. For example, $6\frac{3}{4}\%$ is the same as $6.75\%$ because $3\div4=0.75\text{.}$ We can convert to decimal form by moving the decimal two places to the left to obtain $0.0675\text{.}$

Be careful with small percentages. For example, $0.5\%$ expressed in decimal form is $0.005\text{.}$ The decimal starts out to the left of the 5, and when we shift the decimal two places to the left we must fill in with two additional zeros.

Quick Tips.

Think about changing between dollars and cents. To go from dollars to cents, we multiply by 100. To go from cents to dollars, we divide by 100. The same idea applies to percents and decimals.

Subsection 1.1.3 Calculating Percents

When calculating percents, we divide the part by the whole amount. The whole amount is also called the base of the percentage. First we will look at how to calculate percents and then how to calculate a part.

Calculating Percents.

\begin{equation*} \text{Percent in decimal form} = \frac{\text{part}}{\text{whole}} \end{equation*}

\begin{equation*} \text{Part} = \text{(percent in decimal form)} \cdot \text{whole} \end{equation*}

We can also think of these formulas as sentences. The word is means equals, per means divide, and of often means multiplication. So we can also say, “Percent is part per whole,” and, “Part is percent of whole.”

Example 1.1.4.

In a survey of 1,031 randomly chosen adults, 577 said they would feel better if they got more sleep (source

news.gallup.com/poll/642704/americans-sleeping-less-stressed

). What percent is this?

Solution.

First, we need to identify the part and the whole. The whole is the total number of adults surveyed. The part is the 577 adults who said they would feel better if they got more sleep.

Using the formula

\begin{equation*} \text{Percent} = \frac{\text{part}}{\text{whole}} \end{equation*}

we can calculate

\begin{equation*} \text{Percent} = \frac{577}{1013} \approx 0.570 \text{ or } 57\% \end{equation*}

About 57% of the adults surveyed said they would feel better if they got more sleep.

Example 1.1.5.

First estimate which quantity is the larger percentage. Then calculate both percentages.

A quiz score of 8 out of 10 or 11 out of 15.
A cereal with 2 grams of sugar in a 28 gram serving size or 3 grams of sugar in a 32 gram serving size.

Solution.

We are estimating that 8 out of 10 seems larger compared to 11 out of 15 because missing 2 out of 10 seems like less than 4 out of 15.

\begin{equation*} \frac{8}{10} = 0.80 \text{ or } 80\% \end{equation*}

\begin{equation*} \frac{11}{15} \approx 0.733 \text{ or } 73.3\% \end{equation*}

The score of 8 out of 10 is a higher percentage.
We are estimating that 3 out of 32 is a higher percentage than 2 out of 28 because the added gram of sugar seems high compared with only a 4 gram increase in serving size.

\begin{equation*} \frac{2\text{ g}}{28\text{ g}} \approx 0.07 \text{ or about } 7\% \end{equation*}

\begin{equation*} \frac{3\text{ g}}{32\text{ g}} \approx 0.09 \text{ or about } 9\% \end{equation*}

The cereal with 3 grams of sugar out of 32 grams has a higher percentage of sugar.

Another useful tool is finding the part when we know the percent and the whole amount.

Example 1.1.6.

For example, say you have a high yield online savings account that pays an annual interest rate of 4.5%. If you have $3000 in the account for your emergency fund, how much interest will you earn in a year? What would your total balance be with the interest?

Solution.

To perform calculations with percents, we change them into decimal form first. So we divide 4.5 by 100 or move the decimal two places to the left and we get 0.045.

Then, using the formula

\begin{equation*} \text{Part} = \text{percent} \cdot \text{whole} \end{equation*}

we have

\begin{equation*} \text{Part} = 0.045(\$3000) = \$135 \end{equation*}

You would earn $135 in interest. Interest is added to your account so we will add to find the total balance.

\begin{equation*} \$3000+135 = \$3{,}135 \end{equation*}

Your total balance is $3,135 after one year.

Note: The example above is an example of simple interest. Most accounts use something called "compound interest. Both types of interest are addressed in Chapter 3.

Example 1.1.7.

Find each part described and the total or ending amount.

A smartphone’s battery life of 12 hours lost 20% after a year of use. Find the amount it decreased by and the current battery life.
The sales tax in Vancouver, Washington is currently 8.7%. How much tax would you pay on TV that costs $399 and what is the total price?

Solution 1.

First we will find 20% of 12 hours by multiplying by the decimal form of the percent:

\begin{equation*} 12(0.20)=2.4\text{ hours} \end{equation*}

Then we will subtract since the battery life went down.

\begin{equation*} 12-2.4=9.6\text{ hours} \end{equation*}

The phone battery life went down by 2.4 hours to 9.6 hours.
We will find 8.7% of $399 by multiplying the amount by the decimal form of the percent

\begin{equation*} \$399(0.087)=\$34.71 \end{equation*}

We rounded to the nearest cent. Then we will add since the tax is an additional cost.

\begin{equation*} \$399+34.71=\$433.71 \end{equation*}

The tax on the TV is $34.71, which gives a total price of $433.71.

Solution 2.

There is an optional shorter way to go about this type of problem if you are interested.

If we look at a 20% decrease as the battery retaining 80% of its life (100-20=80)%, we can calculate the new battery life directly using the decimal form
\begin{align*} 12(1.00-0.20) \amp= 12(0.80)\\ \amp= 9.6\text{ hours.} \end{align*}
We can also do this with the TV price using addition for an increase (100+8.7=108.7)%
\begin{align*} \$399(1.00+0.087)\amp=\$399(1.087)\\ \amp=\$433.71. \end{align*}

Subsection 1.1.4 Percents over 100%

In many situations the part must be less than the whole so the percent cannot be more than 100%. In the shirt sale example, a store can’t take off more than the cost of the shirt. If they took off 100% of the price the shirt would be free. In other situations, going over 100% does make sense. So it’s important to use the context and your number sense to make sure your answer is reasonable.

Example 1.1.8.

A company set a goal of $50,000 in profit and actually made $120,000 in their first year. What percent of their goal is this?

Here we can see the “part” is higher than the whole or base amount so this answer should be over 100%.

Solution.

\begin{equation*} \frac{\text{part}}{\text{whole}} = \frac{\$120{,}000}{\$50{,}000} = 2.4 \text{ or } 240\% \end{equation*}

The company’s profit was 2.4 times or 240% of their goal.

Subsection 1.1.5 Absolute and Relative Change

As we saw in the shirt sale prices, we can talk about a difference in absolute terms or relative terms. Absolute change is the actual amount of the change in dollars, people, etc. The change is positive if the value goes up and negative if it goes down. Relative change is the amount of change as a percentage of the starting amount. The relative change would have the same sign as the absolute change. Often we use words like increase or decrease to represent the sign.

Absolute and Relative Change.

\begin{align*} \text{Absolute Change} \amp= \text{Ending Amount} - \text{Starting Amount}\\ \\ \text{Relative Change} \amp = \frac{\text{Absolute Change}}{\text{Starting Amount}} \end{align*}

For example, Portland Community College recently increased its tution from $128 to $133 per credit hour. Find the absolute and relative change. First we will subtract to find out how much the tuition went up.

\begin{equation*} \$133-\$128 = \$5\text{ per credit} \end{equation*}

The absolute change is an increase of $5 per credit. Now we will divide that by the original tuition of $128.

\begin{equation*} \frac{\$5}{\$128}\approx 0.039 \end{equation*}

The relative change is a tuition increase of about 3.9%.

The base of a percent is very important and relative change is calculated as a percentage of the original amount because that was the amount before the change.

Example 1.1.9.

Maria has been thinking about selling her car. The value went from $7400 to $6800 over the last year. What is the absolute and relative change in the car’s value?

Solution.

First we will subtract to calculate the absolute change:

\begin{align*} \text{Absolute Change} \amp= \text{Ending Amount} - \text{Starting Amount}\\ \amp=\$6800 - \$7400\\ \amp=-\$600 \end{align*}

The absolute change is a decrease of $600 in the car’s value.

Now we can calculate the relative change:

\begin{align*} \text{Relative Change} \amp= \frac{\text{Absolute Change}}{\text{Starting Amount}}\\ \amp= \frac{-\$600}{\$7400}\\ \amp\approx -0.081 \end{align*}

The relative change in the price of the car was a decrease of about 8.1% of it’s value.

Notice in the last example we either used the negative sign or the word decrease to show the direction of the change. We could say the car’s value changed by -8.1% or decreased by 8.1%. We wouldn’t use both at the same time because that would be a double negative. Let’s look at another example.

Example 1.1.10.

After raises, Iryna’s wage went from $22 to $25 per hour. Sundar’s hourly wage went from $19 to $21.85. Calculate their absolute and relative raises. Who got a larger raise?

Solution.

First we will calculate the absolute raises:

Iryna:

$\$25 - \$22 = \$3$

Sundar:

$\$21.85 - \$19 = \$2.85$

Iryna got a larger absolute raise of $3 per hour compared with $2.85 for Sundar. Now we will calculate the relative raises.

$\frac{\$3}{\$22} \approx 0.136 \text{ or } 13.6\%$

Iryna’s raise was about 13.6% of her pay.

$\frac{\$2.85}{\$19} = 0.15 \text{ or } 15\%$

Sundar’s raise was 15% of his pay.

Sundar had a relative raise of 15% of his pay and Iryna’s raise was about 13.6% of her pay. Iryna had the larger absolute raise but Sundar had the larger raise relative to their starting wages.

Notice that instead of subtracting first, we could have divided the new pay by the base pay. For Sundar’s pay we get $\frac{\$21.85}{\$19} = 1.15 \text{ or } 115\%\text{.}$

This is another way to see that Sundar had a raise of 15%.

Notice in the last example that Sundar’s new pay was 1.15 times his old pay or an increase of 15%. He is getting 100% of his original salary plus an additional 15%.

Subsection 1.1.6 Doubling, Tripling, Etc.

Now let’s say the company that made $120,000 in profit the first year made $240,000 in profit in their second year. We can say their profit doubled or was two times the previous year. This is a 100% increase because they matched their original profit for 100%, and they also had an increase of 100%.

In order to triple their first year’s profit they would need to make $360,000 in profit. Be precise with your wording when describing percent changes because tripling their profit is an increase of 200%.

Subsection 1.1.7 Multiple Percent Changes

We can also calculate values with multiple percent increases and decreases. We must be careful here because the base changes each time. For example, if a stock price drops by 60% in one week and increases by 75% the next week, this is not the same as an increase of 15%. The base for the drop is the original price and the base for the increase is the lower price. Let’s look at this more closely in the next example.

Example 1.1.11.

A stock valued at $100 dropped in value by 60% one week, then increased in value the next week by 75%. Is the value higher or lower than where it started? What is the ending value and what is the combined relative change?

Solution.

After one week, the value dropped by 60% so we have

\begin{align*} \$100 - \$100(0.60) \amp= \$100 - \$60\\ \amp= \$40 \end{align*}

In the next week, the starting value of the stock is $40, so that is the new base. Computing the 75% increase we get

\begin{align*} \$40 + \$40(0.75) \amp= \$40 + \$30\\ \amp= \$70 \end{align*}

In the end, the stock is $70, which is $30 lower, or 30% lower than it started.

If you enjoyed the shorter method we showed before, it is really useful for multiple percent changes.

\begin{align*} \$100(1-0.60)(1+0.75) \amp= \$100(0.40)(1.75)\\ \amp= \$70 \end{align*}

Exercises 1.1.8 Exercises

Calculations.

Perform the calculations indicated in each exercise.

1.

Convert each percent into decimal form.

15.6%
9.1%
0.07%
135.6%

2.

Convert each percent into decimal form.

1.25%
230%
7%
0.1%

3.

Change each fraction or decimal into percentage form.

0.89
0.043
$\displaystyle \frac{3}{4}$
1.05

4.

Change each fraction or decimal into percentage form.

0.029
$\displaystyle \frac{5}{6}$
1.804
0.72

Applications.

Solve each problem.

5.

Out of 230 racers who started a marathon, 212 completed the race. What percentage is that? Round your percentage to 1 decimal place if needed.

6.

Out of 501 adults surveyed, 390 had a pet. What percentage is that? Round your percentage to 1 decimal place if needed.

7.

A customer left a tip of $9 on a $50 restaurant bill. What percent tip is that?

8.

A customer left a tip of $15 on a $72 restaurant bill. What percent tip is that?

9.

To help alleviate homelessness, voters in the Portland Metro area approved a personal income tax for individuals who make over $125,000 in taxable income. Miguel has a taxable income of $180,000 and the rate for that level of income is 1.5%. How much will Miguel owe for this tax?

10.

Employees pay 7.65% of their gross earning towards Social Security tax (FICA) and employers pay the same amount. James earns $5,100 per month. How much will be deducted from his monthly paycheck for FICA tax?

11.

A project on Kickstarter.com had a goal of $400 to fund the creation of enamel bird pins. They have 21 backers who raised $1,119. What percentage of their goal is this?

12.

A project on Kickstarter.com had a goal of $11,000 to fund a new chai maker. They have 3,658 backers who raised $367,273. What percentage of their goal is this?

13.

A project on Gofundme.com was aiming to raise $5,000 to produce the worlds smallest foldable charger. They have 880 donations raising about 1,637% of their goal. How much did they raise?

14.

A project on Gofundme.com was aiming to raise $50,000 to fund a food truck. They have 1,251 donations raising about 253% of their goal. How much did they raise?

15.

Vancouver, Washington has a sales tax of 8.7% Find the tax amount on a clothing purchase of $120. What is the total cost of the purchase?

16.

Safa managed a company to reduce its carbon emissions by 18.3%. The company originally emitted 920 tons of carbon dioxide annually. How much was the reduction and what are the new emissions?

17.

A company wants to decrease their energy use by 15%. If their electric bill is currently $2,200 per month, what will their bill be if they are successful?

18.

A company starts a social media campaign with a goal of increasing the number of customers by 30%. They currently have about 80 customers a day. How many customers will they have if the campaign is successful?

19.

Portland State University enrollment was 16,423 undergraduate students in Fall, 2023 and 16,864 undergraduate students in Fall, 2022.

20.

A town’s population went from 25,300 residents in 2023 to 24,590 residents in 2024.

21.

Franklin High School enrollment was 1,876 students during the 2022-23 school year and 1,966 students in the 2021-22 school year.

22.

The cost of a restaurant menu item was $19.99 in 2023 and $20.50 in 2024.

23.

A puppy’s weight doubles from 5 pounds to 10 pounds.

24.

A baby’s weight triples from 8 pounds to 24 pounds.

25.

A stock’s value goes from $50 to $40.

26.

A car’s value goes from $15,000 to $10,000.

27.

A company’s sales were $2 million in 2021. They decreased by 9.5% in 2022, then increased by 10% in 2023. How much did they have in sales in 2023? What was the combined relative change?

28.

A new car worth $50,000 dropped in value by 20% in the first year and 15% the second year. What was it worth after the second drop and what was the combined relative change?

29.

A store has clearance items that have been marked down by 60%. They are having a sale, advertising an additional 30% off clearance items. What percent of the original price do you end up paying?

30.

A stock goes up by 20% one day and then down by 18%. What percent of the original price is the stock now?

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