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5.15: Very Small and Large Percent Problems

Difficulty Level: At Grade Created by: CK-12
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Wembley Stadium in London, England, has a seating capacity of 90,000. In contrast, The Georgia Dome in Atlanta, Georgia, has a seating capacity of 71, 250. How much larger is Wembley Stadium than the Georgia Dome?

In this concept, you will learn how to solve statistics-based problems involving percent greater than 100 and less than one.

Statistics

Statistics refers to mathematics involved with data and its interpretation.

You can accumulate large sets of data like from a survey, a poll, an experiment, formal observations, etc. In order to draw any conclusion from the data, you must be able to interpret it. You have seen some statistical measures like mean, median, and mode. You have also used scientific notation to work with large and small numbers. Based on these statistical measures, you might make inferences that reach beyond the set of data or experiment. As you make inferences, percent calculations will be useful. Sometimes percent will be greater than 100 or less than 1 so you need to perform operations carefully.

Let’s start with percent that are greater than one hundred.

Percent is a ratio with a denominator of 100 that represents a part of a whole. One-hundred percent represent one whole. If you have 100% of a pizza, you have an entire pizza. What if you have 200% of a pizza? Then that is 2 whole pizzas. 300% would be 3 whole pizzas. 1000% is 10 times 100% so that would be 10 whole pizzas. You can see, then, that percentages do not stop at 100. Whenever percentages represent more than a whole, they will be greater than 100%.

Let’s look at an example.

A nation-wide survey found that the average home in 1907 had 1.2 bathrooms. A hundred years later, the average home had 2.6 bathrooms. What was the percent of change in the number of bathrooms per home?

First, find the amount of change.

\begin{align*}2.6-1.2=1.4\end{align*}

Next, divide by the original amount and multiply by 100.

\begin{align*}\begin{array}{rcl} \text{percent of increase} &=& \frac{\text{amount of increase}}{\text{original amount}}\times 100 \\ \text{percent of increase} &=& \frac{1.4}{1.2} \times 100 \\ \text{percent of increase} &=& 1.167 \times 100 \\ \text{percent of increase} &=& 116.7 \end{array}\end{align*}

The answer is 116.7.

The number of bathrooms in the average home increased by 116.7% in 100 years. This indicates that the number of bathrooms per home more than doubled.

Percentages can also be very small. The operations do not change but you must be more careful in using decimal places correctly. Scientists often work with very small percentages.

Let’s look at an example.

A researcher was interested in the truth about lucky four-leaf clovers. He surveyed 34,810 clover plants and found that only 18 of them actually had four leaves. All of the others had only 3 leaves. What percent of the plants had four leaves?

First, find the ratio of four-leaf plants to all of the plants.

\begin{align*}\frac{18}{34810}=0.0005\end{align*}

Next, multiply by 100 to get percent.

 \begin{align*}0.0005 \times 100 = 0.05 \%\end{align*}

The answer is 0.05%.

Therefore, four-leaf plants occur only 0.05% of the time.

Examples

Example 1

Earlier, you were given a problem about the sizable stadiums.

Wembley Stadium seats 90,000 people and the Georgia Dome seats 71,250 people. How much bigger is Wembley Stadium than the Georgia Dome?

First, find the difference between the seating capacities of the two stadiums.

\begin{align*}90,000-71,250=18,750\end{align*}

Next, divide by the original amount and multiply by 100.

\begin{align*}\begin{array}{rcl} \text{percent of increase} &=& \frac{\text{amount of increase}}{\text{original amount}}\times 100 \\ \text{percent of increase} &=& \frac{18750}{90000} \times 100 \\ \text{percent of increase} &=& 0.2083 \times 100 \\ \text{percent of increase} &=& 20.83 \end{array}\end{align*}

The answer is 20.83%.

Therefore, Wembley Stadium is 20.83% larger than the Georgia Dome.

Example 2

What percent of 75 is 0.3?

First, find the ratio of four-leaf plants to all of the plants.

\begin{align*}\frac{0.3}{75}= 0.004\end{align*}

Next, multiply by 100 to get percent. 

\begin{align*}0.004 \times 100 = 0.4 \%\end{align*}

The answer is 0.4%.

Example 3

Figure out each percent of increase or decrease from 3.4 to 6.9.

First, find the amount of change.

\begin{align*}6.9-3.4=3.5\end{align*}

Next, divide by the original amount and multiply by 100.

\begin{align*}\begin{array}{rcl} \text{percent of increase} &=& \frac{\text{amount of increase}}{\text{original amount}}\times 100 \\ \text{percent of increase} &=& \frac{3.5}{3.4} \times 100 \\ \text{percent of increase} &=& 1.029 \times 100 \\ \text{percent of increase} &=& 102.9 \end{array}\end{align*}

The answer is 103%.

Example 4

Figure out each percent of increase or decrease from 8.7 to 20.2.

First, find the amount of change.

\begin{align*}20.2-8.7=11.5\end{align*}

Next, divide by the original amount and multiply by 100.

\begin{align*}\begin{array}{rcl} \text{percent of increase} &=& \frac{\text{amount of increase}}{\text{original amount}}\times 100 \\ \text{percent of increase} &=& \frac{11.5}{8.7} \times 100 \\ \text{percent of increase} &=& 1.322 \times 100 \\ \text{percent of increase} &=& 132.2 \end{array}\end{align*}

The answer is 132%.

Example 5

Figure out each percent of increase or decrease from 450,000 to 200,000.

First, find the amount of change.

\begin{align*}45,000-200,000=250,000\end{align*}

Next, divide by the original amount and multiply by 100.

\begin{align*}\begin{array}{rcl} \text{percent of decrease} &=& \frac{\text{amount of decrease}}{\text{original amount}}\times 100 \\ \text{percent of decrease} &=& \frac{250,00}{450,000} \times 100 \\ \text{percent of decrease} &=& 0.5556 \times 100 \\ \text{percent of decrease} &=& 55.56 \end{array}\end{align*}

The answer is 55.56%.

Review

Answer each question and round your answers to the nearest tenth.

1. What percent of 110 is 450?

2. What percent of 32 is 100?

3. What percent of 50 is 200?

4. What percent of 88 is 400?

5. What percent of 10 is 18?

6. What percent of 2 is 4?

7. What percent of 45 is 60?

Answer each question and round your answers to the nearest hundredth.

8. What percent of 50,980 is 325?

9. What percent of 85 is .25?

10. What percent of 90 is 15?

11. What percent of 10 is 4?

12. What percent of 30 is 6?

13. What percent of 385 is 25?

14. What percent of 400 is 3?

15. What percent of 595 is 18?

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.15. 

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