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# 5.15: Solve Statistics-Based Problems Involving Percents

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Practice Small and Large Percents
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Have you ever been to a large stadium? Take a look at this dilemma.

“Wow! We can set 4000 people in our high school football stadium,” Jeremy said while reading the school newspaper.

“Yes, but that isn’t anything compared to National football stadiums,” Cameron said.

“I agree,” Carla chimed in. “I heard that some of those stadiums can seat 70,000 people. My Uncle Tim is a huge fan and he was recently telling us all about it. Think about that, 70,000 is a lot of people.”

“It is, but the new stadium that the Dallas Cowboys is building can beat even that,” Jeremy said.

“Well, the previous stadium seated 80,000 people. This new one will seat 100,000 people and it will become the number one largest stadium that there is!” Jeremy said.

80,000 to 100,000 is quite an increase. We can find the percent of an increase or decrease when working with large numbers too. This Concept will teach you how to work with large numbers or with really small numbers. When finished, you will be able to figure out the percent of the increase in seating for the stadium.

### Guidance

Statistics refers to mathematics involved with data and its interpretation.

Often, we 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, we 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, we might make inferences that reach beyond the set of data or experiment. As we make inferences, percents will be useful. Sometimes percents will be greater than 100 or less than 1 so we will perform operations carefully.

We have worked extensively with percent. We have recognized percent as a ratio with a denominator of 100 and said that the percent represents a part of a whole. One-hundred percent represent one whole. If you have 100% of a pizza, you have an entire pizza—delicious! What if you have 200% of a pizza? Then that is 2 whole pizzas, even better. 300% would be 3 whole pizzas. 1000% is 10 times 100% so that would be 10 whole pizzas. You can see, then, that percents do not stop at 100. Whenever percents represent more than a whole, they will be greater than 100%.

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?

This is a percent of change problem which we will compute as we did in previous lessons—find the amount of change, divide by the original amount, and multiply by 100 to find the percent.

Amount of change: $2.6 - 1.2 = 1.4$

Divide by original amount: $\frac{1.4}{1.2} = 1.167$

Multiply by 100 to get percent: $1.167 \cdot 100 = 116.7\%$

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.

You can see how useful it is to have percents that are greater than 100. They give us a whole new way to measure increases.

Sometimes a percent can be well over 100. Percents can also be very small. The operations do not change but we must be more careful in using decimal places correctly. Scientists often work with very small percents.

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?

Here we have a ratio of four-leaf plants to all of the plants:

$\frac{18}{34810} = .0005$

Multiply by 100 to get percent: $.0005 \times 100 = .05\%$

Only .05% of the plants actually had four leaves. I guess you are lucky if you find one.

Figure out each percent of increase or decrease.

#### Example A

From 3.4 to 6.9.

Solution:  $102\%$

#### Example B

From 8.7 to 20.2

Solution:  $132\%$

#### Example C

From 200,000 to 450,000

Solution:  $125\%$

Now let's go back to the dilemma from the beginning of the Concept.

To find the percent of the increase, first, find the difference between the old seating number and the new seating number.

$100,000 - 80,000 = 20,000$

Now compare the difference to the original number of seats.

$\frac{20,000}{80,000}$

.25 = 25%

The new seating is a 25% increase over the previous seating.

### Vocabulary

Statistics
mathematical that involves data collection and interpretation.
Percent of Change
the percentage that a value changes to increase or decrease over time.

### Guided Practice

Here is one for you to try on your own.

What percent of 75 is .3?

Solution

$\frac{.3}{75} \times 100 = .4\%$

### Practice

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?

1. What percent of 50,980 is 325?
2. What percent of 85 is .25?
3. What percent of 90 is 15?
4. What percent of 10 is 4?
5. What percent of 30 is 6?
6. What percent of 385 is 25?
7. What percent of 400 is 3?
8. What percent of 595 is 18?

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Jan 23, 2013

Aug 21, 2014