5.6: Percent and Statistics
Introduction
A Question of Seating
“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.
“Really? How?” Carla asked.
“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 lesson 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.
What You Will Learn
In this lesson you will learn how to complete the following skills.
- Solve statistics-based problems involving percents greater than one hundred.
- Solve statistics-based problems involving percents less than one.
- Solve statistics-based problems involving scientific notation and percents.
- Solve statistics-based problems involving very large and very small numbers.
Teaching Time
I. Solve Statistics – Based Problems Involving Percents Greater Than One Hundred
Statistics refers to mathematics involved with data and its interpretation. Oftentimes 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.
Let’s start with percents that are greater than one hundred.
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%.
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?
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: \begin{align*}2.6 - 1.2 = 1.4\end{align*}
Divide by original amount: \begin{align*}\frac{1.4}{1.2} = 1.167\end{align*}
Multiply by 100 to get percent: \begin{align*}1.167 \cdot 100 = 116.7\%\end{align*}
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. Let’s look at another example.
Example
What percent of 55 is 70?
With this problem, we can go back to our key words from earlier lessons. We know that we need to find a percent and that is multiplied by 100. But here, the part is larger than the whole. We have 70 of 55. Here is the problem.
\begin{align*}\frac{70}{55}(100) = 127.3\%\end{align*}
This is our answer.
II. Solve Statistics – Based Problems Involving Percents Less Than One
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. Let’s look at an example.
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?
Here we have a ratio of four-leaf plants to all of the plants:
\begin{align*}\frac{18}{34810} = .0005\end{align*}
Multiply by 100 to get percent: \begin{align*}.0005 \times 100 = .05\%\end{align*}
Only .05% of the plants actually had four leaves. I guess you are lucky if you find one.
Example
What percent of 75 is .3?
\begin{align*}\frac{.3}{75} \times 100 = .4\%\end{align*}
The answer is .4%.
III. Solve Statistics – Based Problems Involving Scientific Notation and Percent
Scientific notation is another useful mathematical tool that allows us to work with very large or very small numbers.
What is scientific notation?
Scientific notation is when a number is written as a factor and a power of 10. This means that we are using exponents to represent the power of 10.
Remember that any rational number can be written in scientific notation.
It follows the form:
\begin{align*}a \times 10^b\end{align*}
where \begin{align*}a\end{align*} is a number greater than or equal to 1 but less than 10 and \begin{align*}b\end{align*} is an exponent of 10
When we conduct operations involving percent with numbers in scientific notation, we can use any operation with the \begin{align*}a\end{align*} value as we have seen in these sections. Then we will be sure to write our answer in scientific notation. We may need to adjust the \begin{align*}a\end{align*} and \begin{align*}b\end{align*} values. Let’s look at how this works.
Example
On certain dates, Mars is about \begin{align*}4.9 \times 10^7\end{align*} miles from Earth. If a spacecraft is headed toward Mars and has traveled 30% of the distance, how many miles has it gone?
To solve this, we need to find 30% of \begin{align*}4.9 \times 10^7\end{align*}. The \begin{align*}a\end{align*} value is our factor that is 4.9 so we will find 30% of that and the power \begin{align*}10^7\end{align*} is included in the product once we have multiplied the factor with the percent.
30% of \begin{align*}a\end{align*} value: \begin{align*}.30 \times 4.9 \times 10^7 = 1.47 \times 10^7\end{align*}
The spacecraft has traveled \begin{align*}1.47 \times 10^7 \ miles\end{align*}. We can leave our answer in the form of scientific notation
Our \begin{align*}a\end{align*} value is now 1.47 which is greater than or equal to 1 and less than 10. There is no need to adjust it.
Example
Find 25% of \begin{align*}3 \times 10^{12}\end{align*}.
\begin{align*}.25 \times 3 \times 10^{12} = .75 \times 10^{12}\end{align*}
Our \begin{align*}a\end{align*} value is .75 which is not greater than or equal to 1.
We move the decimal point 1 place to the right on .75 to get 7.5.
Then adjust the exponent 1 integer less—if we make the \begin{align*}a\end{align*} value bigger by a factor of 10, then we make the exponent 1 less.
\begin{align*}7.5 \times 10^{11}\end{align*}
This is our answer.
IV. Solve Statistics – Based Problems Involving Very Large or Very Small Numbers
Very large and very small numbers are not always written in scientific notation. Writing numbers “normally” is called standard notation. We can still work with numbers that are very large but again must be most careful of decimal places. Making an error of 1 decimal place is like multiplying or dividing a number by 10. You would probably agree that there is a big difference between $50 and $500 even though the decimal place is only different by 1 place or we could say by multiplying by 10.
\begin{align*}50 \times 10 = 500\end{align*}
Let’s look at an example.
Example
In the year 2000, the United States had a population of about 280,000,000 people. By 2010, the population is expected to be 308,000,000. What will the percent increase have been in those 10 years?
\begin{align*}\text{Difference}: 308,000,000 - 280,000,000 &= 28,000,000\\ \text{Divide difference by original}: \frac{28,000,000}{280,000,000} &= .10 = 10\%\end{align*}
The population will have grown by 10% in those 10 years.
Let’s review the steps we did here.
- We identified that we are looking for a percent.
- We found the difference between the original population and the new population.
- Then we divided the difference by the original population.
- Finally, we converted this decimal into our percent.
Write these steps down in your notebook.
Example
Find 32% of .00000054.
To work on this problem, we have to change the percent into a decimal first.
32% = .32
Then we notice the key word “of” which means multiply. We are going to multiply the percent times that decimal, which represents a very, very small number.
\begin{align*}.32 \times .00000054 = .0000001728\end{align*}
This is our answer.
Now let’s apply what we have learned to the original problem from the introduction.
Real-Life Example Completed
A Question of Seating
Here is the original problem once again. First, reread it. Then use what you have learned to figure out the percent of the increase in seating.
“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.
“Really? How?” Carla asked.
“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.
Now use what you have learned to figure out the percent of the increase in seating.
Solution to Real – Life Example
To find the percent of the increase, first, find the difference between the old seating number and the new seating number.
\begin{align*}100,000 - 80,000 = 20,000\end{align*}
Now compare the difference to the original number of seats.
\begin{align*}\frac{20,000}{80,000}\end{align*}
.25 = 25%
The new seating is a 25% increase over the previous seating.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Statistics
- mathematical that involves data collection and interpretation.
- Percent of Change
- the percentage that a value changes to increase or decrease over time.
- Scientific Notation
- writing a number as a factor and a power of 10. This involves the use of exponents.
- Standard Notation
- writing numbers in the usual way with all of the zeros accounted for in the value.
Time to Practice
Directions: Answer each question and round your answers to the nearest tenth.
- What percent of 110 is 450?
- What percent of 32 is 100?
- What percent of 50 is 200?
- What percent of 88 is 400?
- What percent of 10 is 18?
- What percent of 2 is 4?
- What percent of 45 is 60?
Directions: Answer each question and round your answers to the nearest hundredth.
- What percent of 50,980 is 325?
- What percent of 85 is .25?
- What percent of 90 is 15?
- What percent of 10 is 4?
- What percent of 30 is 6?
Directions: Round the \begin{align*}a\end{align*} values to the nearest hundredth and place your answers in scientific notation.
- Find 62% of \begin{align*}3.5 \times 10^9\end{align*}.
- Find 5% of \begin{align*}9.1 \times 10^{13}\end{align*}.
- Find 180% of \begin{align*}6.3 \times 10^{-17}\end{align*}.
Directions: Answer each question and leave your answers in standard form.
- What percent of 8,570,000 is 152?
- Find 230% of .00000488
- The number .00036 is 45% of what number?
- A technology company made a great new discovery. It then saw its stock price go from $3.75 in March to $75.52 in May. What was the percent of increase of the stock in those two months?
- A rival company got hit with a big law suit after a scandal was discovered. Its highest stock price was $210. Its stock is now worth $1.62. What percent value does the stock currently have compared to its highest price?
- Margaret loves to watch sad movies. She cries 200 tears in a movie. A single tear has about \begin{align*}1.2 \times 10^{18}\end{align*} atoms in it. So she’ll cry about \begin{align*}2.4 \times 10^{20}\end{align*} atoms during the whole movie. About how many atoms will she have cried after only 13% of the movie?
- The Earth is about 92,960,000 miles from the Sun. Light travels about 186,000 miles per second. What percent of the distance from the Sun to Earth does light travel in 1 second?
- A light year is about 5,880,000,000,000 miles. In one month, it travels about 8.2% of that distance. About how far does it travel in one month?
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