# Calculation of a Percentage

Hi, and welcome to this review of percentages! As with most math concepts, the terminology that is used often holds the key to truly understanding the big idea. We’re going to cover some basic skills to help you translate the language of math so you can grasp the concept of percentages easily after a little practice.

This can be answered by reviewing the concepts of fractions and place value.

## Review of Fractions

A fraction is considered a “part of a whole.” When dealing with percentages, the whole is 100.

The fraction 48/100 is read as, “forty-eight one-hundredths.” The division of 48 by 100 results in the decimal value of 0.48. Note that the decimal place in the whole number, 48, is moved two places to the left.

## Review of Place Value

Here is when the review of place value comes in. For numbers expressed as decimals, the places to the right of the decimal point describe the fraction of the next whole number. As you can see here, the first place to the right of the decimal point is the tenths place. The second place to the right of the decimal point is the hundredths place, which is what determines the percentage. A percentage tells us how many “hundredths” of the next whole number are represented.

## Converting a Decimal to a Percent

To convert a decimal value to a percent, simply move the decimal point two places to the right and be sure to use the percentage notation. Of course, there are situations when a percentage must be rounded to a particular place value. We will cover that idea in the following examples.

### Example 1

Let’s take a minute to apply this information to the effect of the flu.

Let’s say there have been approximately 16,337 people affected out of the 214,589 people who have been tested for the virus. This fraction, 16,377 over 214,589, converts to the decimal, .07613.

We will convert this decimal to a percentage by moving the decimal point two places to the right, and rounding the percentage to the nearest tenth. This gives us 7.6 percent. This tells us that out of all the people tested, 7.6 percent of the tests have been positive.

### Example 2

With this information, we can learn even more about the effects of the virus. Let’s say that 95 percent of the people who tested positive for the flu recover. Typically, this high percentage is enough to know that “most” have recovered, but we can determine the actual number by simply multiplying the decimal equivalent of 95 percent by the number of people who tested positive:

Step 1: Convert the percent to a decimal by moving the decimal point two places to the left. So if we have our 95%, we want to move the decimal point two points to the left, and that gives us 0.95.

Step 2: Multiply .95 by the number of people who tested positive for the flu virus: 0.95 times 16,337; this gives us 15,520.

So, the good news is that 15,520 people who tested positive made a full recovery!

As you can see, this work involves the operations of multiplication and division. The wording of the problem will let you know which operation to use. When asked to find a percentage, you will often notice the phrase, “out of”, as shown in the first example:

16,337 confirmed flu cases out of the 214,589 people who have been tested.

In the second example, the percentage of recovered flu patients is known, and the word of indicates the use of multiplication to determine the actual number:

95% of the people who tested positive for the flu have recovered.

The percent here, in decimal form, is multiplied by what is called the “base” value.

## Using Division

There will be times when you have the information for a straightforward multiplication problem, like the one in our second example. Other times, you will have to solve for the unknown percentage by dividing both sides of the equation by the “base.”

### Example

Let’s move on with an example of a division problem:

Unfortunately, in this hypothetical flu breakout, there also have been 894 deaths. What is the percentage of deaths of those who have tested positive? This time the equation has the number of deaths that resulted from the positive test cases. We will have to solve for the unknown percentage by dividing both sides of the equation by 16,337. 894 over 16,337 is the same as saying, “the percentage of the whole (which is 16,337) over the whole” or 894 over 16,337 is equal to our percentage.

The decimal equivalent of the fraction is 0.00575.

Note that the value in the hundredths place is 0, which indicates that the percentage will be moving less than one percent. Moving the decimal point to the right two places results in this: 0.575 percent. At this point, you will have to determine how precisely you want to represent the percentage.

Showing more decimal places is more accurate, but it is often not necessary to show more than one decimal in a percentage. Because the 7 in the next decimal place (0.575) is greater than 5, the percentage is rounded up to 0.6 percent.

The ability to process data like this is helpful to see the big picture. News of large, increasing numbers and “spikes” of flu cases can be frightening, but relating the numbers correctly is necessary to gain insight to patterns and trends in the data that are available.

• 7.6% of the flu tests were positive
• 95% of the positive cases recovered
• 0.6% of the positive cases resulted in death
• Hopefully, 4.4% of positive cases are in the process of recovering

## General Examples

Let’s move on with a few more examples of calculating and finding percentages of values.

### Example 1

In a group of 169,000 people, 76.5 percent of them prefer apples over bananas. How many people prefer apples?

This is a straightforward case of multiplication using the equation: Value equals percent “of” Base

Step 1: Convert the percentage to decimal form. We do this by moving our decimal point two places to the left, which gives us 0.765.

Step 2: Multiply by the “base,” which is 169,000.

$$.765(169,000)$$$$=129,285$$

In our group of 169,000 people, 129,285 of them prefer apples over bananas.

### Example 2

Here’s an example that you may be familiar with. When dining out, people usually leave a tip for the waitstaff averaging between 15% and 20%, depending on the quality of service.

Mike was so happy with his service, that he left a $20 tip on a bill that totaled$82. What percentage was the $20 tip? This time, the template will look like this: Tip equals percent “of” bill. So 20 equals the percent of 82. Solve for the percent by dividing both sides of the equation by 82, like this: $$\frac{20}{82}=\frac{\%(82)}{82}$$. This gives us that our percentage is equal to 20 over 82. This calculation results in a decimal value of 0.2439. Moving the decimal point two places to the right and rounding to the nearest percent results in a generous tip of 24 percent. For our last two examples, I want you to try to figure out the answer on your own. ### Example 3 The next time Mike went out for dinner with his friend, the service was terrible! Their waiter rarely attended to their table and they waited over an hour for the food to arrive after ordering. Mike decided to leave a 12 percent tip on the bill totaling$76. How much will Mike leave for the tip? Pause the video now and see if you can figure it out.

If we use our tip equation from earlier, we get that the tip is equal to 12 percent of the 76 dollars. This translates to multiplication. So our tip is equal to 12% (which when we turn it into a decimal; we move our decimal two places to the left) and we’re left with 0.12 times $76, which equals$9.12.

$$\text{Tip}= .12(76)$$$$=9.12$$

So, the 12 percent tip on the $76 bill was$9.12.

Here’s our last example:

### Example 4

Last week, at a popular restaurant with a seating capacity of 120 people only had 38 customers in the building at the busiest time of the day. What percent of 120 is 38? Pause the video and see if you can solve it. Let’s see if you’re right! Use the equation 38 equals the percentage “of” 120. In this problem, solve for percent by dividing both sides of the equation by 120, like this:

$$\frac{38}{120}=\frac{\%(120)}{120}$$ or $$\frac{38}{120}=\%$$

$$0.3166=\%$$

If we round up to the nearest tenth, they were only at 32% of capacity at the busiest time of the day. 38 customers equals 32 percent of the total capacity of the restaurant.

That’s all there is to it! As you can see, it is important to know how to calculate percentages so we can put data into proper perspective and create statistics that make sense. I hope this review was helpful!

Thanks for watching, and happy studying!

### How do you find the percentage of a number?

#### A

Find the percentage of a number by turning the percentage into a decimal and multiplying by the number.
Ex. What is 75% of 152?
0.75 × 152 = 144

### How do you calculate percentage increase?

#### A

Calculate percentage increase using this formula:
$$\% increase=\frac{new-old}{old}×100$$
Ex. A student scored a 73 on his first test and a 92 on his second test. What is the percentage increase from his first test score to his second test score?
$$\% increase=\frac{92-73}{73}×100=\frac{19}{73}×100=26\%$$

### How do you calculate percentage change?

#### A

Calculate percentage change using this formula:
$$\% change=\frac{new-old}{old}×100$$
Ex. A house is originally priced at $350,000. After 6 months of not selling, the price is lowered to$315,000. What is the percent change from the original to the new price?
$$\% change= \frac{315,000-350,000}{350,000}×100=-10\%$$

## Fact Sheet ## Practice Questions

Question #1:

What is 79% expressed as a fraction?

$$\frac{79}{50}$$
$$\frac{1}{4}$$
$$\frac{79}{100}$$
$$\frac{79}{10}$$

The correct answer is $$\frac{79}{100}$$. Remember, percent means “per 100,” so 79% is really 79 per 100, which is $$\frac{79}{100}$$ as a fraction.

Question #2:

What is 20% of 42?

2.1

8.4

12.6

20.5

The correct answer is 8.4. In math, the word of means multiply, so 20% of 42 can also be thought of as 20% times 42. Convert 20% to its decimal form by changing the percent sign to a decimal point and moving it two places to the left, 0.20. Then, multiply 0.20 by 42 to get 8.4. 20% of 42 is 8.4.

Question #3:

What is 67% expressed as a decimal?

0.67

6.7

0.067

67.0

The correct answer is 0.67. To convert a percentage to a decimal, change the percent sign to a decimal and move it two places to the left. 67% becomes 0.67.

Question #4:

What percent of 25 is 5?

40%

5%

25%

20%

The correct answer is 20%. To solve this problem, set up congruent fractions.
$$\frac{5}{25}=\frac{x}{100}$$

Since percent means “per 100,” x will be the percentage we are looking for. Solve this equation by cross multiplying.
$$5×100=25x$$
$$500=25x$$
$$20=x$$

5 is 20% of 25.

Question #5:

What is 10% of 210?

2

21

42

2,100

The correct answer is 21. 10% as a fraction is $$\frac{10}{100}$$, so multiply $$\frac{10}{100}$$ by 210 to get 21. 21 is 10% of 210.