# Percent

Here is everything you need to know about percentages. You’ll learn how to find the percent of a number, how to calculate with percent multipliers, and how to increase and decrease a number by a percent. You’ll also learn about percent changes.

Students will first learn about percents as part of ratios and proportional relationships in 6th grade.

## What is a percent?

A percent is a number that is expressed as a fraction of 100. To do this, you need to know that “percent” means “number of parts per hundred.” The symbol we use for percent is the percent sign \%.

43\%=\cfrac{43}{100}=0.43

1\%=\cfrac{1}{100}=0.01

There are different types of questions involving percents.

### How to find a percent of a number

This is when you are asked to find a certain percentage of an amount.

For example,

find 25\% of \$32. This is the same as finding a \cfrac{1}{4} of \$32.

32\div4=8

The answer is \$8. You can also write the percent as a fraction (\cfrac{25}{100}) and then multiply the fraction by the total amount (32). \cfrac{25}{100}\times32=\cfrac{25}{100}\times\cfrac{32}{1}=\cfrac{800}{100}=8 Step-by-step guide: Percent of a number ### Percent conversions Percents, fractions, and decimals can all be used to represent part of a whole. For example, It is useful to be able to convert between percents, fractions, and decimals. To change a percent into a fraction, write the percent as the numerator of a fraction and 100 as the denominator, and then simplify the resulting fraction if possible. For example, 42\%= \cfrac{42 \, \div \, 2}{100 \, \div \,2} = \cfrac{21}{50} To change a fraction into a percent, there are two methods. Method 1 : First, see if it is possible to write an equivalent fraction with a denominator of 100. If so, the numerator will be the percent. For example, \cfrac{7}{20} = \cfrac{7 \, \times \, 5}{20 \, \times \, 5}=\cfrac{35}{100}=35\% Method 2 : Carry out the division represented in the fraction and then multiply by 100. For example, \cfrac{1}{8} = 1 \div 8 =0.125 0.125 \times 100 = 12.5 \cfrac{1}{8} = 12.5 \% To change a percent into a decimal, divide the percent number by 100. For example, 73\% = 73 \div 100 = 0.73 To change a decimal into a percent, multiply the decimal by 100. For example, 0.29 0.29 \times 100 = 29 29\% ### Percent as operator In order to find a percent of an amount, a percent increase, or a percent decrease, you can use a percent multiplier. To do this, you change the percent that you want into a decimal, then multiply the amount by that decimal to calculate the answer. For example, 34\% of 58. Here you want \bf{34\%}, which as a decimal is \bf{0.34}. Therefore, the calculation is: 58 \times 0.34 = 19.72 For example, Increase 78 by 15\%. Here you want an increase of 15\% , which means in total you want \bf{115\%}. This is because you want another 78 \, (100\%) plus another 15\% of 78 \, ( + 15\%). 115\% as a decimal is \bf{1.15}. Therefore, the calculation is: 78 \times 1.15 = 89.7 For example, Decrease 45 by 20\%. Here you want a decrease of 20\%, which means in total you want \bf{80\%}. Since you want to decrease 45 by 20\%, that means you need to determine 80\% of 45. \, 80\% as a decimal is \bf{0.8}. Therefore, the calculation is: 45 \times 0.8 = 36 ### Percent increase This is when you are asked to increase (make bigger) a value by a certain amount. For example, increase 40 \, g by 10\%. You can find 10\% and add it on. 10\% of 40 is 4. 40+4 = 44 The answer is 44 \, g. Step-by-step guide: Percent increase ### Percent decrease This is when you are asked to decrease (make smaller) a value by a certain amount. For example, decrease 60 \, kg by 10\%. You can find 10\% and subtract it. 10\% of 60 is 6. 60-6=54 The answer is 54 \, kg. Step-by-step guide: Percent decrease ### Percent change When values change, you can express this change as a percent of the original value. For example, work out the percent change of 26 \, kg from 25 \, kg. 26-25=1 \cfrac{1 \, \times \, 4}{25 \, \times \, 4}=\cfrac{4}{100}=4\% The answer is 4\%. Step-by-step guide: Percent change ### Percent error This is when you find the difference between an estimated value and the exact or known value. The difference is then divided by the known value and multiplied by 100\%. For example, I estimated that 200 people would attend the concert, but 250 people attended. 250-200=50 50\div250=0.2 0.2\times100=20\% Step-by-step guide: Percent error (coming soon) ### Simple interest This is when you find the percent of an original amount and multiply it by the time period of the investment. Simple interest = principal amount × rate of interest × time period For example, calculate the interest earned on \$3,000 with a simple interest rate of 5\% over 2 years.

\begin{aligned} \text{Simple interest } &= \$3000\times0.05\times2 \\\\ &= \$300 \end{aligned}

Step-by-step guide: Simple interest

### Reverse percents

This is when you are given a certain percent of a number and you have to find the original number.

For example,

20\% of a number is 6, what is the number?

20\%=\cfrac{1}{5}

To find the original number you need to find 100\% or “one whole.”

5\times6=30

### Percent calculations

Let’s look at different methods that can be used to perform percent calculations.

For example,

what is 40\% of 70?

Method 1: The one percent method

Find 1\% first by dividing the amount by 100 and then multiply the amount by the percent you want.

\cfrac{70}{100} \times 40 = 28

Method 2: The decimal multiplier method

Write the percent you want as a decimal and then multiply the amount by this decimal.

40\%=0.4

70 \times 0.4=28

Method 3: Using equivalent fractions

Write the percent you want as a fraction in simplest form and then multiply the amount by this fraction.

40\% = \cfrac{40}{100} =\cfrac{4}{10} =\cfrac{2}{5}

70 \times \cfrac{2}{5} = \cfrac{70\times 2}{5} = \cfrac{140}{5} = 28

Method 4: Building up an answer from simple percent you know

Using simple percents, you can build up the answer to the question.

For this question, if you know 10\% of 70, you can multiply this answer by 4 to find 40\%.

10\% of 70 = 7

40\% of 70 = 7 \times 4 = 28

Note, methods 1 and 2 lend themselves best to questions where you are allowed to use a calculator. Methods 3 and 4 are most helpful when calculators are not permitted.

### What is a percent? ## Common Core State Standards

How does this relate to 6th grade math?

• 6th grade – The Number System (6.RP.3c)
Find a percent of a quantity as a rate per 100 (for example, 30\% of a quantity means \frac{30}{100} times the quantity); solve problems involving finding the whole, given a part and the percent.

## How to find the percent

In order to work out percents, you need to be clear about what you have and what you are trying to find out. Percent questions can vary and there will be links in the other sections with more details.

1. Write down what you have and what you are trying to find.
2. Work out what you need.
3. Write down the final answer.

## Percent examples

### Example 1: percent of a number

Find 23\% of \$160. 1. Write down what you have and what you are trying to find. 100\% is \$160.

You need to find 23\% of \$160. 2Work out what you need. Using the one percent method: \cfrac{160}{100} \times 23 = 1.6 \times 23 = 36.8 3Write down the final answer. 23\% of \$160 is \bf{\$36.80}. ### Example 2: percent multipliers Find 39\% of \$4,700.

100\% is \$4,700. We want to find 39\% of \$4,700.

We can write the percent as a decimal number.

39\%=0.39

This gives you the percent in decimal form. This is the percent multiplier.

4700\times0.39=1833

39\% of \$4,700 is \bf{\$1,833}.

### Example 3: percent increase

Increase \$200 by 30\%. 100\% is \$200.

We want to find 130\% of \$200. 100\%+30\%=130\% 10\% is 20. 200\div10=20 30\% is 60. 3\times20=60 So 130\% is: 200 + 60 = 260 OR You could use the decimal multiplier. 100+30=130 130\%=\cfrac{130}{100}=1.30 200\times1.3=260 \$200 increased by 30\% is \bf{\$260}. ### Example 4: percent decrease Decrease 700 \, g by 20\%. 100\% is 700 \, g. We want to find 80\% of 700 \, g. 100\%-20\%=80\% 10\% is 70. 700\div10=70 20\% is 140. 2\times70=140 So 80\% is: 700-140=560 OR You could use the decimal multiplier. 100-20=80 80\%=\cfrac{80}{100}=0.80 700\times0.80=560 700 \, g decreased by 20\% is \bf{560 \, g}. ### Example 5: percent change A t-shirt went from \$10 to \$12. Work out the percent change. The original amount is \$10.

We need to find the change as a percent of the original amount.

The change is 12-10=2.

The change written as a fraction of the original is \, \cfrac{2}{10}.

The change is the numerator (top number). The original number is the denominator (bottom number).

Convert the fraction to an equivalent fraction where the denominator is 100 (the bottom number is 100 ).

\cfrac{2}{10}=\cfrac{20}{100}=20\%

OR

Write the change as a fraction of the original and multiply by 100.

\cfrac{2}{10}\times100=20

The percent change of \$12 from \$10 is \bf{20\%}.

### Example 6: reverse percent

A coat costs \$40 after a 20\% price cut. Find the original price. Some people call the price after the price cut the “new number.” 100\%-20\%=80\% We have 80\% = \$40.

We need the original price which is 100\%.

80\% is 40.

We can work out 10\%.

40\div8=5

Then you can work out 100\%.

5\times 10 = 50

OR

You can make an equation using x as the original number. Use the decimal multiplier and the new number.

The price after the cut is \$40. The original price was \bf{\$50}.

### Teaching tips for percent

• If needed, provide students with hundreds grids to help them understand percent. This will provide them with a visual of 100 and they can fill in squares to represent the percents.

• Provide students with real-world examples and situations as often as possible. This will help them gain a deeper understanding of the content.

• Create an anchor chart to display in the classroom showing each type of percent problem for students to refer to when needed.

### Easy mistakes to make

• Money needs two digits for the cents
Find 34\% of \$620. 620\times0.34=210.8 The answer is \$210.80.

• The decimal multiplier to work out a percent increase can be greater than \bf{1}
Increase 50 \, km by 3\%.
We can find 103\%.
50\times1.03 =51.5
The answer is 51.5 \, km.

• Percents can be greater than \bf{100\%}
Calculate the percent change of 450 from 200.
The percent change 450-200=250.

\cfrac{250}{200}\times100=125

So the percent change of 450 from 200 is 125\%.

### Percent practice questions

1. Find 40\% of \$300. \$1,200 \$120 \$7.50 \$12 10\% of \$300 is \$30. We can multiply this by 4 to get 40\%. 2. Calculate 12.4\% of \$3,000.

\$3,372 \$37,200 \$372 \$3,720 As a multiplier, 12.4\% is 0.124. To get the answer you can calculate

0.124\times3000

3. Increase 600 \, m by 30\%.

630 \, m 780 \, m 180 \, m 420 \, m 30\% of 600 is 180. We can add this to the original amount to find the quantity after the increase.

4. Decrease 80 \, kg by 5\%.

75 \, kg 95 \, kg 76 \, kg 84 \, kg 5\% of 80 is 4. We can subtract this from the original amount to find the quantity after the decrease.

5. Calculate the percent change from 400 \, kg to 600 \, kg.

50\% 200\% 100\% 150\% The increase is given by:

600-400=200

The percent increase is then given by:

\cfrac{200}{400}\times100=50\%

6. A book costs \$4 after a price reduction of 20\%. What was the original price? \$4.20 \$5.00 \$20.00 \$4.25 \$4 represents 80\% of the original amount, which means 10\% of the original amount is \$0.50. Multiplying by ten to get 100\% means the original price was \$5.

## Practice percent questions – word problems

1. Charlotte invests \$3,000 for 4 years. She gets a simple interest rate of 2\% per year. Find the total interest Charlotte gets. Show answer 3000\times0.02= 4\times{60} =\$240

2. Last year, Ron paid \$450 for his car insurance. This year, he has to pay \$603 for his car insurance. Find the percent increase in his car insurance.

The change is:

603-450=

\cfrac{153}{450}\times 100

=34\%

## Percent FAQs

What is a percent?

A percent is a number that is expressed as a fraction of 100. The symbol we use for percent is the percent sign \%.

How do you find a percent of a number?

To find a percent of a number, you can write the percent as a fraction and then multiply the fraction by the total amount.

How do you convert a percent to a decimal?

To convert a percent to a decimal, divide it by 100. For example, 25\%=0.25.

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