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Simplifying fractions Percent to decimal Percent to fraction

Here you will learn about percent increase and decrease, including how to increase and decrease a value by a given percentage, use decimals to calculate percent increase and percent decrease and find percent change.

Students will first learn about percent increase and decrease as part of ratios and proportions in the 7 th grade.

**Percent increase and decrease** is the change in a value compared to its original value. They are expressed as percentages.

**Percent increase** means to increase (make bigger) a value by a certain amount. To do this, you can either calculate the given percentage of the value and then add it on to the original value or use a percent as a decimal.

For example,

Increase \$ 50 by 30 \%.

Add on the percent | Percent as a decimal |
---|---|

\begin{aligned} & 10 \%=\$ 5 \\\\ & 20 \%=\$ 15 \\\\ & 30 \% = \$ 15 \end{aligned} So \$50 increased by 30\% = \$ 50+\$ 15=\$ 65. | \begin{aligned} & 100 \%+30 \%=130 \% \\\\ & 130 \%=\frac{130}{100}=1.3 \\\\ & \$ 50 \times 1.3=\$ 65 \end{aligned} |

**Percent decrease** means subtracting a given percentage of a value from the original value. To do this, you can either calculate the given percent of the value and then subtract it from the original or use a percent as a decimal.

For example,

Decrease \$ 20 by 20 \%.

Subtract the percent | Percent as a decimal |
---|---|

\begin{aligned} & 10 \%=\$ 2 \\\\ & 20 \%=\$ 4 \end{aligned} So \$20 decreased by 20\% = \$ 20-\$ 4=\$ 16. | \begin{aligned} & 100 \%-20 \%=80 \% \\\\ & 80 \%=\frac{80}{100}=0.8 \\\\ & \$ 20 \times 0.8=\$ 16 \end{aligned} |

When given two values, you can calculate the percent difference or percentage change. When the value goes down this may also be called percent decrease, percent loss, or a markdown.

You can calculate percent change using the percentage change formula:

\text { Percent change }=\frac{\text { amount of change }}{\text { original }} \times 100

How does this relate to 7 th grade math?

**Grade 7: Ratios and Proportional Relationships (7.RP.3)**Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

In order to increase or decrease a value by a given percent:

**Calculate the given percent of the value.****Add or subtract the calculated percent to/from the original number.**

Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

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Increase 500 by 50 \%.

**Calculate the given percent of the value.**

You will first calculate 50 \% of 500.

You can calculate 10 \% (by dividing by 10 ) and then multiply by 5 to get to 50 \%.

10 \% of 500=50

50 \times 5=250

50 \% of 500 is 250.

Note: You can also calculate 50 \% of an amount by dividing it by 2 .

2**Add or subtract the calculated percent to/from the original number.**

Because you are calculating a percent increase, add 250 on to the original value of 500.

500+250=750

The final answer is 750.

500 increased by 50 \% is 750.

Decrease \$ 1,200 by 25 \%.

**Calculate the given percent of the value.**

First, you will calculate 25 \% of \$ 1,200.

You can calculate 25 \% by first finding 10 \% (dividing by 10 ), then multiplying by 2 to find 20 \%.

10 \% of \$ 1,200 is 120.

20 \% of \$ 1,200 is 240.

You can then find 5 \%, by dividing 120 by 2.

120 \div 2=60.

5 \% of \$ 1,200 is 60.

240+60=300.

25 \% of \$ 1,200 is 300.

Note: You can also calculate 25 \% of an amount by dividing it by 4 .

**Add or subtract the calculated percent to/from the original number.**

Because you are calculating a percent decrease, you will subtract \$ 300 from the original value, \$ 1,200.

1,200-300=900

The final answer is \$ 900.

A 25 \% decrease of \$ 1,200 is \$ 900.

In order to increase a value by using a percent as a decimal:

**Add the percent you are increasing by and**\textbf{100\%}**.****Convert the percent to a decimal.****Multiply the original amount by the decimal.**

In order to decrease a value by using a percent as a decimal:

**Subtract the percent you are decreasing from**\textbf{100\%}**.****Convert the percent to a decimal.****Multiply the original amount by the decimal.**

Increase 2,200 \, ml by 20 \%.

**Add the percent you are increasing by and ** \textbf{100\%} **.**

100 \%+20 \%=120 \%

**Convert the percent to a decimal.**

Divide the percentage by 100.

120 \% \div 100=1.2

120 \%=1.2

**Multiply the original amount by the decimal.**

Multiply 2,200 by 1.2.

2,200 \times 1.2=2,640 \mathrm{~ml}

2,200 \, ml increased by 20 \% is 2,640 \, ml.

Kendra had a car payment of \$ 430. When she purchased a new car, her car payment decreased 15 \%. How much is Kendra’s car payment now?

**Subtract the percent you are decreasing from ** \textbf{100\%} **.**

This is a 15 \% decrease, so you will subtract 15 \% from 100 \%.

100 \%-15 \%=85 \%

**Convert the percent to a decimal.**

Now divide the percentage by 100.

\begin{aligned}&85 \div 100=0.85\\\\ &85 \%=0.85\end{aligned}

**Multiply the original amount by the decimal.**

Multiply the original price of \$ 430 by 0.85.

430 \times 0.85=\$ 365.50

Remember to always write money using two decimal places.

Kendra’s car payment decreased from \$ 430 to \$ 365.50.

In order to calculate the percent increase or percent decrease after a percent change:

**Find the amount of change using subtraction**.**Plug numbers into the percent change formula.**

\text { Percentage change }=\cfrac{\text { Change }}{\text { Original }} \times 100**Simplify the fraction, if necessary, using equivalent fractions.****Convert the fraction to a decimal.****Calculate percent change.**

Karla weighed 67 \, kg in January. Over a period of time, her weight had decreased to 55 \, kg. Calculate the percent decrease in her weight.

**Find the amount of change using subtraction**.

Karla’s weight has changed from 67 \, kg to 55 \, kg. You will subtract her original new from her original weight.

67 \mathrm{~kg}-55 \mathrm{~kg}=12 \mathrm{~kg}

**Plug numbers into the percent change formula.**

The change is 12 \, kg and the original amount is 67 \, kg. Plug the numbers into the following formula:

\begin{aligned}& \text { Percent change }=\cfrac{\text { Change }}{\text { Original }} \times 100 \\\\ &\text { Percent change }=\cfrac{12}{67} \times 100\end{aligned}

**Simplify the fraction, if necessary, using equivalent fractions.**

\cfrac{12}{67} \, is already in simplest form.

**Convert the fraction to a decimal.**

\cfrac{12}{67}=0.1791

You can round this decimal number to 0.18 to solve.

\text { Percent change }=0.18 \times 100

**Calculate percent change.**

0.18\times 100=18

\text { Percent change }=18 \%

The percent decrease is about 18 \%.

The cost of a cup of coffee has increased from \$ 2.75 to \$ 3.50. Find the percent increase in cost of a cup of coffee.

**Find the amount of change using subtraction**.

The cost of a cup of coffee has changed from \$ 2.75 to \$ 3.50. You will subtract the old cost from the new cost.

\$ 3.50-\$ 2.75=\$ 0.75

**Plug numbers into the percent change formula.**

Apply the percent change formula. The amount of increase is \$ 0.75 and the original price is \$ 2.75. Plug the numbers into the following formula:

\begin{aligned}& \text { Percent change }=\cfrac{\text { Change }}{\text { Original }} \times 100 \\\\ & \text { Percent change }=\cfrac{0.75}{2.75} \times 100\end{aligned}

**Simplify the fraction, if necessary, using equivalent fractions.**

\cfrac{0.75}{2.75} can be simplified to \cfrac{3}{11}.

\text{ Percent change }=\cfrac{3}{11} \times 100

**Convert the fraction to a decimal.**

\cfrac{3}{11}=0.2727...

You can round this decimal number to 0.27 to solve.

\text { Percent change }=0.27 \times 100

**Calculate percent change.**

0.27\times 100=27

\text { Percent change }=27 \%

The percent increase is about 27 \%.

- Microsoft Excel provides a way to calculate percent change. Students who possess Excel proficiency can further enhance their understanding of percent change by exploring this additional teaching aspect.

- There are times when the figures used to calculate percent change can be large. To help them with calculating, students can use calculators or percentage change calculators.

**Believing that percents cannot exceed**\textbf{100\%}

Percent increase or decrease can be greater than 100 \%. It simply means that the value has more than doubled (in case of an increase) or more than halved (in case of a decrease).

**Incorrectly converting percentages to decimals**

The most common mistakes are with single digit percentages (for example, 7 \% ), multiples of 10 (for example, 50 \% ) and decimal percentages (for example 4.5 \% ). Remember to divide the percentage by 100 to find the decimal. For example,

7 \%=0.07,\, 50 \%=0.5, \, 4.5 \%=0.045.

**Using an incorrect value for the denominator in the percentage increase formula**

Use the new number instead of the original value for the denominator when calculating percent increase.

1. Increase \$ 450 by 25 \%.

\$ 112.50

\$ 562.50

\$ 90.00

\$ 658.00

First, calculate 25 \% of \$ 450.

\begin{aligned}& 10 \% \text { of } 450=45 \\\\ & 20 \% \text { of } 450=90 \\\\ & 25 \% \text { of } 450=112.5\end{aligned}

Because you are calculating a percent increase, add 112.5 on to the original value of 450.

450+112.5=562.5

Remember when representing money, there should be two decimal places.

The final answer is \$ 562.50.

2. Decrease 120 \, m by 60 \%.

72 \, m

60 \, m

48 \, m

12 \, m

First, you will need to calculate 60 \% of 120 \, m.

10 \% of 120 \, m = 12 \, m

12 \times 6=72

60 \% of 120 \, m = 72 \, m

Then you will subtract 72 \, m from the original value.

120-72=48 \mathrm{m}

A 60 \% decrease of 120 \, m is 48 \, m.

3. Use a percent as a decimal to increase 43 by 21 \%.

121.00

52.03

43.32

56.47

Add the percentage you are increasing by, 21 \%, and 100.

100+21 =121

Next, you will convert the percentage to a decimal by dividing.

121 \div 100=1.21

Then multiply the original amount by the decimal.

43 \times 1.21=52.03

43 increased by 21 \% is 52.03.

4. Use a percent as a decimal to decrease \$ 800 by 30 \%.

\$ 491.50

\$ 240.00

\$ 560.00

\$ 506.00

This is a 30 percent decrease so start by subtracting 30 \% from 100 \%.

100 \%-30 \%=70 \%

Next, to convert a percent to a decimal, you divide the percent by 100.

70 \div 100=0.70

70 \% = 0.70

Then, you will multiply the original number, \$ 800, by the decimal.

800 \times 0.70=\$ 560.00

\$ 800 decreased by 30 \% is \$ 560.00.

5. Find the percent increase when 300 \, ml is increased to 560 \, ml.

87 \%

260 \%

300 \%

81 \%

First, you will calculate the change from 300 \, ml to 560 \, ml.

560-300=260

The change is 260 \, ml, so you can now plug the numbers into the percent change formula.

\text { Percent change }=\cfrac{260}{300} \times 100

After simplifying the fraction to \cfrac{13}{15}, you would convert it to the decimal 0.87.

\begin{aligned}& 0.87 \times 100=87 \\\\ & \text { Percentage increase }=87 \%\end{aligned}

6. Find the percent decrease when 725 \, kg is decreased to 575 \, kg.

150 \%

21 \%

32 \%

112 \%

Start by finding the amount of change by subtracting the original number from the new number.

The value has changed from 725 \, kg to 575 \, kg.

725-575=150 \mathrm{~kg}

Next, plug numbers into the percent change formula. The change is 150 \, kg and the original amount is 725 \, kg.

\begin{aligned}& \text { Percent change }=\cfrac{\text { Change }}{\text { Original }} \times 100 \\\\ &\text { Percent change }=\cfrac{150}{725} \times 100\end{aligned}

If the fraction can be simplified, do that next.

\cfrac{150}{725} can be simplified to \cfrac{6}{29}.

\text { Percent change }=\cfrac{6}{29} \times 100

Then you will convert the fraction to a decimal.

\begin{aligned}& \cfrac{6}{29}=0.21 \\\\ & \text { Percent change }=0.21 \times 100\end{aligned}

Then you will calculate the percent change.

0.21 \times 100=21

\text { Percent change }=21 \%

The percent change is a 21 \% decrease.

No, percent increase or decrease cannot be a negative value. The negative sign is used to indicate the direction of change (increase or decrease), but the percent itself is always a positive value.

A positive percent increase indicates growth or expansion, while a percent decrease represents a reduction or contraction. For example, a 20 \% increase means the value has grown by 20 \%, whereas a 10 \% decrease means the value has decreased by 10 \%.

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