Similarly, if we want to decrease 60 by 25\%, we could find 25\% of 60 first, which we can work out by expressing the percent as a fraction \dfrac{25}{100}.

\displaystyle \dfrac{25}{100} \cdot 60	\displaystyle =	\displaystyle \dfrac{1500}{100}
\displaystyle \dfrac{25}{100} \cdot 60	\displaystyle =	\displaystyle 15

The final amount would be the original amount 60 minus 25\% of 60. The final amount would be

60-15 = 45

There is a quicker way to do this, however, if we think completely in percents.

Exploration

Use this applet to explore percent increase and decrease using a single multiplier.

Loading interactive...

Make the whole amount 60. Drag the slider left to decrease by 40\%.
What is 60 reduced by 40\%?
What percent multiplier is used to reduce by 40\%?
Repeat the steps and decrease other amounts by different percents. How do you find a single multiplier when decreasing by a percent?
Make the whole amount 80. Drag the slider right to increase by 30\%.
What is 80 increased by 30\%?
What percent multiplier is used to increase by 30\%?
Repeat the steps and increase other amounts by different percents. How do you find a single multiplier when increasing by a percent?

To find the total amount after the increase, we are essentially finding 100\%+5\%=105\% \text{ of } 40.

Therefore, we can work out the increase this way:

\displaystyle \frac{105}{100}\cdot 40	\displaystyle =	\displaystyle \frac{420}{100}
\displaystyle \frac{420}{100}	\displaystyle =	\displaystyle 42

This gives us the same amount as before, but involved less steps.

Similarly, if we want to decrease 60 by 25\% we can multiply (100\%-25\%) of 60.

That is:

\displaystyle \dfrac{75}{100} \cdot 60	\displaystyle =	\displaystyle \dfrac{4500}{100}
\displaystyle \dfrac{75}{100} \cdot 60	\displaystyle =	\displaystyle 45

So to decrease an amount by a percent, we can multiply the amount by (100\%-\text{ percent}).

Examples

Example 1

Elena and Mikee currently work at the same company and each make \$70\,000 in a year. They both get promoted and received a raise. Mikee receieved a 7\% raise and Elena recieved a 9\% raise.

Estimate Elena's new salary.

Worked Solution

Create a strategy

Use a benchmark of a 10\% raise because it is close to her actual raise of 9\%.

Apply the idea

First we need to find the estimate of Elena's raise..

\displaystyle \text{Estimated raise}	\displaystyle =	\displaystyle 70\,000 \cdot 10\%	Multiply the current salary by 10\%
	\displaystyle =	\displaystyle 70\,000 \cdot \dfrac{10}{100}	Convert the percent to a fraction
	\displaystyle =	\displaystyle 7000	Evaluate the multiplication

To find Elena's estimated new salary, we add the estimated raise to her original salary.

\displaystyle \text{Estimated new salary}	\displaystyle =	\displaystyle 70\,000 + 7000	Add the current salary and the estimated raise
	\displaystyle =	\displaystyle \$77\,000	Evaluate the addition

Elena's estimated new annual salary is approximately \$77\,000.

Estimate Mikee's new salary.

Worked Solution

Create a strategy

Estimate using percentages close to his raise of 7\%. We can quickly approximate 5\% raise and a 10\% raise. Half way between them would be a would be a 7.5\% raise which is very close to 7\%.

Apply the idea

Find a 5\% raise.

\displaystyle 5 \% \text{ raise}	\displaystyle =	\displaystyle 70\,000\cdot 5\%	Multiply the current salary by 5\%
	\displaystyle =	\displaystyle 3500	Evaluate the multiplication

Find a 10\% raise.

\displaystyle 10\% \text{ raise}	\displaystyle =	\displaystyle 70\,000\cdot 10\%	Multiply by 10\%
	\displaystyle =	\displaystyle 7000	Evaluate the multiplication

Now we will find the amount that is half way between \$ 3500 and \$7000.

\displaystyle \text{Average}	\displaystyle =	\displaystyle \dfrac{(3500 + 7000)}{2}
\displaystyle \text{Average}	\displaystyle =	\displaystyle \dfrac{(10 \,500)}{2}	Evaluate the addition
	\displaystyle =	\displaystyle 5250	Evaluate the division

We have found that a 7.5 \% raise is \$ 5250. We can add that to the original salary to estimate the new salary after a 7 \% raise.

\displaystyle \text{New Salary}	\displaystyle =	\displaystyle 70000 + 5250	Add the estimated raise to the original salary
	\displaystyle =	\displaystyle \$75\,250	Evaluate

Mikee's estimated new annual salary is \$75\,250.

Calculate Elena and Mikee's new salaries exactly.

Worked Solution

Create a strategy

Calculate the exact raise of 9\% for Elena and 7\% for Mikee. Multiply their current salary by their raise percentage and then add that to their current salary. This will give them each their new annual salary.

Apply the idea

First we need to find the 9\% raise for Elena.

\displaystyle \text{Raise}	\displaystyle =	\displaystyle 70\,000\cdot 9\%	Multiply the current salary by 9\%
	\displaystyle =	\displaystyle 6300	Evaluate the multiplication

To find Elena's new annual salary, we add the raise to her original salary.

\displaystyle \text{New annual salary}	\displaystyle =	\displaystyle 70000 + 6300	Add the current salary and the raise
	\displaystyle =	\displaystyle \$76\,300	Evaluate the addition

Elena's new annual salary is \$76\,300.

Second we need to find the 7\% raise for Mikee.

\displaystyle \text{Raise}	\displaystyle =	\displaystyle 70\,000\cdot 7\%	Multiply the current salary by 7\%
	\displaystyle =	\displaystyle 4900	Evaluate the multiplication

To find Mikee's new annual salary, we add the raise to his original salary.

\displaystyle \text{New annual salary}	\displaystyle =	\displaystyle 70000 + 4900	Add the current salary and the raise
	\displaystyle =	\displaystyle \$74\,900	Evaluate the addition

Mikee's new annual salary is \$74\,900.

Reflect and check

Recall that we estimated Elena's salary to be \$77 \, 000 and her actual salary is \$76 \, 300. That's a difference of \$77 \, 000 - \$76 \, 300, so we only overestimated by \$700.

We estimated Mikee's salary to be \$75 \, 250 and his actual salary is \$74 \, 900. That's a difference of \$75 \, 250 - \$74 \, 900, so we only overestimated by \$350.

Our estimates were very close to the exact values.

Example 2

A bag of biscuits weighs 120\text{ kg}. The weight of the bag decreases by 48\%.

Estimate the new weight of the bag.

Worked Solution

Create a strategy

Use a 50\% decrease as an estimation for the 48\% decrease.

Apply the idea

50 \% is the same as \dfrac{1}{2} so we just need to find half of the 120 \text{ kg} bag and that is the new weight.

\displaystyle 120 \cdot \dfrac{1}{2}

\displaystyle =

\displaystyle 60

The estimated new weight of the bag is approximately 60\text{ kg}.

Find the new weight of the bag.

Worked Solution

Create a strategy

We can calculate the new weight by determining what is 48 \% of 120\text{ kg} and then subtracting that amount from 120\text{ kg}.

Apply the idea

\displaystyle \text{New Weight}	\displaystyle =	\displaystyle 120 - \left(120 \cdot 48\%\right)	Subtract 48\% of the original weight
\displaystyle \text{New Weight}	\displaystyle =	\displaystyle 120 - \left(120 \cdot \dfrac{48}{100}\right)	Rewrite the percent as a fraction
	\displaystyle =	\displaystyle 120 - 57.6	Evaluate the multiplication
	\displaystyle =	\displaystyle 62.4\operatorname{ kg}	Evaluate the subtraction

The new weight of the bag is 62.4\operatorname{ kg}.

Reflect and check

Decreasing by 48\% is the same as multiplying by a 52\% because 100\%-48\%=52\%. This means that 52\% of the weight will remain after the decrease.

We can also calculate the new weight by multiplying the original weight by 52\%.

\displaystyle \text{New Weight}	\displaystyle =	\displaystyle 120 \cdot 52\%	Multiply current weight by 52\%
	\displaystyle =	\displaystyle 120 \cdot \dfrac{52}{100}	Rewrite the percent as a fraction
	\displaystyle =	\displaystyle 62.4\text{ kg}	Evaluate the multiplication

This confirms the new weight of the bag is 62.4\text{ kg}.

Idea summary

To increase x by y\%, we can calculate: x\cdot \left(100+y\right)\%

To decrease x by y\%, we can calculate: x\cdot(100-y)\%

Calculate percent change

If we want to know the percent by which an amount has increased or decreased, we can calculate the percent change by comparing it to the original amount.

If the new amount is larger than the original amount, the original amount has increased. To find the percent increase, we compare the amount of increase to the original amount using the formula: \text{Percent Increase} = \dfrac{\text{Increase}}{\text{Original Amount}} \cdot 100\%

For example, if a store sold 100 items last month and 120 items this month, the increase in items is 120 - 100=20 items. The percent increase would be \text{Percent Increase} =\dfrac{20}{100} \cdot 100 = 20\%

If the new amount is smaller than the original amount, the original amount has decreased. To find the percent decrease, we compare the amount of decrease to the original amount with the formula:\text{Percent Decrease} = \dfrac{\text{Decrease}}{\text{Original Amount}} \cdot 100\%

For example, if a store sold 200 items last month and 120 items this month, the decrease in items is 200 - 120 =80 items. The percent decrease would be \text{Percent Decrease} =\dfrac{80}{200} \cdot 100 = 40\%

Remember that when calculating percent increase or decrease, the original amount is always the denominator.

Examples

Example 3

Jenna used to earn \$950 per month. Her new monthly salary is \$1235 per month. Determine the percentage increase of Jenna's salary.

Worked Solution

Create a strategy

To calculate the percentage increase, we need to find the difference between Jenna's new salary and her starting salary. Then, we divide this difference by her starting salary and convert it to a percentage.

Apply the idea

\text{Starting salary} = \$950 and \text{New salary} = \$1235

\displaystyle \text{Increase in salary}	\displaystyle =	\displaystyle 1235 - 950	Subtract the starting salary from the new salary
	\displaystyle =	\displaystyle 285	Evaluate the subtraction
\displaystyle \text{Percentage increase}	\displaystyle =	\displaystyle \left(\frac{285}{950}\right) \cdot 100\%	Divide the increase by the previous salary and multiply by 100 \%
	\displaystyle =	\displaystyle 30\%

Jenna's salary increased by 30\%.

Reflect and check

We can check our work by adding Jenna's starting salary to 30 \% of her starting salary to see if it equals her new salary.

\displaystyle \text{New salary}	\displaystyle =	\displaystyle \$950 + \left(\$ 950 \cdot 30\% \right)	Multiply the starting salary by 30 \%
	\displaystyle =	\displaystyle \$950 + \$285	Add the starting salary and amount increased
	\displaystyle =	\displaystyle \$ 1235

This confirms that Jenna's salary increased by 30\%.

Example 4

Last year, a local bookstore sold an average of 300 books per month. This year, the average monthly sales dropped to 216 books. Determine the percentage decrease in the bookstore's monthly book sales.

Worked Solution

Create a strategy

To find the percent decrease, subtract this year's average monthly sales from last year's, then divide the result by last year's average and convert to a percent.

Apply the idea

\text{Last year's sales} = 300 and \text{This year's sales} = 216

\displaystyle \text{Decrease in sales}	\displaystyle =	\displaystyle 300 - 216	Subtract this year's sales from last year's sales
	\displaystyle =	\displaystyle 84	Evaluate the subtraction
\displaystyle \text{Percentage decrease}	\displaystyle =	\displaystyle \left(\frac{84}{300}\right) \cdot 100\%	Divide the decrease by the last year sales and multiply by 100 \%
	\displaystyle =	\displaystyle 28\%	Evaluate

The percentage decrease in the bookstore's monthly book sales is 28\%.

Reflect and check

We can check our work by subtracting 28 \% of last year's sales of 300 books from last year's total amount.

\displaystyle \text{New book sales}	\displaystyle =	\displaystyle 300 - \left(300 \cdot 28\% \right)
	\displaystyle =	\displaystyle 300 - 84	Evaluate the multiplication
	\displaystyle =	\displaystyle 216	Evaluate the subtraction

This confirms that the book store had a 28 \% decrease in sales.

Idea summary

To find the increase as a percent of the original quantity, write it as a fraction and multiply by 100\%: \text{Percent Increase}=\dfrac{\text{Increase}}{\text{Original quantity}}\cdot 100\%

To find the decrease as a percent of the original quantity, write it as a fraction and multiply by 100\%: \text{Percent Decrease}=\dfrac{\text{Decrease}}{\text{Original quantity}}\cdot 100\%

Outcomes

8.CE.1

The student will estimate and apply proportional reasoning and computational procedures to solve contextual problems.

8.CE.1c

Estimate and solve contextual problems that require the computation of the percent increase or decrease.

1.04 Percent increase and decrease

Ideas

Percent increase and decrease

Exploration

Examples

Example 1

Example 2

Calculate percent change

Examples

Example 3

Example 4

Outcomes

8.CE.1

8.CE.1c

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