Let's say we wanted to increase 40 by 5\%.
We could first find 5\% of 40, which we can work out by expressing the percent as the fraction \dfrac{5}{100}:
\displaystyle \frac{5}{100}\cdot 40 | \displaystyle = | \displaystyle \frac{200}{100} |
\displaystyle \frac{200}{100} | \displaystyle = | \displaystyle 2 |
The final amount would then be the original amount 40 plus this extra 5\% of 40. That is, the final amount would be40+2=42
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.
Use this applet to explore percent increase and decrease using a single multiplier.
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}).
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.
Estimate Mikee's new salary.
Calculate Elena and Mikee's new salaries exactly.
A bag of biscuits weighs 120\text{ kg}. The weight of the bag decreases by 48\%.
Estimate the new weight of the bag.
Find the new weight of the bag.
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)\%
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.
Jenna used to earn \$950 per month. Her new monthly salary is \$1235 per month. Determine the percentage increase of Jenna's salary.
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.
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\%