Lesson

Percentages are used in everyday life to represent proportions and to show rates of increases and decreases. Some examples of percentages include:

- Discounts for example $30%$30% off
- Interest rates for example $2%$2% per year
- The amount of battery charge on a device for example $64%$64% remaining

The words "per cent" mean "per $100$100 parts". For example, $2%$2% means $2$2 out of every $100$100 parts. Percentages are equivalent to fractions out of $100$100 and can be written with a $%$% sign, or as a fraction or decimal. For example

$35%=0.35=\frac{35}{100}$35%=0.35=35100

When using percentages in the real world, we often deal with changes to amounts using percentages, whether they're increases or decreases. In both cases, we can go through some shortcuts to find our final amount.

When calculating with percentages, it is usually a good idea to convert the percentage to a fraction or a decimal first.

Let's say we wanted to increase $40$40 by $2%$2% and find the end amount.

We could first find $2%$2% of $40$40, which we can work out here (expressing the percentage as the fraction $\frac{2}{100}$2100):

$\frac{2}{100}\times40=0.8$2100×40=0.8

The final amount would then be the original amount $40$40 plus this extra $2%$2% of $40$40. That is, the final amount would be

$40+0.8=40.8$40+0.8=40.8

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

To find the total amount after the increase, we are essentially finding $100%+2%=102%$100%+2%=102% of $40$40.

Therefore we can just work out the increase this way:

$\frac{102}{100}\times40=40.8$102100×40=40.8

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

Increase by a percentage

To increase $x$`x` by $y%$`y`%, we can calculate

$x\times\left(100+y\right)%$`x`×(100+`y`)%

Decreasing by a percentage has a similar shortcut.

For example, if we want to calculate a $25%$25% discount off $\$60$$60 we could find $25%$25% of $60$60 first, which is $60\times0.25=15$60×0.25=15, and subtract this from $60$60, giving $60-15=45$60−15=45.

But discounting a price by $25%$25% is the same as just paying $75%$75% of the price. So the easier way to find this amount is to calculate

$0.75\times\$60=\$45$0.75×$60=$45,

which is the same as the answer we got before.

So to decrease an amount by a percentage, we can just multiply the amount by ($100%$100% - percentage).

Decrease by a percentage

To decrease $x$`x` by $y%$`y`%, we can calculate

$x\times\left(100-y\right)%$`x`×(100−`y`)%

We want to increase $1300$1300 by $40%$40% by following the steps outlined below.

First find $40%$40% of $1300$1300.

Add the percentage increase to the original amount to find the amount after the increase.

Calculate $140%$140% of $1300$1300.

Is increasing an amount by $40%$40% equivalent to finding $140%$140% of that amount?

Yes

ANo

B

We want to decrease $1500$1500 by $15%$15% by following the steps outlined below.

First find $15%$15% of $1500$1500

Subtract the percentage decrease from the original amount to find the amount after the decrease.

Calculate $85%$85% of $1500$1500

Is decreasing an amount by $15%$15% equivalent to finding $85%$85% of that amount?

Yes

ANo

B

A bag of rice weighs $110$110kg. If the weight of the bag decreases by $40%$40% find the new weight of the bag.

Analyze proportional relationships and use them to model and solve real-world and mathematical problems.

Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease