Percent Change

If you are asked to calculate a percentage change between two values, there is an easy formula. 

$$frac{new-old}{old}$$

Example:

If the population of a certain town was 28,350 in the year 2000, and 25, 785 in the year 2010, by what percent did the population decrease by?

$$frac{25785;left(new;valueright)-28350;(old;value)}{28350;(old;value)}$$

Performing this calculation gives us (-.09), which we can convert to percentage by moving the decimal point two places to the right, (-9%).  Because it is negative, we know it is a decline.  The population of the town decreased by (9%) over the 10 years.

It can get tricky: One way that the SAT likes to try and trick students is by asking you to solve for the old value, from the new value and a percentage.

For Example:

If you buy a t-shirt at Urban Outfitters during a 20% off clearance sale for $20, what was the original price of the t-shirt before it was discounted?

A) 25
B) 24.5
C) 24
D) 23.5

The percent is ALWAYS applied to the old value in order to get the new value.

$$left(mathrm{percentage};mathrm{in};mathrm{decimal};mathrm{form}right)left(mathrm{old};mathrm{value}right)=mathrm{new}$$

Or, in the case of a discount like above:

$$(1-mathrm{percentage};mathrm{in};mathrm{decimal};mathrm{form})(mathrm{old};mathrm{value})=mathrm{new};mathrm{value}$$

Using this formula, we plug in our discount and our new value to solve for (x), which is our old value. Dividing both sides by .8 yields (mathrm x;=frac{20}{.8}), which is 25. The original cost of our t-shirt is 25. A is the correct answer.:

$$left(1-.2right)left(mathrm xright)=20\.8mathrm x=20\mathrm x=20/.8\mathrm x=1.25(20)\mathrm x=25$$

Many people get tripped up and think that they can just find 20% of 20 and add it to 20.

$$mathrm x=20+.2(20)\mathrm x=1.2(20)\mathrm x=24$$

THIS DOES NOT WORK

That method fails because the percentage we get by starting with our old price is going to be different from just applying the percentage to our new price

$$frac1{.8}=1.25neq1.2=1+.2$$

ALWAYS APPLY THE PERCENT TO THE BASE OR OLD VALUE

Discount, Tips, and Taxes

Discounts should be calculated using ((1-%;mathrm{discounted})). The 1 stands for 100% of the item, so we are subtracting the discount from the total item.

Tips and taxes should be calculated using ((1+%;mathrm{tip};mathrm{or};mathrm{tax})). The 1 again stands for 100% of the original item, so we are adding on the tip or tax to that.

Example:

Sarah buys a laptop for 980 dollars, but she also must pay a 6% sales tax. How much did her total purchase cost her? (Round to the nearest dollar)

A) 986
B) 1,006
C) 1,039
D) 1,048