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**

*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

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