Comprehensive Guide to Percents on SAT

Percents in a Nutshell 

Definition:

Percent is commonly defined as some number $x$ for every $100$. This is quite easy to remember if you break up the word percent into its prefix and suffix––per $\cdot$ cent where per means for every and where cent stands for century or $100$. So for example, $50$ per $\cdot$ cent translates to $50$ for every $100$.

Formula:

We can generalize percents into a formula that we can then manipulate. The standard formula is: $$\frac{percent}{100\%} =  \frac{part}{whole}$$ where $\%$ is the unit for percent (in fact the $\%$ symbol literally means percent), where $whole$ is the entirety of a thing (or the denominator of a fraction), and where $part$ is just a section of the $whole$ (or numerator of a fraction). This formula is powerful because it gives you a fractional or decimal equivalent to percent and vice versa. This means that any rational number can be transformed into a percent, and any percent can be transformed into a rational number. So, for instance, the fraction or decimal equivalent of $50 \%$ is: $$\frac{part}{whole} = \frac{50\%}{100\%} = \frac{1}{2} = 0.5$$

Let's try this formula with an example. Suppose you knew that $\frac{1}{5}$ of a pizza has only mushroom side toppings on it. If $\frac{1}{5}$ of the pizza has only mushrooms, then $1$ part of the whole pizza, which is totally $5$ parts, make up the mushroom sidetopping. Using the formula, the percent of the pizza that has only mushrooms is $$percent = \frac{part}{whole} * 100 \% = \frac{1}{5} * 100\% = 20 \%$$

You can also say that $\frac{4}{5}$ of your pizza has no mushrooms on it, and therefore the percent of your pizza with no mushrooms is $$\frac{4}{5} * 100 \% = 80 \%$$

A whole number is a sum of parts. So like how  $\frac{1}{5}$ mushroom toppings and $\frac{4}{5}$ other topics add to 1 pizza with all toppings($\frac{1}{5} + \frac{4}{5} = \frac{5}{5} = 1$), percents work the same way: $$total\:percentage = 20\% + 80\% = 100\%$$

$100\%$ indicates a whole value (the entire pizza in the example) while a percent value $< 100\%$ indicates a part (part of pizza in example). This doesn't mean you can't have a percent $> 100\%$ as if you have a $\frac{part}{whole} > 1$ like $\frac{7}{5}$, its equivalent percent value is $$\frac{7}{5} * 100\% = 140\%$$

Now that we got the formal definition and formula for percents, let's try some basic percent problems.

Basic Percent Problems:

(1) You have two legendary Pokemon cards and 18 common Pokemon cards in your card binder. What percent of all Pokemon cards are legendary in your card binder?

Solution:
Since $percent = \frac{part}{whole} * 100 \%$ then: $$percent = \frac{2}{2 + 18} * 100 \% = \frac{2}{20} * 100 \% = 10 \%$$

(2) What is $5 \%$ of 17? Write your answer as a fraction.

Solution:

By our definition, we know that a percent is $\frac{part}{whole} * 100 \%$.  Here we kinda have to work backward—we have to convert percent to $\frac{part}{whole}$ because we need a numerical answer. So: $$\frac{part}{whole} = \frac{5\%}{100\%} = \frac{1}{20}$$

Since $whole = 17$: $$part = \frac{1}{20} * whole = \frac{1}{20} * 17 = \frac{17}{20}$$

Now try these ones out on your own. The answers are at the bottom of the page.

(3) What percent of $70$ is $15$? 

(4) $100$ is what percent of $65$?

Percent Change in a Nutshell

Percent change is a percent topic you will most likely need to know for the exam. The standard formula is $$\Delta\: percent = \frac{final\:value - initial\:value}{initial\: value} * 100\%$$

Percent Change Key terms to know:

• If $\Delta\: percent > 0$, then we call this a percent increase.

• If $\Delta\: percent < 0$, then we call this a percent decrease.

• Discount: A discount is when we get a certain $\%$ off of our price. It's the same thing to percent decrease! If you see discount, substitute that value for the percent change value. This word is used a lot on the SAT exam and many students get tripped up on it simply because they don't know what it means.

• Sales Tax: This is when you have to pay an extra charge on the total cost. This is the same thing as percent increase! Note that a sales tax is applied on the total cost, which can include the discount if there was any applied. That means, the sales tax is always applied after discount is applied, unless stated otherwise.

Discount and Sales Tax Practice Problem:

Discount Practice Problem: 
If I get a $10 \%$ discount on a shirt worth $\$20$, what is my shirt now worth?

Solution:

(Step 1) Realize that $\Delta\: percent = -10 \%$ because discount is always percent decrease

(Step 2) Use the percent change formula to get the actual price of the shirt: $$final = \frac{\Delta\:percent * initial}{100} + initial = \frac{-10 * 20}{100} + 20 = 18$$

So the shirt which was initially $\$20$, is worth $\$18$ after applying the $10\%$ discount.

Tax Practice Problem: 
The base price of the shoes is $\$10$. If a $\$5\%$ tax is applied upon purchase, what is the total price of the shoes after purchase?

Solution:

(Step 1) Realize that $\Delta\: percent = + 5 \%$ because sales tax is always percent increase

(Step 2) Use the percent change formula to get the final price of the shoes: $$final = \frac{\Delta\:percent * initial}{100} + initial = \frac{5 * 10}{100} + 10 = 10.50$$

So the shoes which was initially $\$10$, is worth $\$10.5$ after being charged a $5\%$ sales tax.


Try this Problem On your Own: (Answer on bottom of the page)
You bought a Hunter X Hunter Gon bobble-head for $\$10$. After a coupon you received, the price you had to pay at checkout was only $\$6.50$. By how much percent did the coupon change your price? Assume no sale tax was added.

SAT Percent Practice Questions:

[[1]] 

$x + 2$ is $25\%$ of $y$. If $y = 2 - z$ what is $x$ in terms of $z$?

(a) $x = \frac{3}{2} - \frac{1}{4}z$

(b) $x = -\frac{3}{2} - \frac{1}{4}z$

(c) $x=\frac{1}{2} - \frac{1}{4}z$

(d) $x=- \frac{1}{2} + \frac{1}{4}z$


[[2]]

Sally decides to buy onions, which costs $x$ dollars per pound, and tomatoes, which costs $w$ dollars per pound. She has a coupon that says if she buys 2 pounds of tomatoes, she will get a $15\%$ discount on her entire purchase. If she buys 3 pounds of onions and 5 pounds of tomatoes and uses her coupon, how much will she have to pay at the register including a sale tax of $5\%$? Write in terms of constants, $x$, and/or $w$?

(a) $(1.05)(0.85)(3x + 5w)$

(b) $(0.95)(1.15)(3x + 5w)$

(c) $\frac{15(1.05)(0.85)}{(x + w)}$

(d) $\frac{15(0.95)(1.15)}{(x + w)}$

[[3]]

If $x$ is increased by $35\%$ and y is decreased by $75\%$, by what percent did $x - y$ change if $x = 2$ and $y = 1$? Round your answer to the nearest integer.

[[4]]

There are $3$ apples in a basket. If $5$ more apples are put into the basket, by how much percent did the number of apples increase in the basket. Round your answer to the nearest integer.

SAT Practice Question Answers:

[[1]] b

[[2]] a

[[3]] $45$

[[4]] $67$

Basic Percent Problem Answers:

(3) $21\frac{3}{7}\%$

(4) $153\frac{11}{13}\%$

Percent Change Answer:

$35\%$ decrease