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

The Must Know SAT Math Words

When I took the math portion of the SAT, there was one type of question I was not looking forward to—the word problems. With symbols, numbers, variables, equations, and words all combined into one, these problems were stacked with different types of information so much so that my head would literally burst trying to figure what the question was even asking me. It wasn't until grinding problem after problem that I began seeing a pattern and figuring out a strategy in getting these word problems correct.

The Strategy that Helped me:

The best strategy that helped me was translating key words into math. When I cyphered though a word problem, I would cross of the key terms I knew referenced a math concept, and replace it with a math symbol I was aware of. That way, I was sort of condensing this long, verbose word problem into this pseudo-algebraic statement that I could understand.

Math Keyword List:

So here is a comprehensive list of the most important SAT key words that I created to get the math word problems under check. I formed the list based on the 8 SAT practice tests and almost all the old math SAT practice exams that I had took in order to provide a comprehensive list to help you conquer the SAT math word problems.

Practice Problem Set 1

$1$.
Suppose John has only green and red apples in his grocery bag. If the ratio of the number of green to red apples in his bag is $(k - 5)$ to $(k - 3)$ where $k$ is an integer greater than $4$, what's the probability John will randomly pick a red apple from the bag in terms of $k$?

(a) $\frac{k - 3}{2(k - 4)}$

(b) $\frac{k - 3}{k - 4}$

(c) $\frac{k-3}{4}$

(d) $\frac{k - 5}{2k - 8}$

$2$.
If $(2n + 1) > (5n + 1)$, then what can we say about $n$?

(a) $n < 0$

(b) $n > 0$

(c) $n = 0$

(d) None of the above

$3$.
Suppose $k$ is an integer such that $x = 2k$ and $y = 2k + 1$. What might $k$ be such that $\frac{y}{x}$ is an integer?

(a) $\frac{1}{2}$

(b) $1$

(c) $0$

(d) None of the above

$4$.
For what value(s) of $n$ is $n^2$ not equal to $2n$?

(a) $n = 0$

(b) $n = 2$

(c) both (a) and (b)

(d) None of the above

$5$.
Given the expressions below, what is the numerical value of $x$?
   (i)   $4$ of $x$ is $5$ more than $y$

   (ii)  $z$ divided by $y$ is $2$ less than $2$ of $y$

   (iii) $y$ divides $z$ is $2$ fewer than the second largest factor of $12$

(a) $x = 1$

(b) $x = 2$

(c) $x = 3$

(d) $x = 4$

For numbers $6$ and $7$ please refer to the description below:

There are $500$ students at the prestigious Redacted University. Recently the flu has been circulating around the area causing many people to get sick. Suppose that $\frac{1}{5}$ of all students are sick due to the flu at school today and the rest are healthy. The next day, a staggering $\frac{3}{5}$ of healthy students become sick and only $\frac{1}{5}$ of sick students become healthy.

$6$.
What is the proportion of all students who are healthy the next day?

(a) $\frac{1}{25}$

(b) $\frac{8}{25}$

(c) $\frac{9}{25}$

(d) $\frac{16}{25}$

$7$.
Exactly how many students are healthy the next day?

(a) $20$ students

(b) $160$ students

(c) $180$ students

(d) $320$ students

$8$.
What answer choice is equivalent to the expression $x^3 - 64$?

(a) $(x - 4)(x^2 + 4x + 16)$

(b) $(x - 4)(x^2 - 4x + 16)$

(c) $(x - 4)^3$

(d) None of the above

$9$.
What answer choice is equivalent to the expression $x^3+ 2x^2 - 4x - 8$?

(a) $(x - 2)(x^2 + 4)$

(b) $(x^2 + 4)(x + 2)$

(c) $(x^2 - 4)(x^2 + 4)$

(d) $(x-2)(x + 2)^2$

Challenge Problem:

$10$.
Suppose k is an integer such that $x = 2k$ and $y = 2k + 1$. What can we say about $x + y$?

(a) $x + y$ is always even

(b) $x + y$ is always odd

(c) Can be both even or odd

(d) Neither even nor odd

Answer key:

1.   a
2.   a
3.   d
4.   d
5.   b
6.   c
7.   c
8.   a
9.   d
10. b