# SAT Formula Sheet

### Is memorizing formulas enough to succeed in the SAT exam?

Memorizing formulas for the SAT exam is not enough!

First of all, you need to understand each formula to decide if it is the right formula needed to solve the question.

Secondly, you need to know how to apply the chosen formula correctly in the context of the question.

Therefore, you need to learn the formulas and practice to apply them in different questions.

### Why it is the best way to study formulas from this page

In addition to the formulas that are given in the SAT math section we teach many additional formulas that are needed for the SAT exam.

The formulas are divided into groups according to the relevant SAT subscore, so you can start practice solving all subscore questions after you finish learning its formulas.

Learn how and why the formula works, so it will be much easier to remember the formula and use it.

Each formula is accompanied by a numeric example for a basic understanding of its components. In addition, each formula is accompanied by exercises to test deeper understanding of the formula and make sure you know how to implement the formula correctly and fast.

### Before you start learning the formulas

#### Formula components

Note that formulas include numbers, letters and variables (x, y, z):

- The numbers have a known value.
- The letters represent unknown quantities.
- The variables can change.

#### The ± symbols in formulas

Note that the ± symbols in formulas have a multiple meaning, so you must choose the top sign (+) or the bottom sign (-).

Once you chose the position of the first symbol, you must continue to choose the same position in all other symbols that follow.

#### Finding which formula to use

To solve a question, you may need to substitute given numbers for variables and letters to find which formula to use and only then solve with this formula.

For example: Solving the expression 35*21-35*20 requires to use the formula ax-bx =(a-b)x.

ax-bx=35*21-35*20 so that x=35 (this is the value that is the same in both parts of the expression so it must be x), a=21 and b=20.

Since the formula is ax-bx=(a-b)x we get 21*35-20*35=(21-20)*35=1*35=35.

### Basic algebra formulas

#### Distributing formula for removing parentheses: (a±b)x=ax±bx

This is a very basic and known formula. The formula has 2 forms (minus or plus):

(a+b)x=ax+bx

(a-b)x=ax-bx

According to this formula (with a plus sign) instead of adding a and b and then multiplying the sum by x you multiply a by x and then multiply b by x and after that sum the 2 outcomes.

Consider the following examples for using this formula with plus and minus signs:

**An example for solving this formula with a plus sign:**

There were 5 boxes in the warehouse, each box contained 2 red balls and an unknown number of blue balls. If the total number of balls was 15, that is the number of blue balls in each box?

(2+x)5=15

10+5x=15

5x=5

x=1

Checking the answer: (2+1)5=3*5=15.

**An example for solving this formula with a minus sign:**

Each box in the warehouse contained 10 products, the worker decided to send less products, so he took an unknown number of products from each box. If the total number of boxes was 5 and the total number of products was 15, that is the number of products the worker took from each box?

(10-x)5=15

50-5x=15

-5x=-35

x=7

Checking the answer: (10-7)5=3*5=15.

#### Combining like terms formula: ax±bx=(a±b)x

This formula is the same as the previous distributing formula for removing parentheses, you just need to switch the left and the right sides of the formula.

Consider the following examples for using this formula with plus and minus signs:

**An example for solving this formula with a plus sign:**

Each table costs 2,000 dollars and each chair costs 500 dollars. If the restaurant needs to purchase 8 tables and there are 6 chairs in each table, what is the total cost for the restaurant?

Note that we can present 2,000 as 500*4.

For the cost of the tables we have the expression (500*4)*8

For the cost of the chairs, we have the expression 500*6*8

To find the total cost we need to solve (500*4)*8 + 500*6*8

We can use the combining like terms formula so that x=500*8 (this are the values that are the same in both parts of the expression so they should be x) getting:

500*4*8 + 500*6*8 = 500*8(4+6) = 500*8*10 = 5,000*8 = 40,000.

Note that we could also define x as x=500*8*2 getting 500*8*2(2+3) = 4,000*2*5 = 8,000*5 = 40,000.

**An example for solving this formula with a minus sign:**

The cost of one computer is 400 dollars. If there were 20 computers in the inventory and 17 were sold, what is the value of the inventory left?

The inventory was at first 400*20

The cost of the computers sold is 400*17

Therefore, the expression we need to solve is 400*20-400*17

We can use the combining like terms formula so that x=400 (400 is the value that is the same in both parts of the expression so it should be x) getting:

400*20-400*17 = 400(20-17) = 400*3 = 1,200

#### Solving with 2 formulas: the formula ax-bx=(a-b)x and the formula (a-b)x=ax-bx

In the following example we need to use the combining like terms formula ax±bx=(a±b)x and then use the distributing formula for removing parentheses (a±b)x=ax±bx.

The cost of one computer is 498 dollars. If there were 20 computers in the inventory and 5 were sold, what is the value of the inventory left?

The inventory was at first 498*20

The cost of the computers sold is 498*5

Therefore, the expression we need to solve is 498*20-498*5

We can use the combining like terms formula so that x=498 (498 is the value that is the same in both parts of the expression so it should be x) getting:

498*20-498*5 = 498(20-5) = 498*15

To calculate the answer fast we can replace 498 with 500-2 getting (500-2)*15

Now we can use the distributing formula for removing parentheses: (a±b)x=ax±bx getting

(500-2)*15 = 500*15 – 2*15 = 500*10+500*5-30 = 5,000+2,500-30 = 7,500-30 = 7,420

### Calculations with fractions formulas

#### Fractions addition and subtraction formula a⁄b±c⁄b=(a±c)⁄b

This formula involves adding or subtracting two fractions with a same denominator.

The formula has 2 forms (minus or plus):

a⁄b+c⁄b=(a+c)⁄b

a⁄b-c⁄b=(a-c)⁄b

Since the denominator is the same, we can add or subtract the nominators first and then divide the result by the denominator.

Consider the following example for using this formula with plus and minus signs:

There pool was 13/15 filled with water. During the first week 3/15 of the water evaporated from the pool, during the second week 1/15 of the water evaporated from the pool. The poolman filled 6/15 of the pool with water. What part of the pool is filled with water?

We need to solve the expression 13/15-3/15-1/15+6/15, we need to use addition and subtraction formula: a⁄b±c⁄b=(a±c)⁄b getting

13/15-3/15-1/15+4/15 = (13-3-1+6)/15 = 15/15 = 1

The pull is full.