## Common Math Tips for the SAT

1. Multiple choice math questions increase in difficulty from the beginning of a section to end. The difficulty restarts for the student-response questions.
2. Always put an answer (even for student-response questions). If you don’t know the answer, put a reasonable answer or 1.
3. All figures are drawn to scale, unless otherwise noted. This means angles and lengths are drawn to scale.
4. You can’t bubble in a negative answer in student-response questions.

## Common Math Tricks

Sometimes to find the answer to a system of equations question, we may be asked to find the value of $x+y$ or $x-y$. In this situation, we likely don’t have to solve for x or y individually. It is possible to find the value of the expression without solving for $x$ or $y$.

Example

If  $3x+2y=30$ and $2x+3y=5$, what is the value of $x+y$?

Explanation & Solution

In this case, we can solve for $x$ and $y$ using elimination. However, we don’t have to.

If we add the equations together, we get $5x+5y=35$.

The question asks us to solve for $x+y$, so we can simply divide both sides of the equation by 5 to get $x+y=7$.

### Use Factors

Similar to the previous tip, we can sometimes find the answer by factoring. If we have to find the value of an expression, the expression may be a factor of a polynomial in the equation.

This type of question commonly include perfect squares. For example, we would factor $x^2 -y^2$ to $(x+y)(x-y)$.

Example

What is the average of $x$ and $y$, if $x^2-y^2=54$ and $x-y=9$?

Explanation & Solution

The question is asking for the average of $x$ and $y$. To find the average of $x$ and $y$, we would have to add the two together and then divide by two, so we need to know what $x+y$ equals.

We can use factors to solve for this.

We are given the equation

$x^2-y^2=54$,

and we can factor the left side of the equation to get

$(x+y)(x-y)=54$.

We are looking for the value of $x+y$, and we are given that $x-y=9$, so we can substitute 9 in to get

$(x+y)(9)=54$,

Divide both sides by 9 to get

$x+y=6$

The average of $x$and $y$ is $6\div2 =3$.

If numerical answers are given, it may be possible to try the answer choices and work backwards. Make sure to pay attention to what the question is asking for, so we know what to test each answer as. For exmplae. if the question asks to find the average cost of sneakers, then when we test the asnwers, we should test them as the average cost of sneakers and work backwards.

Example

The length of a rectangle is 5 more than its width. If the area of the rectangle is 66 sq in, what is the length, in inches, of the rectangle?

1. 5
2. 6
3. 8
4. 11

Explanation & Solution

Lets try choice A.

5 would equal the length. The question states that the length is 5 more than the width, which would make the width 0. That’s not possible, so A is wrong.

Let’s try choice C.

8 would equal the length. The questions states that the length is 5 more than the width, so the width would be 3.

These dimension would give an area of 24. (Area of a rectangle is length times width). However, the question states that the area is 66, so these dimensions don’t work. C is not the answer.

Try choice D.

11 would be the length, and that would make the width 6. The area of this rectangle would be 66 (6 times 11 is 66). This matches what the question states, so D is the answer.

### Pick Numbers

When the answer choices are algebraic expressions, we can pick numbers for the values of the variables. We have to substitute those values in the question to find the numeric answer. After, we substitute the values into the variables in the answer choices to see which answer choice gives us the right numerical value.

Example

Which of the following is equivalent to $\dfrac{3t}{4x-1}-\dfrac{t+1}{2}$, if $x!=-\dfrac{1}/{4}$?

A. $\dfrac{5xt-1}{8x-2}$

B. $\dfrac{-4xt+7t-4x+1}{2(4x-1)}$

C. $\dfrac{2(2xt+1)}{4x-1}$

D. $\dfrac{-4xt+5t}{2}$

Explanation and Solution

Let’s pick 4 for $x$ and 3 for $t$.

Substitute 4 and 3 into the expression and evaluate. $\dfrac{3(3)}{4(4)-1}-\dfrac{(3)+1}{2}$ gives us $\dfrac{9}{15}-\dfrac{4}{2}$, then reduce $\dfrac{3}{5}-2$ becomes $-\dfrac{7}{5}$.

Substitute the value into the answer choices and see which one also gives us $-\dfrac{7}{5}$.

A. $\dfrac{5(4)(3)-1}{8(4)-2}$
which gives us $\dfrac{59}{30}$
B. $\dfrac{-4(4)(3)+7(3)+4(4)-1}{2(4(4)+1)}$ which gives us $\dfrac{-42}{30}$ and that reduces to $-\dfrac{7}{5}$. This matches the answer, so B is the answer.

You should double check to make sure C or D doesn’t also give you the right answer. If one of them does, you need to pick new numbers and do the process over again, but only check the two answers choices that worked .

## Combining Like Terms

Like terms have the same combination of variables and exponents but may have different coefficients.

When combining like terms, simply add the coefficients together.

$4x^2+6x^2$

In the example above, the coefficients are 4 and 6. We simply add them together to get

$10x^2$

More examples:

$9x^3y^2+2x^3y^2=11x^3y^2$$-12a^3b^2c^6+a^3b^2c^6=-11a^3b^2c^6$

We only add coefficients together of like terms. In the example below, there are many terms and not all of them are like terms.

$12x-4y+3x^2-20y+3xy-4x +2xy$

We can combine $12x$ and $-4x$ to get $8x$.

Notice that I treated the minus sign in front of  as a negative sign. Make sure to take the signs in front of the numbers along with the terms when combining like terms.

We can combine $-4y$ and $-20y$ to get $-24y$

We can also combine $3xy$ and $2xy$ to get $5xy$

That’s all of the like terms, so we end up with $8x+5xy+3x^2-24y$

When questions ask us to add and subtract expressions, we are really being asked to combine like terms.

Example
What is the sum of $x^2-2x+7$ and $2x^2-3x-4$?

Explanation & Solution
To solve this let’s add them together.
$(x^2-2x+7)+(2x^2-3x-4)$
Distribute the plus sign $(+1)$ to each of the terms in the second expression. When you do that, all of the terms remain the same and we can drop the parentheses.
$x^2-2x+7+2x^2-3x-4$
Now, just combine like terms to get
$3x^2-5x+3$

Example
Simplify $(4x^2-5x+2)-(2x^2+x-3)$

Explanation & Solution
Distribute the minus sign $(-1)$ to each of the terms in the second expression. When you do that, all of the terms get negated and we can drop the parentheses.
$4x^2-5x+2-2x^2-x+3$
Now, just combine like terms to get
$2x^2-6x+5$

## Solving Equations

### Distribute or Divide

If there is a number outside a set of parentheses, you can choose to distribute it or divide it.

The equation $3(x-4)=15$ can be solved two different ways.

We can either distribute the $3$ into the expressions or divide it.

Example of Distributing

$3(x-4)=15$

distribute the $3$ into the parenthesis and the equation becomes

$3x-12=15$

add $12$ to both sides of the equation

$3x=27$

divide both sides by $3$ to get

$x=9$

Example of Dividing First

$3(x-4)=15$

divide both side by $3$ to get

$x-4=5$

add $5$ to both sides of the equation to get

$x=9$

With both methods we are able to get $x=9$. However, on the SAT there may be questions in which one method works better than another.

Example

$9(2x+3)=36$. What is the value of $2x+3$?

Explanation & Solution

In this question, the expression we are solving for can be found directly.

If we divide both sides by $9$, we will have the answer immediately. It would be a waste of time to distribute the $9$.

$9(2x+3)=36$

divide both sides by $9$ to get

$2x+3=4$

The answer is $4$.

### Multiply by Least Common Denominator (LCD) to Eliminate Fractions in Equations

The following equation

$\dfrac{2}{3}x-\dfrac{4}{5}=\dfrac{1}{6}$

can be intimidating for students.

Many students are not comfortable working with fractions. Plus, there is room to make mistakes with fractions.

We can multiply all of the terms in the equation by the LCD to cancel out the denominators, so we are left with integers.

Example

What is the value of $x$ in the following equation?
$\dfrac{2}{3}x-\dfrac{4}{5}=\dfrac{1}{6}$

Explanation & Solution

$\dfrac{2}{3}x-\dfrac{4}{5}=\dfrac{1}{6}$
The denominators in the equation above are 3, 5, and 6.
To find the least common denominator, we need the least common multiple of the denominators.

In this case, the LCD is $30$.

We should multiple each term in the equation by 30 to get
$(30)\dfrac{2}{3}x-(30)\dfrac{4}{5}=(30)\dfrac{1}{6}$

Reduce each term to get

$(10)2x-(6)4=(5)1$

Simplifying further gives us

$20x-24=5$

Now, we can solve like normal.

Add $24$ to both sides to get

$20x=29$

Divide both sides by $20$ to get

$x=\dfrac{29}{20}$

### Cross Multiply When a Fraction Equals a Fraction

If you have an equation in which one fraction equals another fraction, it is a proportion. To solve a proportion you should cross-multiply.

It is best to simplify each fraction as much as possible before cross-multiplying.

Example

Solve for $x$ in the equation below.

$\dfrac{2x-7}{4}=\dfrac{2x-1}{12}$

Explanation & Solution

$\dfrac{2x-7}{4}=\dfrac{2x-1}{12}$

Cross-multiply to get

$(2x-7)(12)=(2x-1)(4)$

Simplify both sides

$24x-84=8x-4$

Since there is an $x$-term on both sides of the equation, we should get them both on one side.

Subtract $8x$ from both sides. (You can subtract $24x$ from both sides instead. It is up to you)

$16x-84=-4$

Add $84$ to both sides to get

$16x=80$

Divide both sides by $16$

$x=\dfrac{80}{16}$

Reduce

$x=5$