Fraction math

Understanding how to perform basic operations (addition, subtraction, multiplication, and division) on fractions is an essential GRE skill. If you’re already comfortable with fractions, you can skip to the quiz at the end.

Definitions

Numerator

The top part of a fraction. For example:

• The $1$ of $\frac{1}{4}$
• The $x$ of $\frac{x}{y}$
• The $1+x$ of $\frac{1+x}{4+y}$

Denominator

The bottom part of a fraction. For example:

• The $4$ of $\frac{1}{4}$
• The $y$ of $\frac{x}{y}$
• The $4+y$ of $\frac{1+x}{4+y}$

How to add and subtract fractions

Adding and subtracting fractions works the same way as adding and subtracting whole numbers. Subtraction is just adding a negative.

There are two key steps for fraction addition/subtraction:

1. Adjust the fractions so they have the same denominators
2. Add (or subtract) the numerators

To adjust a fraction without changing its value, multiply it by $1$ written as a fraction (like $\frac{2}{2}$ or $\frac{6}{6}$). For example, to rewrite $\frac{1}{2}$ with a denominator of $4$, multiply by $\frac{2}{2}$:

$$\frac{1}{2}\ast \frac{2}{2}=\frac{2}{4}$$

This doesn’t change the value of the number (because you’re multiplying by $1$). It only changes how the same value is written.

Let’s consider a simple example:

What is $\frac{1}{2}+\frac{3}{4}$?

Because the denominators are different, first rewrite $\frac{1}{2}$ with a denominator of $4$, then add.

$$\begin{array}{rl}x& =\frac{1}{2}+\frac{3}{4}\\ & =(\frac{1}{2}\ast 1)+\frac{3}{4}\\ & =(\frac{1}{2}\ast \frac{2}{2})+\frac{3}{4}\\ & =\frac{2}{4}+\frac{3}{4}\\ & =\frac{5}{4}\end{array}$$

Now try another example, this time with variables.

$3y/2+y/3-2y/12=1$

Solve for $y$.

Do you know the answer?

First, rewrite each term so all fractions have a common denominator. It’s often best to use the least common multiple (LCM), but any common multiple works. Here, we’ll use $12$, so we need to adjust the first two terms.

Start with $3y/2$. To change the denominator from $2$ to $12$, multiply by $6/6$:

$$3y/2\ast 6/6=18y/12$$

Next, adjust $y/3$. To change the denominator from $3$ to $12$, multiply by $4/4$:

$$y/3\ast 4/4=4y/12$$

Now substitute these equivalent fractions into the equation. With matching denominators, you can combine the numerators directly.

$$\begin{array}{rl}3y/2+y/3-2y/12& =1\\ 18y/12+4y/12-2y/12& =1\\ (18y+4y-2y)/12& =1\\ 20y/12& =1\\ 20y& =12\\ y& =12/20\\ y& =3/5\end{array}$$

How to multiply fractions

To multiply fractions, multiply straight across:

• multiply the numerators together
• multiply the denominators together

For example:

$$\frac{3}{5}\ast \frac{2}{7}=\frac{3\ast 2}{5\ast 7}=\frac{6}{35}$$

As a reminder, you might see fractions written horizontally, but they mean the same thing:

$$\frac{3}{5}=3/5$$

Let’s work through an example that uses fraction multiplication.

$\sqrt{(3x/y^2)(3x/4)}=3$

What is the ratio of $x$ to $y$?

Know the answer?

The ratio of $x$ to $y$ is the same as the value of $x/y$. Here’s the step-by-step algebra:

$$\begin{array}{rl}\sqrt{(3x/y^2)(3x/4)}& =3\\ \sqrt{(3x\ast 3x)/(y^2\ast 4)}& =3\\ \sqrt{9x^2/4y^2}& =3\\ 3x/2y& =3\\ x/2y& =1\\ x/y& =2\end{array}$$

Since $x/y=2$, you can write this as $x/y=2/1$. That means the ratio of $x$ to $y$ is $2$ to $1$.

How to divide fractions

Dividing one fraction by another is closely related to multiplication:

1. Swap the numerator and denominator of the divisor (the second fraction).
2. Multiply.

For example:

$$\frac{2}{7}÷\frac{5}{3}=\frac{2}{7}\ast \frac{3}{5}=\frac{6}{35}$$

You’ll often hear this described as multiplying by the reciprocal. The reciprocal is the swapped fraction. For example, the reciprocal of $\frac{5}{3}$ is $\frac{3}{5}$.

Here’s an example using fraction division:

The $n$th term of a sequence is $(n/(n+1))/(n/(n+1))$

What is the sum of the first 7 terms in the sequence?

If you haven’t read the chapter on sequences yet, don’t worry. Here’s another way to phrase the same question:

$f(n)=(n/(n+1))/(n/(n+1))$

What is the sum of $f(n)$ for each integer $n$ where $1<=n<=7$?

Don’t let the horizontal format, variables, and parentheses distract you from the fraction structure.

Start by finding the first term, which is the value when $n=1$:

$$\begin{array}{rl}f(1)& =(n/(n+1))/(n/(n+1))\\ & =(1/(1+1))/(1/(1+1))\\ & =(1/2)/(1/2)\\ & =(1/2)\ast (2/1)\\ & =2/2\\ & =1\end{array}$$

Now look at the expression written vertically:

$$f(n)=\frac{n/(n+1)}{n/(n+1)}$$

The numerator is divided by the exact same expression in the denominator, so the value is always $1$ (for any $n$ where the expression is defined). Therefore, the sum of the first 7 terms is $1+1+1+1+1+1+1=7$.