fraction is a number of the form ±a/b, where a and b are positive integers. The number a is called the numerator and b is called the denominator. The numerator tells you the number of equal parts, and the denominator tells you how many of those parts make up a whole. Often fractions are numbers that fall between integers.  Fractions are also used to show division.

Some examples:

3/5 of the cake means 3 pieces of a cake which is divided into 5 equal pieces.

The fraction 7/4 is a number between 1 and 2.

6x/2 = 6x ÷ 2 = 3x

If a fraction’s numerator and denominator are equal (e.g. 5/5), the fraction is equal to 1.
(All 5 pieces of a cake that is cut into 5 pieces is 1 whole cake.)

A fraction that has zero as its numerator (e.g. 0/5) is equal to zero.  (Zero pieces of the cake that is cut into 5 pieces.)

A fraction that has zero as its denominator (e.g. 8/0) is undefined.

Note: On the GRE, fractions may be shown in either of two formats. You will need to understand both.

Fractions can be on one line, like
3/5 = (x + 2)/10.
They can also be “built up,” like:
\\[1ex]\dfrac{3}{5} = \dfrac{\textit{x}+2}{10}

Fractions and Decimals

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## Mixed Numbers

Mixed numbers are numbers that are an integer plus a proper fraction.

The number 4\dfrac{\,2\,}{3} is the integer 4, plus the fraction \dfrac{\,2\,}{3}

Any mixed number can be written as a fraction, and any fraction greater than 1 can be written as a mixed number.

Proper fractions have a value between 0 and 1.

Improper fractions have a value greater than 1. The numerator is greater than the denominator. An improper fraction is another way to write a mixed number.

To express a mixed number as an improper fraction, write the integer as a fraction, then add the fractions.

##### Example

Convert 4\dfrac{\,2\,}{3} into an improper fraction.

### Solution

One whole is \dfrac{\,3\,}{3},
so 4 wholes  = 4 × \dfrac{\,3\,}{3} = \dfrac{\,12\,}{3}

4\dfrac{\,2\,}{3} = \dfrac{\,12\,}{3} + \dfrac{\,2\,}{3} = \dfrac{\,14\,}{3}

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The remainder will be the numerator of the fraction.

##### Example

Convert 79/9 into a mixed number.

### Solution

Divide the numerator by the denominator.
79 ÷ 9 = (8 × 9) + 7
The integer is 8, and the numerator of the fraction is the remainder, 7.
\\[1ex]79/9 = 8\dfrac{\,7\,}{9}

## Equivalent Fractions

A fraction that has a common factor in both the numerator and denominator is equal to the fraction with the common factor canceled. The fraction 6/10 is equivalent to the fraction 3/5 since they are equal with the common factor 2 in both numerator and denominator of 6/10.

Multiplying the numerator and denominator of a fraction by the same (non-zero) number also gives a new fraction which is equivalent to the original fraction. The fractions
3/5 = 6/10 = 9/15 = 12/20 are equivalent.
\\[2ex]\dfrac{3 × 2}{5 × 2} = \dfrac{\,6\,}{10} = \dfrac{6 × 2}{10 × 2} = \dfrac{\,12\,}{20}

A fraction with a negative numerator or denominator is equivalent to a negative fraction.
\\[2ex]-\dfrac{\,3\,}{5} = \dfrac{\,-3\,}{5} = \dfrac{\,3\,}{-5}

If both numerator and denominator are negative, the fraction is positive.
\\[2ex]\dfrac{\,-3\,}{-5} = \dfrac{\,3\,}{5}

## Simplifying Fractions

#### Always reduce fractions to the simplest terms.

The GRE always puts fractions in the lowest possible terms, so you should reduce any fractions to the simplest terms before going to the answer choices. If the GRE answers weren’t in simplest form, there could be confusion about the right answer because there could be two answer choices that were equivalent (e.g., 1/2 and 2/4).

To simplify fractions, one method is to use the GCF (greatest common factor)

Divide the numerator and denominator by the GCF to reduce the fraction.

##### Example

Reduce 275/525 to lowest possible terms.

### Solution

Look for the GCF.

\dfrac{275}{525} = \dfrac{11×25}{21×25} = \dfrac{11}{21}

Another method is to find the prime factors and cancel.

##### Example

Reduce 220/594 to simplest terms.

### Solution

Do the prime factorization of the numerator and denominator. Cancel the common factors.

\dfrac{220}{594} = \dfrac{2×2×5×11}{2×3×3×3×11}

= \dfrac{2×5}{3×3×3} = \dfrac{10}{27}

##### Example

Express 26/16 as a mixed number in lowest terms.

### Solution

Find the integer, then simplify the fraction.

26/16 = 1\dfrac{\,10\,}{16} = 1\dfrac{\,5\,}{8}

## Multiplying Fractions

To multiply fractions, multiply the numerators, then multiply the denominators, and then reduce the fraction. There is a shortcut that will make fraction multiplication less tedious.

Shortcut: Multiplying Fractions

To multiply fractions:

1. Cancel out any common factors that appear in both numerators and denominators.
2. Multiply all numerators to form one numerator and all denominators to form one denominator.

Remember, you DON’T need a common denominator to multiply fractions!

##### Example

Multiply:  (6/35)(5/18)

### Solution

Factor the numerators and denominators. Cancel the common factors.

\Big(\,\dfrac{6}{35}\,\Big) \Big(\,\dfrac{5}{18}\,\Big) = \dfrac{6 × 5}{7 × 5 × 3 × 6}

= \dfrac{1}{7 × 3} = \dfrac{1}{21}

To multiply mixed numbers, first change the numbers to improper fractions.

##### Example

Multiply: 3\dfrac{\,1\,}{2} × \dfrac{\,4\,}{5}

### Solution

Change 3\dfrac{\,1\,}{2} to an improper fraction. Multiply, then cancel the common factors.

3\dfrac{\,1\,}{2} × \dfrac{\,4\,}{5} = \dfrac{\,7\,}{2} × \dfrac{\,4\,}{5} = \dfrac{7 × 2 × 2}{2 × 5}

= \dfrac{14}{5} = 2\dfrac{\,4\,}{5}

##### Example

Multiply:  (5x2/6)(9/2x)

### Solution

Factor the numerators and denominators. Cancel the common factors.

\Big(\dfrac{5\textit{x}^{\displaystyle{2}}}{6}\Big) \Big(\dfrac{9}{2\textit{x}}\Big) = \dfrac{5 × \textit{x} × \textit{x} × 3 × 3}{2 × 3 × 2 × \textit{x}}

= \dfrac{15\textit{x}}{4}

## Dividing Fractions

When dividing fractions, use the reciprocal. Informally, the reciprocal of a fraction is the fraction flipped upside down.

The reciprocal of 2/3 is 3/2.  The reciprocal of 5/4 is 4/5.

To divide fractions, change the divisor to its reciprocal, then multiply the fractions. (Remember that the divisor is the second number.) When multiplying, cancel out any common factors that appear in both numerators and denominators.

##### Example

Divide:  (5/6) ÷ (7/2)

### Solution

Use the reciprocal.  Multiply and cancel the common factors.

\dfrac{\,5\,}{6} ÷ \dfrac{\,7\,}{2} = \dfrac{\,5\,}{6} × \dfrac{\,2\,}{7} = \dfrac{\,5\,}{21}

To divide mixed numbers, first change the numbers to improper fractions.

##### Example

Divide: \dfrac{3}{5} ÷ 2\dfrac{1}{10}

### Solution

Change 2\dfrac{1}{10} to an improper fraction.  Multiply by the reciprocal, then cancel the common factors.

\dfrac{3}{5} ÷ 2\dfrac{1}{10} = \dfrac{3}{5} ÷ \dfrac{21}{10} = \dfrac{3}{5} × \dfrac{10}{21}

= \dfrac{3 × 2 × 5}{5 × 3 × 7} = \dfrac{2}{7}

## Complex Fractions

A complex fraction is a fraction that has a fraction in the numerator and/or denominator. In other words, it is a fraction divided by a fraction. Complex fractions contain variable expressions. To simplify, use the reciprocal of the divisor, then multiply.

##### Example

Simplify: (x/y)/(2x/3)

### Solution

Use the reciprocal then multiply.

\dfrac{\dfrac{\textit{x}}{\textit{y}}}{\dfrac{2\textit{x}}{3}} = \dfrac{\textit{x}}{\textit{y}} × \dfrac{3}{2\textit{x}} = \dfrac{3}{2\textit{y}}

To add or subtract fractions that have the same denominator, add or subtract the numerators. Check to see that the answer is in simplest terms.

For example, 3/8 – 1/8 = 2/8 = 1/4.

To add or subtract fractions that have different denominators, the first step is to write equivalent fractions that have the same, or a common, denominator.

To write all fractions with the same denominator, a quick choice is to multiply the denominators. For example,
1/3 + 1/4 = 4/12 + 3/12 = 7/12.

But multiplying the denominators may give a rather large denominator. To avoid a large denominator, use the least common denominator (LCD). The LCD is the least common multiple (LCM) of all the denominators.

##### Example

Subtract: 7/12 – 1/16

### Solution

Find the LCD. Factor the denominators.
12 = 2 × 2 × 3
16 = 2 × 2 × 2 × 2
LCD = (2 × 2 × 3) × (2 × 2) = 48

Find equivalent fractions that have 48 as the denominator.
\\[2ex]\dfrac{7}{12} × \dfrac{4}{4} = \dfrac{28}{48}          \dfrac{1}{16} × \dfrac{3}{3} = \dfrac{3}{48}

Subtract. \dfrac{28}{48} \,-\, \dfrac{3}{48} = \dfrac{25}{48}

A final rule about adding and subtracting fractions is that you can split up the numerator, but you can never split up the denominator.

\dfrac{7 + 14}{7} = \dfrac{7}{7} + \dfrac{14}{7} = 1 + 2 = 3  Splitting the numerator works gives you the same value.

\dfrac{7}{7 + 14}\dfrac{7}{7} + \dfrac{7}{14} = 1 + \dfrac{1}{2} Splitting the denominator gives you a different value, and is therefore incorrect.

## Cross Multiplication

One method of comparing fractions is to find the common denominator.

##### Example

Which fraction is greater, 3/4 or 8/11?

### Solution

The LCD is 44, so compare
3/4 = 33/44 and 8/11 = 32/44.

So 3/4 is greater than 8/11,
or 3/4 > 8/11.

An easier way to compare fractions is cross multiplication. Multiply the numerator of one fraction by the denominator of the other fraction and compare the products.

##### Example

Which fraction is greater, 7/30 or 21/91?

### Solution

Set the fractions next to each other and cross multiply.

\dfrac{\,\,7\,\,}{30}  \dfrac{\,\,21\,\,}{91}

30 × 21 = 630
7 × 91 = 637

637        630         …Put the products under the corresponding numerator.

637 > 630, so 7/30 > 21/91.

Cross multiplication is also used to solve rational equations. Rational equations have fractions set equal with variable expressions in the numerators and denominators. There will be more about rational equations in Chapter 6 Algebra.

##### Example

Solve: x/16 = 5/2

### Solution

2x = 5 × 16     …Cross multiply.

x = 40          …Divide both sides by 2.

##### Example

Solve: \dfrac{\textit{x}}{\textit{x} + 4} = \dfrac{\,2\,}{3}

### Solution

3x = 2(x + 4)     …Cross multiply.

3x = 2x + 8      …Distribute the 2.

= 8             …Subtract 2x from both sides.