A fraction is a number of the form ±\dfrac{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:

\dfrac{3}{5} of the cake means 3 pieces of a cake which is divided into 5 equal pieces.

The fraction \dfrac{7}{4} is a number between 2 and 3. \dfrac{6x}{2} = 6x ÷ 2 = 3x

If a fraction’s numerator and denominator are equal (e.g. \dfrac{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. \dfrac{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. \dfrac{8}{0}) is undefined.

Note: On the GMAT, fractions may be shown in either of two formats. You will need to understand both. Fractions can be on one line, like \dfrac{3}{5} = \dfrac{(x+2)}{10} They can also be “built up,” like: \dfrac{3}{5} = \dfrac{x+2}{10}

Mixed numbers are numbers that are an integer plus a proper fraction. The number 4^{2}/_{3} is the integer 4, plus the fraction 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.

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. 79/9 = 8^{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.

A fraction with a negative numerator or denominator is equivalent to a negative fraction. (3 × 2)/(5 × 2) = 6/10 = (6 × 2)/(10 × 2) = 12/20 -(3/5) = -3/5 = 3/-5 If both numerator and denominator are negative, the fraction is positive. -3/-5 = 3/5

Simplifying Fractions

Always reduce fractions to the simplest terms. The GMAT 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 GMAT answers weren’t in simplest form, there could be confusion about the right answer, 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.

Find the integer, then simplify the fraction. 26/16 = 1^{10}⁄_{16} = 1^{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:

Cancel out any common factors that appear in both numerators and denominators.

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!

Factor the numerators and denominators. Cancel the common factors. (5x^{2}/6)(9/2x) = (5 × x × x × 3 × 3)/(2 × 3 × 2 × x) = 15x/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.

Change 2^{1}⁄_{10} to an improper fraction. Multiply by the reciprocal, then cancel the common factors. 3/5 ÷ 2^{1}⁄_{10 }= 3/5 ÷ 21/10 = 3/5 × 10/21 = (3×2×5)/(5×3×7) = 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.

Use the reciprocal then multiply. (x/y)/(2x/3) = x/y × 3/2x = 3/2y

Adding and Subtracting Fractions

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.

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.

Set the fractions next to each other and cross multiply.

^{7}/_{30}^{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.