ParaPro Math Study Guide: Fractions

This page of our free ParaPro study guide covers fractions. It starts by describing the basics of fractions, and then moves on to adding, subtracting, multiplying and dividing fractions. Finally, it shows how to compare fractions.

What is a Fraction?

A fraction is made up of two numbers: the numerator on top and denominator on the bottom:

$\dfrac{\text{Numerator}}{\text{Denominator}}$

The numerator is the number of parts we have, and the denominator is the number of parts we divide the whole into. Look at the following examples. The number on top of the fraction represents how many pieces are shaded out of the whole (the denominator). For example, $\frac{1}{3}$ represents $1$ shaded piece of $3$ total.

Adding and Subtracting Fractions

To add/subtract fractions with the same denominator, simply add/subtract the numbers in the numerator. The denominator stays the same.

Example 1

\(\dfrac{3}{8} + \dfrac{2}{8} = \dfrac{3+2}{8} = \dfrac{5}{8}\)

Example 2

\(\dfrac{1}{6} + \dfrac{2}{6} = \dfrac{1+2}{6} = \dfrac{3}{6}\)

\(⇒\) We can write \(\dfrac{3}{6}\) as \(\dfrac{1}{2}\) since the two fractions are equivalent.

Example 3

\(\dfrac{7}{9} + \dfrac{4}{9} = \dfrac{7-4}{6} = \dfrac{3}{9}\) which, when we divide both the numerator and denominator by \(3\), can be reduced to \(\dfrac{1}{3}\)

To add/subtract fractions with different denominators, we have to make the denominators the same. To do this, we can multiply each fraction by the denominator of the other:

\(\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a \times d}{b \times d} + \dfrac{c \times b}{d \times b} = \dfrac{ad + bc}{bd}\)

Example 4

\(\dfrac{2}{3} + \dfrac{1}{4} = \dfrac{2 \times 4}{3 \times 4} + \dfrac{1 \times 3}{4 \times 3} = \dfrac{8}{12} + \dfrac{3}{12} = \dfrac{8+3}{12} = \dfrac{11}{12}\)

Example 5

\(\dfrac{5}{6} – \dfrac{2}{4} = \dfrac{5 \times 4}{6 \times 4} – \dfrac{2 \times 6}{4 \times 6} = \dfrac{20}{24} – \dfrac{12}{24} = \dfrac{20-12}{24} = \dfrac{8}{24}\)

Sometimes, we subtract a fraction from a whole number. To do this, change the whole number \((1, 2, 3, …)\) into a fraction such that the denominator matches the fraction you are subtracting:

\( \dfrac{2}{2}, … \dfrac{4}{4}, … \dfrac{10}{10}, … \) are all equal to \(1\)

\( \dfrac{4}{2}, … \dfrac{8}{4}, … \dfrac{10}{5}, … \) are all equal to \(2\)

Example 6

\( 1 – \dfrac{3}{8} = \dfrac{8}{8} – \dfrac{3}{8} = \dfrac{8-3}{8} = \dfrac{5}{8}\)

Example 7

\( 3 + \dfrac{5}{6} = \dfrac{3}{1} + \dfrac{5}{6} = \dfrac{3 \times 6}{1 \times 6} + \dfrac{5}{6} = \dfrac{18+5}{6} = \dfrac{23}{6}\)

Multiplying Fractions

To multiply fractions, multiply the numerators and denominators straight across:

\( \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}\)

When multiplying a whole number with a fraction, multiply the whole number with the numerator and keep the same denominator:

$ \# \times \dfrac{a}{b} = \dfrac{\# \times a}{b}$

Example 8

\( \dfrac{3}{4} \times \dfrac{2}{3} = \dfrac{3 \times 2}{4 \times 3} = \dfrac{6}{12}\)

Example 9

\( 5 \times \dfrac{2}{11} = \dfrac{5 \times 2}{11} = \dfrac{10}{11}\)

Dividing Fractions

Dividing fractions uses the ”keep-change-flip” method:

\( \dfrac{a}{b} \qquad \div \qquad \dfrac{c}{d} \)

\(↑ \ \qquad ↑ \ \qquad ↑\)

\(Keep \quad change \quad flip\)

\(↓ \ \qquad ↓ \ \qquad ↓\)

\( \dfrac{a}{b} \qquad \times \qquad \dfrac{d}{c} \)

The first fraction stays the same, change the division symbol to multiplication, and flip the second fraction. Essentially, we are turning a division problem into a multiplication one!

Example 10

\( \dfrac{3}{5} \div \dfrac{2}{7}\)

Solution:

\(\dfrac{3}{5} \qquad \qquad \div \qquad \qquad \ \dfrac{2}{7} \)

\(↑ \qquad \qquad \ ↑ \qquad \qquad \ ↑\)

\(Keep \qquad change \qquad \ \ \ flip \ \ ⇒ \ \ \dfrac{3 \times 7}{5 \times 2} = \dfrac{21}{10}\)

\( \dfrac{3}{5} \qquad \qquad \times \qquad \qquad \ \dfrac{7}{2} \)

Comparing Fractions

We can compare fractions to see which number is bigger by finding a common denominator between the two fractions. This is similar to how we added or subtracted numbers with different denominators: we first found a common denominator. We will use this same approach to compare fractions.

Some symbols we will be using to compare fractions:

  • \(\lt\) less than
  • \(\gt\) greater than
  • \(\le\) less than or equal to
  • \(\ge\) greater than or equal to

Example 11

Which fraction is bigger: \( \dfrac{2}{3} \) or \( \dfrac{4}{5}\)

Solution: The first step is to make the denominators the same number on both fractions by multiplying \( \dfrac{2}{3} \) by \(5\) and \( \dfrac{4}{5} \) by \(3\) :

\(⇒ \dfrac{10}{15} \) O \( \dfrac{12}{15} \)

\(⇒\) Now, it is easy to compare these two fractions because we can look at the number in the numerator and see which is bigger.

\(⇒ \dfrac{10}{15} \lt \dfrac{12}{15} \)

\(\dfrac{4}{5}\) is the bigger fraction.

Example 12

Which fraction is smaller: \( \dfrac{3}{8} \) or \( \dfrac{5}{12}\)

Solution: To make the denominators the same, we can multiply the first fraction by \(3\) and the second fraction by \(2\), which gives \(24\) in the denominator on both.

\(⇒ \dfrac{3}{8} \times \dfrac{3}{3} \) O \( \dfrac{5}{12} \times \dfrac{2}{2}\)

\(⇒ \dfrac{9}{24} \) O \( \dfrac{10}{24}\)

\(⇒\) Since the denominators are now the same, we can compare the numbers in the numerator:

\(⇒ \dfrac{9}{24} \lt \dfrac{10}{24} \)

\(\dfrac{3}{8}\) is smaller.

Example 13

Sort the following fractions from smallest to largest: \( \dfrac{2}{3}, \dfrac{5}{9}, \dfrac{16}{18}, \dfrac{5}{6} \)

Solution: The first step is find a common denominator between all \(4\) fractions. Since \(18\) is the Least Common Denominator (smallest common multiple of \(3, 9, 6, 18\)), we multiply each fraction to make the denominator equal to \(18\):

\(⇒ \dfrac{2}{3} \times \dfrac{6}{6} = \dfrac{12}{18} \qquad \dfrac{5}{9} \times \dfrac{2}{2} = \dfrac{10}{18} \qquad \dfrac{16}{18} \qquad \dfrac{5}{6} \times \dfrac{3}{3} = \dfrac{15}{18} \)

\(⇒\) Since the denominators are now the same, we can write the numbers in the numerator from smallest to largest: \(10, 12, 15, 16\).

The original fractions from smallest to largest:

\(⇒ \dfrac{5}{9}, \dfrac{2}{3}, \dfrac{5}{6}, \dfrac{16}{18}\)

At this point you should know a lot about fractions! You should be able to add, subtract, multiply, and divide them. Hopefully you reviewed all 13 examples and understand how to solve problems of this type. Before moving on to the next part of this study guide, be sure to try our fractions review test.


Fractions Review Test


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