Reduction of Fractions

To reduce a fraction, also called simplifying it, means to write the same number with a smaller numerator and denominator: $\frac{18}{24}$ reduces to $\frac{3}{4}$. This page shows how to reduce a fraction to its simplest form, how to change an improper fraction into a mixed number, how to bring fractions to a common denominator, and how to cancel factors in fractions with letters.

Equivalent fractions

Multiplying or dividing the numerator and the denominator by the same non-zero number doesn't change the value of a fraction. $$\frac{a}{b}=\frac{a\cdot k}{b\cdot k}\qquad\frac{a}{b}=\frac{a\div k}{b\div k}$$

Fractions with the same value are called equivalent. $\frac{3}{4}$ and $\frac{6}{8}$ are the same amount; in $\frac{6}{8}$ the parts are just cut twice as small:

$$\frac{3}{4}=\frac{3\cdot 2}{4\cdot 2}=\frac{6}{8}$$

Reducing a fraction uses the second half of the rule: divide the numerator and the denominator by a common factor.

Reducing to simplest form

A fraction is in simplest form, or lowest terms, when its numerator and denominator have no common factor except 1.

Method 1: divide step by step

Divide by any common factor you can see, and repeat until none is left. The divisibility rules help you spot the factors.

$$\begin{aligned}\frac{18}{24}&=\frac{18\div 2}{24\div 2}=\frac{9}{12}\\&=\frac{9\div 3}{12\div 3}=\frac{3}{4}\end{aligned}$$

Method 2: divide by the greatest common factor

Dividing by the greatest common factor (GCF) of the numerator and the denominator gives the simplest form in one step. The factors of 18 are 1, 2, 3, 6, 9, 18 and the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The greatest common one is 6:

$$\frac{18}{24}=\frac{18\div 6}{24\div 6}=\frac{3}{4}$$

Method 3: prime factors

For bigger numbers, write both as products of primes and cancel the primes they share. The prime factorization calculator can do the factoring for you.

$$\begin{aligned}\frac{84}{126}&=\frac{2\cdot 2\cdot 3\cdot 7}{2\cdot 3\cdot 3\cdot 7}\\&=\frac{\cancel{2}\cdot 2\cdot\cancel{3}\cdot\cancel{7}}{\cancel{2}\cdot 3\cdot\cancel{3}\cdot\cancel{7}}=\frac{2}{3}\end{aligned}$$

Zeros at the end: if both numbers end in zeros, strike the same number of zeros from each. That's dividing both by 10, 100 and so on: $\frac{300}{1200}=\frac{3}{12}=\frac{1}{4}$.

Is it in simplest form?

Check whether the numerator and the denominator share a prime factor. In $\frac{8}{15}$, $8=2\cdot 2\cdot 2$ and $15=3\cdot 5$ share none, so $\frac{8}{15}$ can't be reduced, even though neither 8 nor 15 is prime.

Improper fractions and mixed numbers

A fraction whose numerator is greater than or equal to its denominator is called improper. To write it as a mixed number, divide the numerator by the denominator. The quotient is the whole part and the remainder is the new numerator:

$$\frac{29}{6}=4\frac{5}{6},\quad\text{because } 29=4\cdot 6+5$$

Reduce the fraction first if you can: $\frac{38}{8}=\frac{19}{4}=4\frac{3}{4}$.

To change a mixed number back into an improper fraction, multiply the whole part by the denominator and add the numerator: $3\frac{2}{5}=\frac{3\cdot 5+2}{5}=\frac{17}{5}$. You can practise both directions in the mixed fractions worksheet.

Reducing to a common denominator

To add, subtract or compare fractions, rewrite them with the same denominator by multiplying each fraction's numerator and denominator by the same number. The best choice is the least common multiple of the denominators. For example, to compare $\frac{5}{6}$ and $\frac{7}{8}$: the least common multiple of 6 and 8 is 24, so

$$\begin{aligned}\frac{5}{6}&=\frac{5\cdot 4}{6\cdot 4}=\frac{20}{24}\\\frac{7}{8}&=\frac{7\cdot 3}{8\cdot 3}=\frac{21}{24}\end{aligned}$$

20 is less than 21, so $\frac{5}{6}\lt\frac{7}{8}$. More in comparing fractions and adding fractions with different denominators.

Algebraic fractions

Cancel the factors that the whole numerator and the whole denominator have in common. Factor both first.

Example 1: $\frac{6dm}{8dy}$

$$\frac{6dm}{8dy}=\frac{2d\cdot 3m}{2d\cdot 4y}=\frac{3m}{4y}$$

Example 2: $\frac{3am+ay}{ad+ah}$

Take the common factor $a$ out of the numerator and out of the denominator:

$$\frac{3am+ay}{ad+ah}=\frac{a(3m+y)}{a(d+h)}=\frac{3m+y}{d+h}$$

Example 3: $\frac{x^2-9}{x^2+3x}$

$$\frac{x^2-9}{x^2+3x}=\frac{(x-3)(x+3)}{x(x+3)}=\frac{x-3}{x}$$

for $x\ne 0$ and $x\ne -3$, where the original fraction isn't defined.

Example 4: $\frac{a-b}{b-a}$

Since $b-a=-(a-b)$, for every $a\ne b$:

$$\frac{a-b}{b-a}=\frac{a-b}{-(a-b)}=-1$$

Cancel factors, never terms. In $\frac{x+3}{x}$ nothing cancels: $x$ is a factor of the denominator, but only a term of the numerator. Try a number: with $x=2$, $\frac{2+3}{2}=\frac{5}{2}$, not 3.

Common mistakes

Stopping too early: $\frac{18}{24}=\frac{9}{12}$ is right but not finished, because 9 and 12 still share the factor 3.
Dividing only one part: $\frac{18}{24}$ is not $\frac{9}{24}$. Divide the numerator and the denominator by the same number.
Subtracting instead of dividing: $\frac{5}{7}\ne\frac{5-4}{7-4}=\frac{1}{3}$. Only multiplying or dividing both parts keeps the value.
Cancelling terms: $\frac{x+3}{x+5}$ is already in simplest form.

Practice

Try each problem, then open it to check your answer.

1. $\frac{12}{18}$

The GCF is 6: $\frac{12\div 6}{18\div 6}=\frac{2}{3}$

2. $\frac{35}{49}$

The GCF is 7: $\frac{5}{7}$

3. $\frac{42}{56}$

The GCF is 14: $\frac{3}{4}$

4. $\frac{150}{375}$

The GCF is 75: $\frac{2}{5}$

5. $\frac{27}{6}$ as a mixed number

$\frac{27}{6}=\frac{9}{2}=4\frac{1}{2}$

6. $\frac{10x^2y}{15xy^3}$

Cancel $5xy$: $\frac{2x}{3y^2}$

7. $\frac{x^2-1}{x^2+x}$

$\frac{(x-1)(x+1)}{x(x+1)}=\frac{x-1}{x}$

You can check the numeric ones with the fraction calculator below.