Equivalent fractions

A fraction is not a unique writing. The amount called "half" can be written as $\frac{1}{2}$, $\frac{2}{4}$, $\frac{3}{6}$, $\frac{50}{100}$, and on and on. All of these are equivalent fractions , same amount, different costumes.

Concept

Two fractions are equivalent if they represent the same amount. The classic example: $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{5}{10}$. Fold a strip of paper in half, then in quarters, then in eighths , you'll see that one half of the strip is exactly the same length as two quarters or four eighths.

The rule that produces equivalent fractions:

$\frac{p}{q} = \frac{p \times k}{q \times k}$

for any whole number $k > 0$. You multiply both top and bottom by the same number. The fraction "scales up" , more pieces, smaller each, same total.

Going the other way, you can simplify a fraction by dividing both top and bottom by a common factor:

$\frac{p \div k}{q \div k}$

You keep dividing by common factors until there are no more. A fraction is in lowest terms (also called simplest form) when the only common factor of numerator and denominator is $1$ , that is, when they are coprime.

For example, $\frac{12}{18}$. Common factor $6$. $\frac{12 \div 6}{18 \div 6} = \frac{2}{3}$. Now $2$ and $3$ are coprime, so $\frac{2}{3}$ is in lowest terms.

Why simplify?

• Smaller numbers are easier to work with.
• Comparison becomes easier.
• The simplest form is unique , there is only one way to write a fraction in lowest terms.

Quick check for equivalence

To test if $\frac{a}{b} = \frac{c}{d}$, check if the cross products are equal: $a \times d \stackrel{?}{=} b \times c$. If yes, the fractions are equivalent.

For example, is $\frac{3}{5} = \frac{12}{20}$? Cross products: $3 \times 20 = 60$, $5 \times 12 = 60$. Yes , equivalent.

Worked examples

Example 1. Write four equivalent fractions for $\frac{3}{4}$.

• Multiply both by $2$: $\frac{6}{8}$.
• By $3$: $\frac{9}{12}$.
• By $4$: $\frac{12}{16}$.
• By $5$: $\frac{15}{20}$.

Example 2. Simplify $\frac{18}{24}$ to lowest terms.

• $\gcd(18, 24) = 6$.
• $\frac{18 \div 6}{24 \div 6} = \frac{3}{4}$.

Example 3. Simplify $\frac{45}{60}$.

• $\gcd(45, 60) = 15$.
• $\frac{45 \div 15}{60 \div 15} = \frac{3}{4}$.

Example 4. Are $\frac{6}{9}$ and $\frac{8}{12}$ equivalent?

• Simplify $\frac{6}{9} = \frac{2}{3}$ (divide by $3$).
• Simplify $\frac{8}{12} = \frac{2}{3}$ (divide by $4$).
• Both simplify to $\frac{2}{3}$. Yes, equivalent. (Or cross-multiply: $6 \times 12 = 72 = 9 \times 8$. ✓)

Example 5. Fill in the blank: $\frac{5}{7} = \frac{?}{42}$.

• $7 \to 42$ is a multiplication by $6$.
• So numerator also multiplied by $6$: $5 \times 6 = 30$.
• Answer: $\frac{30}{42}$.

Example 6. Reduce $\frac{144}{216}$ to lowest terms.

• $\gcd(144, 216) = 72$ (using prime factorisation: $144 = 2^4 \times 3^2$, $216 = 2^3 \times 3^3$, common $= 2^3 \times 3^2 = 72$).
• $\frac{144}{216} = \frac{2}{3}$.

Try it yourself

1. Write three equivalent fractions for $\frac{2}{5}$.
2. Simplify $\frac{16}{20}$.
3. Simplify $\frac{48}{72}$.
4. Are $\frac{4}{6}$ and $\frac{10}{15}$ equivalent?
5. Fill in: $\frac{3}{8} = \frac{?}{40}$.
6. Fill in: $\frac{?}{9} = \frac{24}{27}$.
7. Reduce $\frac{36}{48}$.
8. Investigate: a fraction equals $\frac{1}{2}$. Its denominator minus its numerator is $11$. Find the fraction.

Activity

Fraction strip ladder. Cut $4$ paper strips, all of the same length. Leave the first as $1$ whole. Fold the second into halves, the third into thirds, the fourth into fourths, fifth into sixths, sixth into twelfths. Line them up. Now match equivalents: $\frac{1}{2}$ aligns with $\frac{2}{4}$ aligns with $\frac{3}{6}$ aligns with $\frac{6}{12}$. Confirm them visually.