Fraction Shortcuts and Tricks for Faster Maths

Fractions can feel tedious to work with, especially under time pressure during exams or when doing mental arithmetic. But with the right shortcuts and tricks, you can speed up your fraction calculations significantly. This guide covers practical techniques for adding, multiplying, simplifying, and converting fractions faster.

The Butterfly Method for Adding Fractions

When adding two fractions with different denominators, the butterfly method provides a quick visual shortcut. For $\frac{a}{b} + \frac{c}{d}$ :

Cross-multiply diagonally: $a \times d$ and $b \times c$
Add the two products to get the numerator: $ad + bc$
Multiply the denominators: $b \times d$

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Example: $\frac{2}{3} + \frac{3}{5}$

\frac{(2 \times 5) + (3 \times 3)}{3 \times 5} = \frac{10 + 9}{15} = \frac{19}{15}

For subtraction, use the same method but subtract instead: $\frac{ad - bc}{bd}$ .

Cross-Cancellation Before Multiplying

When multiplying fractions, you can simplify before you multiply by cancelling common factors diagonally. This keeps the numbers small and avoids having to simplify a large result afterwards.

Example: $\frac{8}{15} \times \frac{5}{12}$

Before multiplying, notice that 8 and 12 share a common factor of 4, and 5 and 15 share a common factor of 5:

\frac{\cancel{8}^{\,2}}{\cancel{15}_{\,3}} \times \frac{\cancel{5}^{\,1}}{\cancel{12}_{\,3}} = \frac{2}{3} \times \frac{1}{3} = \frac{2}{9}

Without cross-cancellation, you would compute $8 \times 5 = 40$ and $15 \times 12 = 180$ , then simplify $\frac{40}{180}$ by dividing by 20. Much harder.

The Flip-and-Multiply Rule for Division

Dividing by a fraction is the same as multiplying by its reciprocal:

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Example: $\frac{3}{4} \div \frac{6}{7} = \frac{3}{4} \times \frac{7}{6}$

Cross-cancel: 3 and 6 share a factor of 3:

\frac{1}{4} \times \frac{7}{2} = \frac{7}{8}

Quick Fraction-to-Percentage Conversions

Memorising common fraction-to-percentage equivalences saves time:

$\frac{1}{2} = 50\%$
$\frac{1}{3} \approx 33.3\%$ , $\frac{2}{3} \approx 66.7\%$
$\frac{1}{4} = 25\%$ , $\frac{3}{4} = 75\%$
$\frac{1}{5} = 20\%$ , $\frac{2}{5} = 40\%$ , $\frac{3}{5} = 60\%$ , $\frac{4}{5} = 80\%$
$\frac{1}{8} = 12.5\%$ , $\frac{3}{8} = 37.5\%$ , $\frac{5}{8} = 62.5\%$ , $\frac{7}{8} = 87.5\%$
$\frac{1}{10} = 10\%$ , $\frac{1}{6} \approx 16.7\%$

To convert any fraction to a percentage, divide the numerator by the denominator and multiply by 100: $\frac{a}{b} \times 100\%$ .

Simplifying with GCD

To simplify a fraction, divide both the numerator and denominator by their greatest common divisor (GCD):

\frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}

Quick trick: If both numbers are even, keep halving. If both end in 0 or 5, divide by 5. Check divisibility by 3 (digit sum divisible by 3).

Example: Simplify $\frac{84}{126}$ .

Both even: $\frac{42}{63}$
Both divisible by 3 (digit sums: 6 and 9): $\frac{14}{21}$
Both divisible by 7: $\frac{2}{3}$

Alternatively, $\gcd(84, 126) = 42$ , so $\frac{84}{126} = \frac{2}{3}$ in one step.

Mixed Numbers and Improper Fractions

Mixed to improper: Multiply the whole number by the denominator, add the numerator:

3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}

Improper to mixed: Divide numerator by denominator. The quotient is the whole number, the remainder is the new numerator:

\frac{23}{4} = 5\frac{3}{4} \quad \text{(since } 23 \div 4 = 5 \text{ remainder } 3\text{)}

Adding Mixed Numbers Quickly

Instead of converting to improper fractions first (which creates large numbers), add the whole number parts and fraction parts separately:

Example: $2\frac{3}{4} + 1\frac{5}{6}$

Whole numbers: $2 + 1 = 3$
Fractions: $\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12} = 1\frac{7}{12}$
Combine: $3 + 1\frac{7}{12} = 4\frac{7}{12}$

The Benchmark Method for Comparing Fractions

To quickly compare two fractions without finding a common denominator, compare each to a benchmark like $\frac{1}{2}$ :

$\frac{3}{7}$ is less than $\frac{1}{2}$ (since $3 < 3.5$ )
$\frac{5}{9}$ is greater than $\frac{1}{2}$ (since $5 > 4.5$ )

Therefore $\frac{3}{7} < \frac{5}{9}$ without any calculation.

For fractions on the same side of $\frac{1}{2}$ , cross-multiply: $\frac{a}{b}$ vs $\frac{c}{d}$ — compare $a \times d$ with $b \times c$ .

Fraction Arithmetic Cheat Sheet

Addition: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
Subtraction: $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$
Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
Division: $\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$

Try It Yourself

Practice these shortcuts with our free Fraction Calculator which handles addition, subtraction, multiplication, and division with automatic simplification. For simplifying large fractions, our GCD/LCM Calculator can help you find the greatest common divisor quickly.