How to Simplify Fractions to Lowest Terms

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After working through this page, you will be able to simplify any fraction to its lowest terms quickly and confidently. You will master two reliable methods -- repeated division and the GCD shortcut -- and understand why simplified fractions are the standard form expected in math class.

Theory

What does it mean to simplify a fraction?

Simplifying a fraction means rewriting it as an equivalent fraction with the smallest possible numerator and denominator. The value of the fraction does not change -- only the numbers used to express it.

$\frac{12}{18} = \frac{2}{3}$

Both fractions represent the exact same amount, but $\frac{2}{3}$ is in lowest terms because 2 and 3 share no common factor other than 1.

Method 1: Repeated division by common factors

Find any common factor of the numerator and denominator, divide both by it, and repeat until no common factor remains.

$\frac{24}{36}$

• 24 and 36 are both even, so divide by 2: $\frac{12}{18}$
• 12 and 18 are both even, so divide by 2: $\frac{6}{9}$
• 6 and 9 are both divisible by 3: $\frac{2}{3}$

This method is intuitive and works well when the common factors are easy to spot (especially 2, 3, and 5).

Method 2: The GCD shortcut (one-step method)

The Greatest Common Divisor (GCD) -- also called the Greatest Common Factor (GCF) -- is the largest number that divides both the numerator and denominator evenly. Dividing both by the GCD simplifies the fraction in a single step:

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

For $\frac{24}{36}$:

• $\gcd(24, 36) = 12$
• $\frac{24 \div 12}{36 \div 12} = \frac{2}{3}$

How to find the GCD

Listing factors:

Write out all factors of each number and pick the largest one they share.

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Common factors: 1, 2, 3, 4, 6, 12
• GCD = 12

Prime factorization:

Break each number into prime factors. Multiply together the primes they share, using the lowest power of each.

$24 = 2^3 \times 3, \quad 36 = 2^2 \times 3^2$

$\gcd = 2^2 \times 3^1 = 4 \times 3 = 12$

Euclidean algorithm (for larger numbers):

Repeatedly divide and take remainders until the remainder is 0. The last non-zero remainder is the GCD.

$\gcd(48, 180):$ $180 = 3 \times 48 + 36$ $48 = 1 \times 36 + 12$ $36 = 3 \times 12 + 0$

$\gcd(48, 180) = 12$

When is a fraction already in lowest terms?

A fraction $\frac{a}{b}$ is in lowest terms when $\gcd(a, b) = 1$. Quick checks:

• If one number is prime and does not divide the other, the fraction is already simplified.
• If the numerator and denominator are consecutive numbers (like 7 and 8), they are always coprime.
• If the numerator is 1, the fraction is always in lowest terms.

Simplifying improper fractions and mixed numbers

Improper fractions are simplified the same way -- find the GCD and divide. You may also convert to a mixed number afterward:

$\frac{45}{30} = \frac{45 \div 15}{30 \div 15} = \frac{3}{2} = 1\frac{1}{2}$

For mixed numbers, simplify only the fraction part:

$3\frac{8}{12} = 3\frac{8 \div 4}{12 \div 4} = 3\frac{2}{3}$

Worked Examples

Example 1: Simple even numbers (easy)

Problem: Simplify $\dfrac{10}{16}$.

Step 1: Find the GCD of 10 and 16.

• Factors of 10: 1, 2, 5, 10
• Factors of 16: 1, 2, 4, 8, 16
• GCD = 2

Step 2: Divide both by 2. $\frac{10 \div 2}{16 \div 2} = \frac{5}{8}$

Step 3: Check -- 5 is prime and does not divide 8, so the fraction is fully simplified.

Answer: $\dfrac{5}{8}$

Example 2: Using prime factorization (medium)

Problem: Simplify $\dfrac{36}{48}$.

Step 1: Prime factorize both numbers. $36 = 2^2 \times 3^2, \quad 48 = 2^4 \times 3$

Step 2: Find the GCD. $\gcd = 2^2 \times 3^1 = 12$

Step 3: Divide both by 12. $\frac{36 \div 12}{48 \div 12} = \frac{3}{4}$

Answer: $\dfrac{3}{4}$

Example 3: Larger numbers with the Euclidean algorithm (medium)

Problem: Simplify $\dfrac{84}{126}$.

Step 1: Use the Euclidean algorithm to find the GCD. $126 = 1 \times 84 + 42$ $84 = 2 \times 42 + 0$

$\gcd(84, 126) = 42$

Step 2: Divide both by 42. $\frac{84 \div 42}{126 \div 42} = \frac{2}{3}$

Answer: $\dfrac{2}{3}$

Example 4: Simplifying an improper fraction (medium)

Problem: Simplify $\dfrac{72}{54}$ and write as a mixed number.

Step 1: Find the GCD. $72 = 2^3 \times 3^2, \quad 54 = 2 \times 3^3$ $\gcd = 2 \times 3^2 = 18$

Step 2: Divide both by 18. $\frac{72 \div 18}{54 \div 18} = \frac{4}{3}$

Step 3: Convert to a mixed number. $\frac{4}{3} = 1\frac{1}{3}$

Answer: $\dfrac{4}{3} = 1\dfrac{1}{3}$

Example 5: Simplifying within an expression (challenging)

Problem: After multiplying $\dfrac{9}{14} \times \dfrac{7}{12}$, simplify the result.

Step 1: Multiply straight across. $\frac{9 \times 7}{14 \times 12} = \frac{63}{168}$

Step 2: Find the GCD of 63 and 168 using prime factorization. $63 = 3^2 \times 7, \quad 168 = 2^3 \times 3 \times 7$ $\gcd = 3 \times 7 = 21$

Step 3: Divide both by 21. $\frac{63 \div 21}{168 \div 21} = \frac{3}{8}$

Answer: $\dfrac{3}{8}$

(Tip: Cross-cancellation before multiplying would have given $\frac{3}{8}$ directly with smaller numbers.)

Common Mistakes

Mistake 1: Dividing by a common factor but not the GCD

❌ $\frac{18}{24} = \frac{9}{12}$ (divided by 2, then stopped)

✅ $\frac{18}{24} = \frac{3}{4}$ (GCD is 6; or continue dividing $\frac{9}{12}$ by 3)

Why this matters: If you divide by a factor that is not the GCD, the fraction is smaller but not fully simplified. Always check whether the result can be reduced further, or use the GCD from the start.

Mistake 2: Dividing numerator and denominator by different numbers

❌ $\frac{15}{20} = \frac{15 \div 5}{20 \div 4} = \frac{3}{5}$ (wrong divisors!)

✅ $\frac{15}{20} = \frac{15 \div 5}{20 \div 5} = \frac{3}{4}$

Why this matters: You must divide both the numerator and denominator by the same number to create an equivalent fraction. Using different divisors changes the value of the fraction.

Mistake 3: Thinking 1 is a common factor that simplifies further

❌ Student keeps trying to simplify $\frac{7}{9}$ because "they can both be divided by 1."

✅ $\frac{7}{9}$ is already in lowest terms because $\gcd(7, 9) = 1$.

Why this matters: Every pair of numbers shares the factor 1, but dividing by 1 does not change anything. A fraction is fully simplified when the GCD is 1.

Practice Problems

Try these on your own before checking the answers:

1. Simplify $\dfrac{14}{21}$.
2. Simplify $\dfrac{45}{60}$.
3. Simplify $\dfrac{56}{98}$.
4. Simplify $\dfrac{120}{84}$ and write as a mixed number.
5. Is $\dfrac{29}{53}$ already in lowest terms? Explain.

Click to see answers

1. $\gcd(14, 21) = 7$. $\frac{14}{21} = \frac{2}{3}$
2. $\gcd(45, 60) = 15$. $\frac{45}{60} = \frac{3}{4}$
3. $\gcd(56, 98) = 14$. $\frac{56}{98} = \frac{4}{7}$
4. $\gcd(120, 84) = 12$. $\frac{120}{84} = \frac{10}{7} = 1\frac{3}{7}$
5. Yes. 29 is a prime number and does not divide 53, so $\gcd(29, 53) = 1$. The fraction is already in lowest terms.

Summary

• Simplifying a fraction means dividing both the numerator and denominator by their common factors until no common factor remains except 1.
• The fastest method is to find the GCD and divide both numbers by it in one step.
• Use prime factorization or the Euclidean algorithm for larger numbers.
• A fraction is in lowest terms when $\gcd(\text{numerator}, \text{denominator}) = 1$.
• Always simplify your final answer in math class -- and simplify before multiplying (cross-cancel) to save time.

Related Topics

• Fractions -- Complete Guide
• How to Add Fractions with Unlike Denominators
• How to Multiply and Divide Fractions Step by Step

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