How to Simplify Ratios — Methods and Examples

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After this lesson you will be able to simplify any ratio to its lowest terms, including ratios with large numbers, decimals, and fractions. You will understand the GCD method, know how to handle three-part ratios, and recognize when a ratio is already in simplest form.

Theory

What does "simplify a ratio" mean?

Simplifying a ratio means rewriting it with the smallest possible whole numbers while keeping the same relationship between the quantities. The ratio $12 : 8$ and $3 : 2$ describe the same comparison, but $3 : 2$ is in simplest form.

A ratio is in simplest form when the parts share no common factor other than 1 — just like a fraction in lowest terms.

The GCD method

The most reliable technique is dividing every part of the ratio by the Greatest Common Divisor (GCD):

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

How to find the GCD:

• List factors: Write out all factors of each number and pick the largest one they share.
• Prime factorization: Break each number into primes and multiply the common prime factors.

For example, simplify $36 : 48$:

• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
• GCD = 12

$36 : 48 = \frac{36}{12} : \frac{48}{12} = 3 : 4$

Ratios with decimals

If a ratio contains decimals, multiply all parts by 10, 100, or 1000 to convert them to whole numbers first, then simplify normally.

$0.6 : 1.5 \xrightarrow{\times 10} 6 : 15 = 2 : 5$

Ratios with fractions

If a ratio contains fractions, multiply all parts by the LCD of the fractions to clear them:

$\frac{1}{2} : \frac{3}{4} \xrightarrow{\times 4} 2 : 3$

Three-part ratios

The same GCD method works — find the GCD of all three numbers and divide each part:

$12 : 18 : 30 \quad \text{GCD} = 6 \quad \Rightarrow \quad 2 : 3 : 5$

Worked Examples

Example 1: Basic two-part ratio (easy)

Problem: Simplify $20 : 35$.

Step 1: Find the GCD of 20 and 35.

• Factors of 20: 1, 2, 4, 5, 10, 20
• Factors of 35: 1, 5, 7, 35
• GCD = 5

Step 2: Divide both parts by 5.

$20 : 35 = \frac{20}{5} : \frac{35}{5} = 4 : 7$

Answer: $4 : 7$

Example 2: Larger numbers (medium)

Problem: Simplify $54 : 90$.

Step 1: Use prime factorization.

• $54 = 2 \times 3^3$
• $90 = 2 \times 3^2 \times 5$
• GCD = $2 \times 3^2 = 18$

Step 2: Divide both parts by 18.

$54 : 90 = \frac{54}{18} : \frac{90}{18} = 3 : 5$

Answer: $3 : 5$

Example 3: Ratio with decimals (medium)

Problem: Simplify $2.4 : 3.6$.

Step 1: Multiply both parts by 10 to remove decimals.

$2.4 : 3.6 \xrightarrow{\times 10} 24 : 36$

Step 2: Find the GCD of 24 and 36.

• $24 = 2^3 \times 3$
• $36 = 2^2 \times 3^2$
• GCD = $2^2 \times 3 = 12$

Step 3: Divide both parts by 12.

$24 : 36 = 2 : 3$

Answer: $2 : 3$

Example 4: Ratio with fractions (medium)

Problem: Simplify $\dfrac{2}{3} : \dfrac{5}{6}$.

Step 1: Find the LCD of the fractions. LCD of 3 and 6 is 6.

Step 2: Multiply both parts by 6.

$\frac{2}{3} \times 6 : \frac{5}{6} \times 6 = 4 : 5$

Step 3: Check if 4 and 5 share a common factor. GCD = 1, so the ratio is already simplified.

Answer: $4 : 5$

Example 5: Three-part ratio (challenging)

Problem: In a class, there are 16 students who walk to school, 24 who take the bus, and 8 who cycle. Simplify the ratio of walkers to bus riders to cyclists.

Step 1: Write the ratio.

$16 : 24 : 8$

Step 2: Find the GCD of 16, 24, and 8.

• Factors of 8: 1, 2, 4, 8
• 8 divides into 16 (yes) and 24 (yes)
• GCD = 8

Step 3: Divide each part by 8.

$16 : 24 : 8 = 2 : 3 : 1$

Answer: $2 : 3 : 1$

Common Mistakes

Mistake 1: Not dividing by the greatest common divisor

❌ $18 : 24 = 9 : 12$ (divided by 2 instead of 6)

✅ $18 : 24 = 3 : 4$ (GCD is 6)

Why this matters: Dividing by any common factor reduces the ratio, but it is only fully simplified when you divide by the GCD. If your result still has a common factor, you need to simplify further.

Mistake 2: Forgetting to remove decimals first

❌ Trying to find the GCD of 0.6 and 1.5 directly.

✅ Multiply by 10 first: $6 : 15$, then simplify to $2 : 5$.

Why this matters: GCD works on whole numbers. Convert decimals to whole numbers before simplifying.

Mistake 3: Simplifying each part of a three-part ratio separately

❌ $12 : 18 : 30 \rightarrow 12 : 9 : 15$ (divided middle by 2 and last by 2, but not first)

✅ Divide all parts by the same number: $\text{GCD} = 6$, giving $2 : 3 : 5$.

Why this matters: Every part must be divided by the same GCD. Dividing different parts by different numbers changes the relationship and produces a ratio that means something entirely different.

Practice Problems

Try these on your own before checking the answers:

1. Simplify $15 : 25$.
2. Simplify $72 : 108$.
3. Simplify $1.2 : 0.8$.
4. Simplify $\dfrac{3}{4} : \dfrac{9}{8}$.
5. Simplify $30 : 45 : 15$.

Click to see answers

1. GCD of 15 and 25 is 5. $15 : 25 = 3 : 5$
2. GCD of 72 and 108 is 36. $72 : 108 = 2 : 3$
3. Multiply by 10: $12 : 8$. GCD is 4. $12 : 8 = 3 : 2$
4. LCD is 8. $\frac{3}{4} \times 8 : \frac{9}{8} \times 8 = 6 : 9$. GCD is 3. $6 : 9 = 2 : 3$
5. GCD of 30, 45, and 15 is 15. $30 : 45 : 15 = 2 : 3 : 1$

Summary

• Simplifying a ratio means dividing all parts by their GCD to get the smallest whole numbers.
• For decimals, multiply all parts by 10 (or 100) first to convert to whole numbers.
• For fractions, multiply all parts by the LCD to clear the fractions.
• For three-part ratios, find the GCD of all three numbers and divide each part by it.
• A ratio is in simplest form when no common factor (other than 1) is shared by all parts.

Related Topics

• Ratios and Proportions — Complete Guide with Examples
• How to Solve Proportions Using Cross Multiplication
• How to Simplify Fractions to Lowest Terms

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