Simplifying Ratios: Definition, Method and Examples

What does simplifying a ratio mean?

A ratio compares the size of two or more quantities. When you simplify a ratio, you write it in its lowest terms so that the numbers share no common factors other than $1$.

Simplifying does not change the mathematical relationship. Instead, it creates equivalent ratios that are easier to understand and work with. You can represent this visually using a bar model. For example, grouping shares equally shows why the ratio $4:6$ simplifies to $2:3$.

Find a common factor

To simplify a ratio, you must divide both sides by the same whole number. You can start by identifying any common factor shared by the terms.

If the terms are large, you can reduce ratios by dividing by smaller numbers in multiple steps. For instance, you could simplify $12:18$ by first dividing by $2$ to get $6:9$, and then dividing by $3$ to reach $2:3$.

This step-by-step approach always works, but you must keep dividing until no common factors remain.

Use the greatest common factor

You can simplify any ratio in a single step by dividing its terms by their greatest common factor (GCF). The GCF is the largest whole number that divides exactly into all parts of the ratio.

For example, to simplify $20:32$:

1. List the factors of $20$: $1, 2, 4, 5, 10, 20$.
2. List the factors of $32$: $1, 2, 4, 8, 16, 32$.
3. Identify the GCF: $4$.
4. Divide both terms by $4$ to find the simplest ratio.

Alternatively, you can use a GCF ladder or prime factorization to find the common factor systematically.

Simplify ratios with more than two terms

Ratios can compare three or more parts at once. The rule for simplifying a three-term ratio is exactly the same: you must divide every term by the GCF shared by all of them.

If you find a common factor that works for two terms but not the third, you cannot use it. All terms in the ratio must be divisible by the chosen factor.

Handle ratios with units

When a ratio compares measurements, the units must be identical before you can simplify or write the final ratio. If the units differ, convert the larger unit into the smaller unit first.

For example, to find the ratio of $50\text{ cm}$ to $2\text{ m}$:

1. Convert $2\text{ m}$ to $200\text{ cm}$.
2. Write the ratio in the same units: $50 : 200$.
3. Divide both parts by the GCF of $50$.

Worked examples

Example 1: Simplifying a two-term ratio

Question: Write the ratio $45:60$ in simplest form.

Method:

1. Find the GCF of $45$ and $60$.
2. The factors of $45$ are $1, 3, 5, 9, 15, 45$.
3. The factors of $60$ are $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$.
4. The GCF is $15$.
5. Divide both parts of the ratio by $15$.

Answer: $45:60 = 3:4$.

Check: Multiply $3$ and $4$ by $15$ to ensure you return to $45$ and $60$.

Example 2: Simplifying a three-term ratio

Question: Simplify the ratio $18 : 24 : 36$.

Method:

1. Identify the GCF for all three numbers.
2. The numbers $18$, $24$, and $36$ share the GCF of $6$.
3. Divide every term by $6$.
4. $18 \div 6 = 3$.
5. $24 \div 6 = 4$.
6. $36 \div 6 = 6$.

Answer: $3:4:6$.

Check: Ensure no common factor remains among $3$, $4$, and $6$. The only common factor is $1$, so it is fully simplified.

Example 3: Simplifying with different units

Question: Write the ratio of $40\text{ minutes}$ to $2\text{ hours}$ in simplest form.

Method:

1. Convert the larger unit (hours) into the smaller unit (minutes).
2. $2\text{ hours} = 120\text{ minutes}$.
3. Write the ratio using the common unit: $40 : 120$.
4. Find the GCF of $40$ and $120$, which is $40$.
5. Divide both terms by $40$.

Answer: $1:3$.

Check: Since $40\text{ minutes}$ is exactly one-third of $120\text{ minutes}$, the ratio $1:3$ is correct.

Common mistakes

Using subtraction instead of division

A ratio represents a multiplicative relationship. You can only simplify by dividing (or multiplying) all terms by the same number. If you subtract the same number from every term, you break the proportion and create an incorrect ratio.

Not fully simplifying

If you divide by a common factor that is not the GCF, the ratio will become smaller, but it will not be in ratio lowest terms. Always check your final answer to see if another common factor exists. Similar reasoning applies when finding equivalent fractions.

Leaving decimals in a simplified ratio

A fully simplified ratio should only contain whole numbers. If you have a ratio like $1.5 : 2$, multiply both sides by $2$ to remove the decimal, resulting in the whole-number ratio $3 : 4$.

Frequently asked questions

What does it mean if the greatest common factor is $1$?

If the GCF of the terms is $1$, they share no other factors. This means the ratio is already in simplest form and cannot be reduced further.

Can you write a ratio as a fraction?

Yes, a two-term ratio can be written as a fraction where the first term is the numerator and the second term is the denominator. Simplifying a ratio uses the same mathematical rules as simplifying a fraction.

What is the highest common factor?

The highest common factor (HCF) is another term for the greatest common factor (GCF). They refer to exactly the same mathematical concept.

How do I apply these skills next?

Once you are confident with simplifying, you can build ratio tables to solve proportion problems and scale mixtures up or down efficiently.