How to simplify ratios

Read,3 minutes

To simplify a ratio, divide every term of the ratio by the same nonzero number until the terms have no common factor greater than \(1\). The comparison stays equivalent because each term is scaled by the same amount.

For a two-term ratio, this is the same idea as reducing a fraction. For example, \(24:36\) simplifies by the greatest common factor \(12\): \(24 \div 12 = 2\) and \(36 \div 12 = 3\), so \(24:36 = 2:3\).

Simplifying Ratios

Think of this lesson as more than a rule to memorize. Simplifying Ratios is about comparisons, scaling, and equal ratios. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

A ratio compares quantities, and a proportion says two ratios are equal. Cross products help because \(\frac{a}{b}=\frac{c}{d}\) implies \(ad=bc\).

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Read what is given and what is being asked.
Choose the rule that connects them.
Substitute carefully and simplify in small steps.
Check the final answer against the original question.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

When units are different, convert to matching units before simplifying. When decimals or fractions appear, clear them first by multiplying every term by the same power of \(10\) or by the least common denominator.

Free printable Worksheets

Exercises for Simplifying Ratios

1) \(12:18 = \)

2) \(15:35 = \)

3) \(24:32 = \)

4) \(45:60 = \)

5) \(28:49 = \)

6) \(64:96 = \)

7) \(18:27:45 = \)

8) \(0.4:1.2 = \)

9) \(\frac{3}{4}:\frac{9}{8} = \)

10) \(36\) inches to \(3\) feet

11) \(2.5:7.5 = \)

12) \(84:126 = \)

13) \(72:108:180 = \)

14) \(1.25:2 = \)

15) \(\frac{5}{6}:\frac{7}{9} = \)

16) \(150\) centimeters to \(2\) meters

17) \(96:144:216 = \)

18) \(3.6:4.8 = \)

19) \(\frac{2}{3}:\frac{5}{12}:\frac{7}{6} = \)

20) \(0.45:1.2:2.25 = \)

Answers

1) Find the greatest common factor of \(12\) and \(18\), which is \(6\). Divide both terms: \(12 \div 6 = 2\) and \(18 \div 6 = 3\). So \(12:18 = \color{red}{2:3}\).

2) The greatest common factor of \(15\) and \(35\) is \(5\). Divide both terms by \(5\): \(15 \div 5 = 3\), \(35 \div 5 = 7\). So \(15:35 = \color{red}{3:7}\).

3) The greatest common factor is \(8\). Divide both terms: \(24 \div 8 = 3\) and \(32 \div 8 = 4\). So \(24:32 = \color{red}{3:4}\).

4) The greatest common factor of \(45\) and \(60\) is \(15\). Divide both terms: \(45 \div 15 = 3\), \(60 \div 15 = 4\). So \(45:60 = \color{red}{3:4}\).

5) The greatest common factor is \(7\). Divide both terms: \(28 \div 7 = 4\), \(49 \div 7 = 7\). So \(28:49 = \color{red}{4:7}\).

6) The greatest common factor of \(64\) and \(96\) is \(32\). Divide both terms: \(64 \div 32 = 2\), \(96 \div 32 = 3\). So \(64:96 = \color{red}{2:3}\).

7) For three terms, divide every term by the same greatest common factor. The GCF of \(18\), \(27\), and \(45\) is \(9\). Then \(18 \div 9=2\), \(27 \div 9=3\), and \(45 \div 9=5\). So the ratio is \(\color{red}{2:3:5}\).

8) Clear decimals by multiplying both terms by \(10\): \(0.4:1.2 = 4:12\). Divide both terms by \(4\): \(4 \div 4 = 1\), \(12 \div 4 = 3\). So the ratio is \(\color{red}{1:3}\).

9) Clear fractions by multiplying both terms by the least common denominator, \(8\). We get \(\frac{3}{4}\cdot 8=6\) and \(\frac{9}{8}\cdot 8=9\). The ratio is \(6:9\), which simplifies by \(3\) to \(\color{red}{2:3}\).

10) Use the same units first: \(3\) feet is \(36\) inches. The ratio is \(36:36\). Divide both terms by \(36\) to get \(\color{red}{1:1}\).

11) Clear decimals by multiplying by \(10\): \(2.5:7.5 = 25:75\). Divide by \(25\): \(25 \div 25=1\), \(75 \div 25=3\). So the ratio is \(\color{red}{1:3}\).

12) The greatest common factor of \(84\) and \(126\) is \(42\). Divide both terms: \(84 \div 42 = 2\), \(126 \div 42 = 3\). So \(84:126 = \color{red}{2:3}\).

13) The greatest common factor of all three numbers is \(36\). Divide each term: \(72 \div 36=2\), \(108 \div 36=3\), \(180 \div 36=5\). So the ratio is \(\color{red}{2:3:5}\).

14) Clear the decimal by multiplying both terms by \(100\): \(1.25:2 = 125:200\). Divide by \(25\): \(125 \div 25=5\), \(200 \div 25=8\). So the ratio is \(\color{red}{5:8}\).

15) Clear fractions by multiplying both terms by the least common denominator, \(18\). Then \(\frac{5}{6}\cdot 18=15\) and \(\frac{7}{9}\cdot 18=14\). Since \(15\) and \(14\) share no common factor greater than \(1\), the ratio is \(\color{red}{15:14}\).

16) Convert \(2\) meters to \(200\) centimeters. The ratio is \(150:200\). Divide by \(50\): \(150 \div 50=3\), \(200 \div 50=4\). So the ratio is \(\color{red}{3:4}\).

17) The greatest common factor of \(96\), \(144\), and \(216\) is \(24\). Divide each term: \(96 \div 24=4\), \(144 \div 24=6\), \(216 \div 24=9\). So the ratio is \(\color{red}{4:6:9}\).

18) Clear decimals by multiplying both terms by \(10\): \(3.6:4.8 = 36:48\). Divide by \(12\): \(36 \div 12=3\), \(48 \div 12=4\). So the ratio is \(\color{red}{3:4}\).

19) Clear fractions by multiplying every term by the least common denominator, \(12\). We get \(8:5:14\). These numbers have no common factor greater than \(1\), so the simplified ratio is \(\color{red}{8:5:14}\).

20) Clear decimals by multiplying each term by \(100\): \(45:120:225\). The greatest common factor is \(15\). Divide each term: \(45 \div 15=3\), \(120 \div 15=8\), \(225 \div 15=15\). The ratio is \(\color{red}{3:8:15}\).

How to simplify ratios