Word Problems Involving Writing a Ratio

Writing a ratio from a word problem is a two-step skill: first you read carefully to identify the two quantities and their correct order, then you write the ratio in the form the problem requires. This lesson gives you a reliable strategy and plenty of practice with GED-style questions.

What Makes a Good Ratio Problem Strategy?

Many students rush past the wording and mix up the order of the quantities. A strong strategy forces you to slow down and identify exactly what is being compared to what.

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Key vocabulary to recognize:

• “for every”, “per”, “to”, “for each”, signal a ratio relationship.
• “out of”, “out of every”, often signal a part-to-whole ratio.
• “compared to”, “relative to”, signal a comparison ratio.

Strategy for Writing a Ratio from a Word Problem

Step 1: Read and underline the two quantities being compared

The problem will name two things. Underline each one.

Step 2: Identify the order

The first quantity mentioned in the ratio description goes in front (or in the numerator if written as a fraction). Pay close attention to the phrasing: “the ratio of A to B” means A comes first.

Step 3: Write the ratio

Use colon, fraction, or word form as the question requests.

Step 4: Simplify if possible

Divide both numbers by their GCF.

Step-by-Step Summary

1. Identify the two quantities.
2. Determine the correct order (which is first).
3. Write the ratio as \(\color{blue}{a : b}\) or \(\color{blue}{\frac{a}{b}}\).
4. Simplify by dividing by the GCF.

Watch: Ratio Word Problem Examples (Video Lesson)

Khan Academy works through several ratio word problem examples:

Worked Examples

Example 1: A box contains 8 red pens and 12 blue pens. Write the ratio of red pens to blue pens in simplest form.

Red to blue: \(\color{blue}{8 : 12}\). \(\color{blue}{\text{ GCF }(8, 12) = 4}\). Simplified: \(\color{blue}{2 : 3}\).

Example 2: At a school, 180 students play sports and 60 do not. What is the ratio of students who play sports to the total number of students?

\(\color{blue}{\text{ Total } = 180 + 60 = 240}\). Ratio: \(\color{blue}{180 : 240}\). \(\color{blue}{\text{ GCF }(180, 240) = 60}\). Simplified: \(\color{blue}{3 : 4}\).

Example 3: A recipe uses 2 cups of milk for every 3 cups of flour. If you use 9 cups of flour, how many cups of milk do you need? Write the original ratio and use it to solve.

Ratio milk to flour: \(\color{blue}{2 : 3}\). Scale factor: \(\color{blue}{9 \div 3 = 3}\). Milk: \(\color{blue}{2 \times 3 = 6}\) cups.

Example 4: On a test, a student answered 35 questions correctly out of 40 total. Write the ratio of correct to incorrect answers in simplest form.

\(\color{blue}{\text{ Incorrect } = 40 – 35 = 5}\). Correct to incorrect: \(\color{blue}{35 : 5}\). \(\color{blue}{\text{ GCF }(35, 5) = 5}\). Simplified: \(\color{blue}{7 : 1}\).

More Practice: Introduction to Ratios Video

This Khan Academy video walks through what a ratio is and how to identify it from real-world situations:

Exercises

1. A parking lot has 24 cars and 16 motorcycles. Write the ratio of motorcycles to cars in simplest form.
2. Out of 50 workers, 30 work full-time and 20 work part-time. Write the ratio of part-time to total workers in simplest form.
3. A store sold 45 fiction books and 30 non-fiction books. Write the ratio of fiction to non-fiction in simplest form.
4. A juice mixture uses 3 parts orange juice to 2 parts grapefruit juice. If you use 12 parts orange juice, how much grapefruit juice do you need?
5. In a survey, 56 people preferred coffee and 21 preferred tea. Write the ratio of coffee to tea in simplest form.
6. There are 150 students and 6 teachers. Write the ratio of students to teachers in simplest form.

Answers

1. \(\color{blue}{16 : 24}\); \(\color{blue}{\text{ GCF } = 8}\); \(\color{blue}{2 : 3}\)
2. \(\color{blue}{20 : 50}\); \(\color{blue}{\text{ GCF } = 10}\); \(\color{blue}{2 : 5}\)
3. \(\color{blue}{45 : 30}\); \(\color{blue}{\text{ GCF } = 15}\); \(\color{blue}{3 : 2}\)
4. Scale factor: \(\color{blue}{12 \div 3 = 4}\); Grapefruit juice = \(\color{blue}{2 \times 4 = 8}\) parts.
5. \(\color{blue}{56 : 21}\); \(\color{blue}{\text{ GCF } = 7}\); \(\color{blue}{8 : 3}\)
6. \(\color{blue}{150 : 6}\); \(\color{blue}{\text{ GCF } = 6}\); \(\color{blue}{25 : 1}\)

Frequently Asked Questions

How do you solve ratio word problems step by step?

First underline the two quantities being compared. Second, decide which comes first using phrases like the ratio of A to B. Third, write the ratio as a : b, a/b or a to b. Finally, simplify by dividing by the GCF. For example, 12 pens to 20 pencils gives 12 : 20, which simplifies to 3 : 5.

How do you write ratios in 3 ways?

Any ratio can be written with a colon, as a fraction, or with the word to. The ratio of 4 dogs to 9 cats can be written as 4 : 9, 4/9 or 4 to 9. All three mean the same thing, so use whichever form the word problem or answer choices ask for.

How do you write ratios in simplest form?

Divide both terms of the ratio by their greatest common factor. If a class has 14 girls and 21 boys, the ratio of girls to boys is 14 : 21. The GCF of 14 and 21 is 7, so the simplest form is 2 : 3. A ratio is fully simplified when the only common factor left is 1.

How do you write ratios in words?

Use the word to between the two quantities, keeping the order given in the problem. A ratio of 5 : 2 is written in words as 5 to 2. In a word problem you might say there are five teachers to every two aides. Phrases like for every and per signal the same comparison.

How do you write a ratio of 3 quantities?

List all three numbers in the order the problem names them, separated by colons, such as 2 : 3 : 5. To simplify, divide every term by the GCF of all three. For example, 6 red, 9 blue and 15 green marbles give 6 : 9 : 15, and dividing by 3 gives 2 : 3 : 5.

How do you write ratios in the form 1 : n?

Divide both parts of the ratio by the first number so the first term becomes 1. For a ratio of 4 : 10, divide each part by 4 to get 1 : 2.5. The second number can be a decimal. This form is handy for comparing several ratios quickly, since each one is scaled to a starting value of 1.

Related Topics

• Write a Ratio
• Using a Fraction to Write Down a Ratio
• Ratio Tables
• Finding Equivalent Ratio
• Word Problems Involving Comparing Ratio