Ratios and Proportions

Grade 6 · Math · Study Guide

Learn how to write, simplify, and use ratios and proportions to solve real-world problems.

A ratio is a comparison of two quantities. If a class has 12 girls and 8 boys, the ratio of girls to boys is 12 to 8, which can also be written as 12:8 or 12/8. Ratios can always be simplified — 12:8 simplifies to 3:2, meaning for every 3 girls there are 2 boys.

A proportion is a statement that two ratios are equal. For example, 1/2 = 3/6 is a proportion. Proportions are incredibly useful for solving problems where you know three of the four numbers and need to find the missing one. You can solve a proportion using cross multiplication: if a/b = c/d, then a × d = b × c.

Ratios and proportions show up constantly in real life. Recipes scale up using proportions — if a recipe for 4 servings needs 2 cups of flour, then 8 servings need 4 cups. Maps use ratios called scale factors. Unit rates, like miles per hour or price per item, are special ratios that compare a quantity to exactly 1 unit. Mastering ratios and proportions builds a foundation for percent, slope, and almost all of algebra.

Key terms

Ratio
A comparison of two quantities, written as a:b, a to b, or a/b.
Proportion
An equation stating that two ratios are equal, like 3/4 = 6/8.
Cross Multiplication
A method to solve proportions: multiply the numerator of each fraction by the denominator of the other.
Unit Rate
A ratio with a denominator of 1, such as 60 miles per 1 hour.
Equivalent Ratios
Ratios that represent the same comparison, like 1:2 and 3:6.

Flashcards

Ratio
A comparison of two numbers written as a:b, a to b, or a/b.
Proportion
Two equal ratios — for example, 2/4 = 1/2.
Cross Multiplication
To solve a/b = c/d, multiply a × d and b × c and set them equal.
Unit Rate
A ratio per 1 unit, like $2.50 per gallon or 30 miles per hour.
Equivalent Ratios
Ratios that simplify to the same value, like 2:4 and 1:2.
Scale Factor
The ratio used to enlarge or reduce a figure or map.
Simplify a Ratio
Divide both parts of a ratio by their greatest common factor.

Questions

  1. A bag contains 4 red marbles and 6 blue marbles. What is the ratio of red to blue marbles in simplest form?

    1. 4:6
    2. 2:3
    3. 3:2
    4. 6:4
    Show answer

    Answer: 2:3

    4:6 simplifies by dividing both by 2, giving 2:3.

  2. If 3/5 = x/20, then x = ___.

    Show answer

    Answer: 12

    Cross multiply: 3 × 20 = 5 × x → 60 = 5x → x = 12.

  3. Which proportion is correct?

    1. 2/3 = 6/8
    2. 1/4 = 3/12
    3. 5/10 = 2/3
    4. 3/7 = 6/15
    Show answer

    Answer: 1/4 = 3/12

    1/4 = 3/12 because 1 × 12 = 12 and 4 × 3 = 12. Cross products are equal.

  4. A unit rate always has a denominator of 1.

    Show answer

    Answer: True

    A unit rate compares a quantity to exactly 1 unit, so the denominator is always 1.

  5. A car travels 150 miles in 3 hours. What is the unit rate?

    1. 50 miles per hour
    2. 45 miles per hour
    3. 60 miles per hour
    4. 153 miles per hour
    Show answer

    Answer: 50 miles per hour

    150 miles ÷ 3 hours = 50 miles per hour.

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