Lesson 2: Proportions
Lesson 2: Proportions
Solve proportions and ratio problems.
A proportion is an equivalent relationship of quantities. It is a name we give to a statement that two ratios are equal. It can be represented in two ways: two equal fractions or as a : b = c : d. We can determine whether a proportion is true or false. We can also find an unknown in a proportion or solve real life problems using proportions.
We often use proportions to convert to or from percentages.
For example, what is 10% of 45?
We can set up a proportion: 45 corresponds to 100%, then what corresponds to 10%?
10 : 100 = n : 45
n = (45 x 10)/100
n = 4.5
There are many examples in math in which the exact answer is required.
For example, if a doctor prescribed to a patient a drug requiring a specific measure, the doctor would have to prescribe the exact amount; if the doctor simply estimated the amount, a patient could have severe side effects or it could even lead to a fatal outcome. Thus, understanding proportions could save lives!
Proportions can be solved using visual models. However, it is important to understand and learn how to quickly set them up and solve algebraically.
Key ideas: Proportions
A ratio compares two quantities. A proportion says two ratios are equal: a/b = c/d. To solve a proportion, use equivalent fractions or cross-multiply: a × d = b × c.
A unit rate is a rate per one unit, like $0.75 per item or 80 km per hour. It is handy for comparing prices.
Worked examples
Example 1
Solve x/12 = 3/4.
- 12 = 4 × 3, so x = 3 × 3.
- x = 9
Example 2
A recipe uses 3 cups of flour for 24 cookies. How much flour is needed for 40 cookies?
- 3/24 = f/40, so 24f = 120.
- f = 5 cups
Example 3
On a map, 1 cm represents 25 km. Two towns are 7.2 cm apart. How far apart are they really?
- 7.2 × 25 = 180 km
Example 4
6 granola bars cost $4.50. What is the unit price, and how much do 10 cost?
- 4.50 ÷ 6 = $0.75 per bar.
- 10 × 0.75 = $7.50
Common mistakes
- Setting up the ratios with the quantities in different orders (cups/cookies on one side, cookies/cups on the other).
- Forgetting units in the answer.
Questions students ask
How do I know a situation is proportional?
If doubling one quantity doubles the other (and zero gives zero), it is proportional.
Grade 8 Proportions Quiz
Check your understanding
Try the quiz for this lesson. You’ll see right away if each answer is correct.