How to use this guide
Read each section carefully and follow the worked examples. Pay attention to how quantities are compared and how proportions are solved step by step.
Big idea: Ratios compare quantities, while proportions show that two ratios are equal.
What is a Ratio?
\[
2 : 3
\]
A ratio compares two quantities of the same kind.
Example:
\[
\text{Boys : Girls} = 2 : 3
\]
This means for every 2 boys, there are 3 girls.
If the ratio of cats to dogs is 2:3 and there are 15 dogs, find the number of cats.
\[ \text{Cats : Dogs} = 2 : 3 \] \[ \frac{x}{15} = \frac{2}{3} \] \[ 3x = 30 \] \[ x = 10 \] \[ \text{Number of cats} = 10 \]Simplifying Ratios
Simplify:
\[
12 : 18
\]
Step 1: Find the greatest common factor.
\[
\text{GCF} = 6
\]
Step 2: Divide both terms by 6.
\[
12 \div 6 : 18 \div 6
\]
\[
2 : 3
\]
The simplified ratio is:
\[
12 : 18 = 2 : 3
\]
Try This Question
Equivalent Ratios
\[
2 : 5 = 4 : 10 = 6 : 15
\]
Equivalent ratios have the same value.
Multiply both terms by the same number:
\[
2 : 5 \times 2 = 4 : 10
\]
Sharing in a Ratio
Divide \$60 in the ratio
\[
2 : 3
\]
Step 1: Find total parts.
\[
2 + 3 = 5
\]
Step 2: Find value of one part.
\[
60 \div 5 = 12
\]
Step 3: Multiply by each part.
\[
2 \times 12 = 24
\]
\[
3 \times 12 = 36
\]
Answer: The amounts are \$24 and \$36.
What is Proportion?
\[
\frac{2}{3} = \frac{4}{6}
\]
A proportion states that two ratios or fractions are equal.
\[
2 : 3 = 4 : 6
\]
Solving Proportions
Solve:
\[
\frac{3}{5} = \frac{x}{20}
\]
Step 1: Cross multiply.
\[
3 \times 20 = 5x
\]
\[
60 = 5x
\]
Step 2: Divide by 5.
\[
x = 12
\]
Answer:
\[
x = 12
\]
Unit Rates
A car travels 180 km in 3 hours.
Find the unit rate:
\[
180 \div 3 = 60
\]
The car travels at:
\[
60 \text{ km/h}
\]
A unit rate compares a quantity to 1 unit of another quantity.
Direct and Inverse Proportion
Direct proportion:
\[
y \propto x
\]
If
\[
x = 2, \quad y = 6
\]
then
\[
y = 3x
\]
Inverse proportion:
\[
y \propto \frac{1}{x}
\]
In direct proportion, both quantities increase together. In inverse proportion, one increases while the other decreases.
Mixed Practice
Simplify:
\[
15 : 25
\]
\[
3 : 5
\]
Solve:
\[
\frac{4}{7} = \frac{x}{21}
\]
\[
x = 12
\]
Share \$80 in the ratio
\[
1 : 3
\]
\[
20, \quad 60
\]