Finding New Ratios
Learn to combine ratios and work with three-term ratios
1. Combining Two Ratios
The Strategy
When you have two ratios with a common term, you can combine them into a three-term ratio.
- Identify the common term in both ratios
- Make the common term equal in both ratios
- Combine into one three-term ratio
Example 1: Money Problem
The ratio of Ken's money to Suzy's money is 2 : 3.
The ratio of Ken's money to Ming's money is 6 : 5.
Find the ratio of Ken's money to Suzy's money to Ming's money.
Common term: Ken's money (appears in both ratios)
Ken : Suzy = 2 : 3
Ken is 2 in this ratio and 6 in the other. 2 × 3 = 6, so multiply both terms by 3 → 6 : 9
Ken : Ming = 6 : 5
(already has Ken = 6)
Ken : Suzy : Ming = 6 : 9 : 5
Example 2: Squares, Triangles, and Circles
Anna has some shapes.
The ratio of squares to triangles is 1 : 4.
The ratio of squares to circles is 2 : 5.
Find the ratio of squares to triangles to circles.
Common term: Number of squares
Squares are 1 in one ratio and 2 in the other. Make squares 2 in both ratios.
Squares : Triangles = 1 : 4
1 × 2 = 2, so multiply both terms by 2 → 2 : 8
Squares : Circles = 2 : 5
(already has squares = 2)
Squares : Triangles : Circles = 2 : 8 : 5
What is the ratio of triangles to circles? 8 : 5
2. Working with Three-Term Ratios
Extracting Two-Term Ratios
From any three-term ratio A : B : C, you can find:
- A : B (first two terms)
- B : C (last two terms)
- A : C (first and last terms)
Example 3: Extracting from 6 : 9 : 5
Given Ken : Suzy : Ming = 6 : 9 : 5, find Ken : Suzy, Suzy : Ming and Ken : Ming, each in its simplest form.
Ken : Suzy = 6 : 9 = 2 : 3 (simplify by ÷3)
Suzy : Ming = 9 : 5 (already in simplest form)
Ken : Ming = 6 : 5 (already in simplest form)
Example 4: Finding Values from Three-Term Ratio
Red, blue and green paint are mixed in the ratio 3 : 5 : 2 by volume.
If there are 15 litres of blue paint, find the total amount of paint.
Step 1: Blue = 5 units = 15 L
Step 2: 1 unit = 15 ÷ 5 = 3 L
Step 3: Total units = 3 + 5 + 2 = 10 units
Step 4: Total paint = 10 × 3 = 30 L
3. Complex Problems
Example 5: Art Exhibition
At an art exhibition:
• Ratio of adults to boys = 3 : 2
• Ratio of boys to girls = 1 : 4
There are 30 boys. Find the total number of people.
Common term: Boys
Adults : Boys = 3 : 2 (Boys = 2)
Boys : Girls = 1 : 4 → ×2 → 2 : 8 (Boys = 2)
Solution:
2 units = 30 → 1 unit = 15
Total = 3 + 2 + 8 = 13 units
13 × 15 = 195 people
Key Points to Remember
- ✓ Identify the common term between two ratios
- ✓ Make the common term equal by scaling
- ✓ Combine to form a three-term ratio
- ✓ You can extract two-term ratios from any three-term ratio
- ✓ Find the value of 1 unit to solve for actual quantities