Proportional Reasoning-1 · Lesson 2 of 8
Ratios and Their Simplest Form
“Write ordered ratios, form equivalent ratios, simplify using the HCF and recognise proportion through equal simplest forms.”
• Read a ratio as a comparison in a stated order. • Form equivalent ratios by multiplying or dividing both terms. • Simplify whole-number ratios using the HCF. • Interpret the proportion symbol ::.
You found that several image sizes can preserve the same proportions. Writing every size separately can hide what they have in common: 60 mm by 40 mm and 90 mm by 60 mm look like different pairs of numbers. We need a compact way to describe their shared relationship. A ratio does this by comparing the first quantity with the second in a fixed order, without requiring either quantity to have one particular size.
The order and the meaning of each quantity matter. As you work, keep the labels beside the numbers until you are sure what each term represents.
7.2 Ratios
For image A, width is 60 mm and height is 40 mm. Write its width-to-height ratio as 60 : 40, read “60 to 40.” The numbers 60 and 40 are the terms of the ratio.
A ratio a : b means that for every a units of the first quantity, there are b units of the second. It describes a comparison, not the total and not the difference. Here every 60 mm of width goes with 40 mm of height.
An ordered comparison written a : b. The first term refers to the first named quantity and the second term to the second named quantity.
Height-to-width reverses the order: it is 40 : 60. You cannot switch the numbers while keeping the original labels. Ratios such as coffee : milk and milk : coffee describe opposite comparisons.
Multiplying both terms by the same positive factor makes an equivalent ratio. From 60 : 40, multiplying both by 1/2 gives 30 : 20. Multiplying both by 3/2 gives 90 : 60. Changing only one term generally changes the relationship.
Problem
There are 8 red counters and 12 blue counters. Write red : blue and blue : red.
Problem
Give two ratios equivalent to 4 : 9.
7.3 Ratios in their Simplest Form
The simplest form removes common whole-number factors. For 60 : 40, the highest common factor is 20. Divide both terms by 20 to obtain 3 : 2. The numbers 3 and 2 have no common factor larger than 1.
You could divide in smaller stages: 60 : 40 → 6 : 4 → 3 : 2. Dividing by the HCF reaches the simplest whole-number form in one step. If you stop at 6 : 4, the ratio is equivalent but not yet simplest.
For 90 : 60, the HCF is 30, giving the same simplest ratio 3 : 2. The original amounts differ, but their comparisons match. Image B gives 40 : 20 = 2 : 1, and image E gives 60 : 60 = 1 : 1, so those do not match A.
For a ratio of positive whole numbers, divide both terms by their HCF. In simplest form the terms have no common factor greater than 1.
| Ratio | HCF | Simplest form |
|---|---|---|
| 60 : 40 | 20 | 3 : 2 |
| 30 : 20 | 10 | 3 : 2 |
| 90 : 60 | 30 | 3 : 2 |
| 40 : 20 | 20 | 2 : 1 |
| 60 : 60 | 60 | 1 : 1 |
Problem
Simplify 72 : 96.
When two ratios have the same simplest form, they are in proportion. The notation a : b :: c : d states that a : b and c : d represent the same comparison. Read the middle symbol as “is in proportion to.”
Thus 60 : 40 :: 30 : 20 is true. However, 60 : 40 :: 40 : 20 is false because the simplest forms are 3 : 2 and 2 : 1. Do not decide by whether one or two numbers happen to appear in both ratios.
A statement that two ratios represent the same relationship. We write a : b :: c : d when a : b and c : d are equivalent.
Problem
Are 21 : 6 and 35 : 10 proportional?
A ratio of 3 red counters to 2 blue counters gives 5 counters in one complete group. The 2 refers to blue counters, not to the total. Later, this distinction will help you share a whole in a given ratio.
Quiz
What does a width-to-height ratio of 3 : 2 mean?
What is the simplest form of 84 : 126?
Which ratio is equivalent to 4 : 9?
What does a : b :: c : d assert?
Which step is valid when simplifying 18 : 24?
Practice Problems
- Write the ratio of 15 green counters to 10 yellow counters in simplest form.
- Simplify 48 : 72, showing the common divisor.
- Give three ratios proportional to 4 : 9.
- Are 8 : 3 and 24 : 6 proportional? Explain.
- A rectangle is 45 cm wide and 30 cm high. Another is 18 cm wide and 12 cm high. Compare their width-to-height ratios and then reverse both ratios.
1. Green : yellow = 15 : 10. 2. Divide both by HCF 5 to get 3 : 2.
1. The HCF is 24. 2. 48 ÷ 24 : 72 ÷ 24 = 2 : 3.
1. Use common factors 2, 3 and 4. 2. The ratios are 8 : 18, 12 : 27 and 16 : 36.
1. 8 : 3 is already simplest. 24 : 6 simplifies to 4 : 1. 2. The simplest forms differ. Equivalently, 8 becomes 24 by ×3 but 3 becomes 6 by ×2.
1. 45 : 30 and 18 : 12 both simplify to 3 : 2. 2. Their width-to-height ratios are proportional. 3. Reversing both gives height-to-width ratios 2 : 3. Reversing just one would compare unlike orders.
Key Takeaways
• A ratio compares quantities in a stated order. • Its terms are not automatically the total or the difference. • Multiply or divide both terms by the same factor to preserve a ratio. • Divide whole-number terms by their HCF to reach simplest form. • Ratios in proportion have the same simplest form and may describe different actual sizes.