Comparing Ratios

Truffles

To compare two ratios with different terms, you need to find equivalent ratios.

If one box of chocolates has a ratio of 3 truffles to 2 toffees (3:2) and another box has a ratio of 7 truffles to 5 toffees (7:5). Which has the higher ratio of truffles?

One way to find the solution is to change each ratio to equivalent ratios with the same second term.

For 3:2 and 7:5 find the lowest common multiple of (second terms) 2 and 5 (10)

Multiples of 2: 2, 4, 6, 8, 10

Multiples of 5: 5, 10, 15, 20

The lowest common multiple of 2 and 5 is 10.

Change each ratio to an equivalent ratio with 10 as the second term.

For the ratio 3:2, multiply each term by 5.

3:2 = 15:10

For the ratio 7:5, what will you multiply each term by?

7:5 = ? : 10

7:5 = 14:10 (multiply each term by 2)

They both have 10 as the 2nd term so you can compare the first term.

15:10 compared to 14:10

Since truffles are my favourite I'd prefer the box with the 3:2 (15:10) ratio! Which would you choose?

A second way to solve this is to write equivalent ratios with 1 as the second term.

To write 3:2 with the second term as 1 you divide both terms by 2

so 3:2 = 3/2 : 1 or 1.5:1

To write 7:5 with 1 as the second term you divide both terms by 5

so 7:5 = 7/5 : 1 or 1.4:1

Now you can compare the first term to see the 1.5>1.4. And it looks like I'd still take the first box! Yum.

Example

The Canucks scored 2 goals with 8 shots on goal and the Bruins scored 3 goals with 11 shots on goal. Which team had the better record of goals compared to shots on goal?

There are several ways to solve this type of problem. Here are two ways.

Solution 1

Express the ratios with the same second term.

2:8

3:11

A common multiple of both 8 and 11 is 88.

2:8 = 22:88 (multiply each term by 11)

3:11 = 24:88 (multiply each term by 8)

It's clear that 24:88 it's better than 22:88 so the Bruins have the better record.

Solution 2

Express the ratios with 1 as the second term.

2:8 = ? : 1

Divide both terms by 8

2:8 = 0.25:1

3:11 = ? : 1

Divide both terms by 11

3:11 = 0.27:1 (rounded)

Again, it's clear that 0.27:1 is better than 0.25:1.