Simplifying Ratios and Sharing in a Ratio

Let’s share 24 lollies between Alice and Bob in the ratio 1 to 3. For every 1 Alice gets, Bob gets 3. Bob ends up with 3 times as many. Now let’s share the same 24 lollies in the ratio 2 to 6. For every 2 Alice gets, Bob gets 6. Bob still has 3 times as many! So 2 to 6 is the same ratio as 1 to 3. Divide both sides by 2 and you’re back to 1 to 3.

Simplifying a ratio is just like simplifying a fraction. Take six fifteenths. Both 6 and 15 divide by 3. 6 divided by 3 is 2, 15 divided by 3 is 5. So six fifteenths simplifies to two fifths. Ratios work the same way — 6 to 15 simplifies to 2 to 5.

Simplify 24 to 36. Both are even, so divide by 2. That gives 12 to 18. Still both even — divide by 2 again: 6 to 9. Now divide by 3: 2 to 3. Fully simplified. You could also do it in one go — the highest common factor of 24 and 36 is 12, so dividing both by 12 gives 2 to 3 straight away.

Sometimes we need to find a missing number in an equivalent ratio. 2 to 5 equals 6 to what? What do we multiply 2 by to make 6? That’s times 3. So we multiply 5 by 3 as well. 5 times 3 is 15. The answer is 15.

Here’s a trickier one. 8 to 12 equals 6 to what? 6 doesn’t divide evenly into 8, so we can’t find a simple multiplier. Instead, let’s simplify first. 8 to 12 — divide both by 4 — gives us 2 to 3. Now, what do we multiply 2 by to make 6? That’s times 3. So multiply 3 by 3 as well. 3 times 3 is 9. The answer is 9.

Now let’s look at sharing in a ratio.

Share 12 apples between Alice and Bob in the ratio 1 to 3. First, find the total number of parts. 1 plus 3 equals 4 parts. Now divide the apples into 4 equal groups. 12 divided by 4 equals 3 apples per group. Alice gets 1 part — that’s 3 apples. Bob gets 3 parts — that’s 9 apples.

Share 20 sweets between Alice and Bob in the ratio 2 to 3. First, find the total number of parts. 2 plus 3 equals 5 parts. Now divide the sweets into 5 equal groups. 20 divided by 5 equals 4 sweets per group. Alice gets 2 parts — that’s 8 sweets. Bob gets 3 parts — that’s 12 sweets.

Now let’s try one without a picture. Share 45 dollars in the ratio 4 to 5. How many parts is that? 4 plus 5 gives us 9 parts. So each part is worth 45 divided by 9, which is 5 dollars. That means the first person gets 4 lots of 5 dollars — that’s 20 dollars. And the second person gets 5 lots of 5 dollars — 25 dollars. Quick check: 20 plus 25 equals 45. Perfect.