Dividing in a Given Ratio - How to share that cake

Cake. The fruit makes it ok.

Not everything in life is equal.

And not everything is shared equally - you don’t really want to split your chocolate cake and share it out… but sometimes you have to.

So how do you split things up when they aren’t equal.

For example, how do you split a cake in the ratio 1 : 3 (in your favour)?

Let’s look at how we would split something equally, i.e 1 : 1.

If you add both sides of the ratio, 1 + 1 = 2, so there are 2 parts in total.

So you take your cake and cut it into 2 equal parts (or shares).

Then the ratio shows you how many parts each person gets.

In this case, each person gets 1 part each or 1 share each.

Its the same process for when the ratio is different.

Let’s look at how to share a cake in the ratio: 1 : 3

1 + 3 = 4, so there are 4 shares.

Split the cake into 4 equal parts.

Then you multiply to work out how many parts everyone gets.

1 person gets 1 out of the 4 pieces and the other person gets 3 out of the 4 pieces.

Of course your maths questions aren’t always going to be that easy, let’s look at some other examples.

Let’s say you are given £120 for some work you did on the weekend.

You completed that work with the help of a good friend.

You’ve both agreed that you did 6 hours of work and your friend did 4, so you will split the money according to how many hours you both did.

So you need to split £120, in the ratio 6 : 4

6 + 4 = 10, so there are 10 parts.

£120 / 10 = £12. So the value of each part is £12 - this bit is super important.

Now we know that 1 part, which is 1 hour in this case, is worth £12.

If you worked 6 hours, you get 6 x 12, £72 out of the £120.

And then your friend will get 4 x 12 = £48.

Example 2:

A recipe needs to be divided into portions in the ratio 2:3. If there are 20 grams of sugar in the smaller portion, how much sugar is there in the larger portion?

Solution:

1. Set up the problem:
   • Total parts in the ratio = 2 + 3 = 5 parts.
   • The smaller portion (2 parts) has 20 grams of sugar.
2. Find the value of one part:
   20 ÷ 2 = 10 grams per part.
3. Calculate the amount in the larger portion:
   3 × 10 = 30 grams

Answer: There is 30 grams of sugar in the larger portion.

Here's a harder example:

Anna and Ben are sharing a sum of money in the ratio 3:5. After they split the money, Anna receives £120 less than Ben. How much money did they share in total?

Solution:

1. Set up the problem:
   • Let the total amount of money be T.
   • According to the ratio 3:5, Anna’s share will be ³/₈ T and Ben’s share will be ⁵/₈ T.
2. Translate information into an equation:
   • We know that Ben’s share is £120 more than Anna’s share.
   • So, ⁵/₈ T − ³/₈ T = 120.
3. Solve the equation:
   • ²/₈ T = 120
   • Simplify ²/₈ to ¹/₄, so:
   • ¹/₄ T = 120
   • Multiply both sides by 4:
   • T = 120 × 4 = 480

Answer: The total amount of money they shared is £480.

Check the answer:

• Anna’s share: ³/₈ × 480 = 180
• Ben’s share: ⁵/₈ × 480 = 300
• Difference: 300 − 180 = 120, which matches the information given.

Example 4:

Anna and Ben share 560 ml of juice in the ratio 5:3. How much juice does each person get?

Solution:

1. Set up the problem:
   • Total parts in the ratio = 5 + 3 = 8 parts.
   • Total amount to be divided = 560 ml.
2. Find the value of one part:
   560 ÷ 8 = 70 ml per part.
3. Calculate each person’s share:
   • Anna’s share = 5 × 70 = 350 ml
   • Ben’s share = 3 × 70 = 210 ml

Answer: Anna gets 350 ml and Ben gets 210 ml.

Example 5:

James and Lily have £420, which they want to share in the ratio 3:4. How much does each person receive?

Solution:

1. Set up the problem:
   • Total parts in the ratio = 3 + 4 = 7 parts.
   • Total amount to be divided = £420.
2. Find the value of one part:
   420 ÷ 7 = £60 per part.
3. Calculate each person’s share:
   • James’s share = 3 × 60 = £180
   • Lily’s share = 4 × 60 = £240

Answer: James receives £180 and Lily receives £240.

Practice Quiz: Dividing Amounts in Ratios

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