# Problems Involving Ratio

**Problems involving Ratio**

The ratio problem outlined below is represented with double sided counters. These can be used as an alternative representation to drawing a bar. The advantage of this is that they can be moved around to represent a change in relationship.

**Sam and Tom have football stickers in the ratio of 2 to 3. Altogether they have 25 stickers. If Sam gives half of his stickers to Tom, how many will Tom have?**

Sam

Tom

25

**Step 1**

Represent the ratio

**Step 2**

Recognise that if together the counters have a value of 25 then one has a value of 5

**Step 3**

Give half of Sam's stickers to Tom

Sam

Tom

25

**Step 4**

Multiply 5 by 4 to give the total value of Tom's stickers

**Answer:**

5 × 4 = 20

Notice how the many-to-one correspondence, as discussed above, allows this problem to be modelled efficiently. Children may at first use a one-to-one correspondence (i.e. each counter having a value of 1) and represent the problem using 25 counters, setting them out in a two to three ratio as illustrated below.

Children can then be supported in transferring their understanding to many-to-one correspondence as illustrated above (i.e. that every 5 counters in each column can be replaced with one counter worth 5). The flexibility of appreciating that the value of one counter can change depending on the context and the total quantity helps to develop the pupils’ algebraic reasoning.

Notice how the strategy explored above supports this KS2 SATS question

A gardener plants tulip bulbs in a flower bed.

She plants 3 red bulbs for every 4 white bulbs.

She plants 60 red bulbs.

**How many white bulbs does she plant?**

60

**Answer = 80 white bulbs**