Find a missing term in a ratio

What you will learn

• Find a missing term in a ratio

Models used: number sentences, stories, tables.

Before you start

• Dividing a quantity in a ratio

Find the value of one ratio unit

Red : blue is 2 : 3. If there are 4 red counters, how many blue counters are there?

1. All the small boxes represent equally sized units, even when they belong to different rows.

   Red has two equal units and blue has three. We know the two red units together are worth four counters.

   2 red units = 4 counters

   The counts of equal units describe the comparison. Their actual size comes from a given quantity or total.

   Red

   Blue

   Red : Blue = 2 : 3

2. All the small boxes represent equally sized units, even when they belong to different rows.

   Share the four red counters between its two units. Each unit is worth two counters in both colours.

   4 ÷ 2 = 2 counters per unit

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Red224 altogether

   Blue2226 altogether

   Red : Blue = 2 : 3 · 5 units = 10 · 1 unit = 2

3. All the small boxes represent equally sized units, even when they belong to different rows.

   Blue has three of those same-sized units, so it has six counters. Fill the missing ratio term with six.

   3 × 2 = 6; 2 : 3 = 4 : 6

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Red224 altogether

   Blue2226 altogether

   Red : Blue = 2 : 3 · 5 units = 10 · 1 unit = 2

Worked example: A new example, together

Complete 4:2 = 8:□.

1. Both quantities must grow by the same factor. Read each value with the heading above its column.

   Keep the quantity labels in order. Distinguish one part from the sum of all ratio parts.

   Equivalent ratios scale every term by the same factor.

   Both quantities must grow by the same factor.

   First quantity	Second quantity
   4	2
   8	?

2. Both quantities must grow by the same factor. Read each value with the heading above its column.

   4 becomes 8, so the scale factor is 2.

   Both quantities must grow by the same factor.

   First quantity	Second quantity
   4	2
   8	?

   Complete 4:2 = 8:□.

   4

3. Both quantities must grow by the same factor. Read each value with the heading above its column.

   4 becomes 8, so the scale factor is 2.

   Both quantities must grow by the same factor.

   First quantity	Second quantity
   4	2
   8	?

4. Both quantities must grow by the same factor. Read each value with the heading above its column.

   Scale the second term by the same factor.

   Both quantities must grow by the same factor.

   First quantity	Second quantity
   4	2
   8	?

5. Both quantities must grow by the same factor. Read each value with the heading above its column.

   2 × 2 = 4.

   Both quantities must grow by the same factor.

   First quantity	Second quantity
   4	2
   8	?

6. Both quantities must grow by the same factor. Read each value with the heading above its column.

   Check the result against the question we started with.

   8 ÷ 4 = 4 ÷ 2 = 2.

   We found 4. Equivalent ratios scale every term by the same factor.

   Both quantities must grow by the same factor.

   First quantity	Second quantity
   4	2
   8	?

Common mistakes to discuss

• Using the difference between the terms. Ratios keep equal units, not equal gaps. If 2 : 3 becomes 6 : ?, each unit trebles, so the answer is 9.

Check your understanding

• Find a missing term in a ratio
• explain a ratio through labelled equal units
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.