Divide a quantity in a given ratio

What you will learn

• Divide a quantity in a given ratio

Models used: number sentences, stories, diagrams.

Before you start

• What a ratio compares

One unit is the same size everywhere

Red has 2 equal units. Blue has 3 of the same units.

1. A small box represents one equal unit. Read the number inside to find the value of that unit.

   Red has 2 equal units. Blue has 3 of the same units.

   red : blue = 2 : 3

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

2. A small box represents one equal unit. Read the number inside to find the value of that unit.

   Together they have five units. If there are 20 counters, each unit is 4 counters.

   20 ÷ 5 = 4

   Keep the number of units separate from the value of one unit. Multiplying them gives the amount in that row.

3. A small box represents one equal unit. Read the number inside to find the value of that unit.

   Red gets 8 counters. Blue gets 12. Their shares still have the ratio 2 to 3.

   2 × 4 = 8; 3 × 4 = 12

   Keep the number of units separate from the value of one unit. Multiplying them gives the amount in that row.

Worked example: A new example, together

Share 28 counters in the ratio 4:3. How many belong to the first share?

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

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

   Divide by the total unit count before multiplying by the requested share.

   Red

   Blue

   Total amount: 28

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

   There are 7 equal ratio units.

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

   Red

   Blue

   Total amount: 28

   Share 28 counters in the ratio 4:3. How many belong to the first share?

   16

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

   There are 7 equal ratio units.

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

   Red

   Blue

   Red : Blue = 4 : 3 · 7 units = 28

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

   One unit = 28 ÷ 7 = 4.

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

   Red

   Blue

   Red : Blue = 4 : 3 · 7 units = 28

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

   The first share is 4 × 4 = 16.

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

   Red

   Blue

   Red : Blue = 4 : 3 · 7 units = 28

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

   Check the result against the question we started with.

   Shares 16 + 12 total 28.

   We found 16. Divide by the total unit count before multiplying by the requested share.

   Red

   Blue

   Red : Blue = 4 : 3 · 7 units = 28

Common mistakes to discuss

• Dividing by one side of the ratio. Divide by the total number of units. Sharing $40 in 3 : 5 uses 8 units, so one unit is $5.

Check your understanding

• Divide a quantity in a given 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.