Mathematics • Year 8 • Unit 1 • Lesson 17
Dividing a Quantity in a Given Ratio
Use the unitary method, sum of parts, value of one part, then multiply for each share, to split money, mass and time. One worked example, one guided example with blanks, then eight independent problems including reverse problems.
1. I do, fully worked example
Watch the unitary method in action. Sum the parts, find the value of ONE part, then multiply.
Problem. Three friends share $480 in the ratio 1 : 2 : 3. How much does each receive?
Step 1, Add the ratio parts (total parts).
1 + 2 + 3 = 6 parts
Reason: this tells us the $480 is being split into 6 equal "parts" (even though the shares are unequal).
Step 2, Find the value of 1 part.
$480 ÷ 6 = $80 per part
Reason: total ÷ number of parts.
Step 3, Multiply each ratio number by the 1-part value.
Friend A: 1 × $80 = $80
Friend B: 2 × $80 = $160
Friend C: 3 × $80 = $240
Reason: each share is (ratio number) × (value of 1 part).
Step 4, Sum check.
$80 + $160 + $240 = $480 ✓
Reason: the shares MUST add back to the original total, if they don't, you have an arithmetic mistake.
Answer: $80, $160, $240.
2. We do, fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. Split $600 in the ratio 2 : 3. Find each share.
Step 1, Total parts: 2 + 3 = ______
Step 2, Value of 1 part:
$600 ÷ ______ = $______
Step 3, Each share:
First share: 2 × $______ = $______
Second share: 3 × $______ = $______
Step 4, Sum check:
$______ + $______ = $______ (should be $600) ✓
3. You do, independent practice
Show your working in the space under each problem. The first four are foundation (two-part splits). The middle two are standard (three-part splits). The last two are extension (reverse problems, given one share, find the total).
Foundation, two-part splits
3.1 Split $60 in the ratio 1 : 2. 1 mark
3.2 Split $140 in the ratio 3 : 4. 1 mark
3.3 Split 24 m in the ratio 3 : 5. 1 mark
3.4 Split $300 in the ratio 2 : 3. 1 mark
Standard, three-part splits
3.5 Split $210 in the ratio 2 : 3 : 2. Show your sum check at the end. 2 marks
3.6 A 36 kg bag of dry concrete mix is in the ratio 2 : 3 : 4 (cement : sand : gravel). How much of each? 2 marks
Extension, reverse problems
3.7 Two people share a sum of money in the ratio 4 : 5. The smaller share is $160. (a) Find the value of 1 part. (b) Find the total amount shared. 2 marks
3.8 Two friends split a pizza in the ratio 3 : 2. The smaller share is 4 slices. How many slices in total? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (faded $600 in 2 : 3)
Step 1: 2 + 3 = 5 parts.
Step 2: $600 ÷ 5 = $120 per part.
Step 3: First = 2 × $120 = $240; Second = 3 × $120 = $360.
Step 4: $240 + $360 = $600 ✓.
3.1, $60 in 1 : 2
Parts = 3; 1 part = $60 ÷ 3 = $20. Shares: $20 and $40. Check: $20 + $40 = $60 ✓.
3.2, $140 in 3 : 4
Parts = 7; 1 part = $140 ÷ 7 = $20. Shares: $60 and $80. Check: $60 + $80 = $140 ✓.
3.3-24 m in 3 : 5
Parts = 8; 1 part = 24 ÷ 8 = 3 m. Shares: 9 m and 15 m. Check: 9 + 15 = 24 ✓.
3.4, $300 in 2 : 3
Parts = 5; 1 part = $300 ÷ 5 = $60. Shares: $120 and $180. Check: $120 + $180 = $300 ✓.
3.5, $210 in 2 : 3 : 2
Parts = 7; 1 part = $210 ÷ 7 = $30. Shares: $60, $90, $60. Check: $60 + $90 + $60 = $210 ✓.
3.6-36 kg in 2 : 3 : 4
Parts = 9; 1 part = 36 ÷ 9 = 4 kg. Cement: 8 kg; sand: 12 kg; gravel: 16 kg. Check: 8 + 12 + 16 = 36 ✓.
3.7, Reverse: 4 : 5, smaller share $160
(a) Smaller share is 4 parts, so 1 part = $160 ÷ 4 = $40.
(b) Total parts = 4 + 5 = 9. Total amount = 9 × $40 = $360. (Or: bigger share = 5 × $40 = $200, total = $160 + $200 = $360.)
3.8, Reverse: pizza in 3 : 2, smaller = 4 slices
Smaller share is 2 parts = 4 slices, so 1 part = 2 slices. Total parts = 3 + 2 = 5. Total = 5 × 2 = 10 slices.