Word Problems — Part 2
4A · Chapter 8 · Lesson 8 · MATH-4A-C08-L08
Learning goal. Solve two-stage problems where the second fractional share may refer to a remainder.
Before you begin
Recall: 4A Ch6/7: equal partitions, mixed numbers and fraction operations; Word Problems — Part 1. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
A teacher has 8 cards. She gives 2/4 of them away. How many cards remain?
Show answer & reasoning
2/4 of 8 is 4; subtract from the original 8.
A teacher has 28 cards. She gives 3/4 of them away. How many cards remain?
Show answer & reasoning
3/4 of 28 is 21; subtract from the original 28.
A teacher has 42 cards. She gives 1/6 of them away. How many cards remain?
Show answer & reasoning
1/6 of 42 is 7; subtract from the original 42.
Key vocabulary
- unit fraction
- fraction with numerator one
- fraction of a set
- part of a discrete collection
- reference whole
- quantity to which a fraction applies
- remaining amount
- quantity after some part is removed
Understand the concept
Two-stage fraction stories can change the reference whole. A shop sells 1/3 of 60 items, leaving 40; then it sells 1/4 of the remaining 40, not 1/4 of the original 60. After the second sale, 30 remain. Always draw a new bar for the remaining whole before taking another fraction.
Model it: before-and-after fractional bars. Explain what every part of the model represents, then write a matching equation.
Fully worked examples
Read the problem first and predict the answer. Then study how each quantity is represented and calculated.
Worked example 1
A shop starts with 24 items and sells 5/6. It then sells 1/2 of what remains. How many items remain?
First 20 sell, leaving 4. Taking 1/2 of that remainder removes 2, so 2 remain.
Check your understanding 1
A shop starts with 16 items and sells 1/2. It then sells 1/4 of what remains. How many items remain?
Show answer & reasoning
First 8 sell, leaving 8. Taking 1/4 of that remainder removes 2, so 6 remain.
Worked example 2
A shop starts with 18 items and sells 3/6. It then sells 1/3 of what remains. How many items remain?
First 9 sell, leaving 9. Taking 1/3 of that remainder removes 3, so 6 remain.
Check your understanding 2
A shop starts with 50 items and sells 2/10. It then sells 1/2 of what remains. How many items remain?
Show answer & reasoning
First 10 sell, leaving 40. Taking 1/2 of that remainder removes 20, so 20 remain.
Worked example 3
A shop starts with 56 items and sells 2/8. It then sells 1/3 of what remains. How many items remain?
First 14 sell, leaving 42. Taking 1/3 of that remainder removes 14, so 28 remain.
Check your understanding 3
A shop starts with 24 items and sells 2/3. It then sells 1/2 of what remains. How many items remain?
Show answer & reasoning
First 16 sell, leaving 8. Taking 1/2 of that remainder removes 4, so 4 remain.
Guided practice
A shop starts with 60 items and sells 2/10. It then sells 1/2 of what remains. How many items remain?
Show answer & reasoning
First 12 sell, leaving 48. Taking 1/2 of that remainder removes 24, so 24 remain.
A shop starts with 12 items and sells 4/6. It then sells 1/4 of what remains. How many items remain?
Show answer & reasoning
First 8 sell, leaving 4. Taking 1/4 of that remainder removes 1, so 3 remain.
A shop starts with 4 items and sells 1/2. It then sells 1/2 of what remains. How many items remain?
Show answer & reasoning
First 2 sell, leaving 2. Taking 1/2 of that remainder removes 1, so 1 remain.
A shop starts with 6 items and sells 1/2. It then sells 1/3 of what remains. How many items remain?
Show answer & reasoning
First 3 sell, leaving 3. Taking 1/3 of that remainder removes 1, so 2 remain.
A shop starts with 16 items and sells 2/4. It then sells 1/2 of what remains. How many items remain?
Show answer & reasoning
First 8 sell, leaving 8. Taking 1/2 of that remainder removes 4, so 4 remain.
Independent practice
A shop starts with 20 items and sells 1/4. It then sells 1/3 of what remains. How many items remain?
Show answer & reasoning
First 5 sell, leaving 15. Taking 1/3 of that remainder removes 5, so 10 remain.
A shop starts with 15 items and sells 2/3. It then sells 1/5 of what remains. How many items remain?
Show answer & reasoning
First 10 sell, leaving 5. Taking 1/5 of that remainder removes 1, so 4 remain.
A shop starts with 18 items and sells 5/6. It then sells 1/3 of what remains. How many items remain?
Show answer & reasoning
First 15 sell, leaving 3. Taking 1/3 of that remainder removes 1, so 2 remain.
A shop starts with 42 items and sells 1/6. It then sells 1/5 of what remains. How many items remain?
Show answer & reasoning
First 7 sell, leaving 35. Taking 1/5 of that remainder removes 7, so 28 remain.
A shop starts with 16 items and sells 6/8. It then sells 1/4 of what remains. How many items remain?
Show answer & reasoning
First 12 sell, leaving 4. Taking 1/4 of that remainder removes 1, so 3 remain.
A shop starts with 25 items and sells 1/5. It then sells 1/4 of what remains. How many items remain?
Show answer & reasoning
First 5 sell, leaving 20. Taking 1/4 of that remainder removes 5, so 15 remain.
A shop starts with 36 items and sells 3/4. It then sells 1/3 of what remains. How many items remain?
Show answer & reasoning
First 27 sell, leaving 9. Taking 1/3 of that remainder removes 3, so 6 remain.
A shop starts with 54 items and sells 2/6. It then sells 1/2 of what remains. How many items remain?
Show answer & reasoning
First 18 sell, leaving 36. Taking 1/2 of that remainder removes 18, so 18 remain.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A shop has 84 boxes and sells 1/3 of its stock on Monday and 1/2 of what remains on Tuesday. Did it sell 5/6 of its original stock? Explain.
Show answer & reasoning
After Monday 2/3 remains; on Tuesday half of that is 1/3 of original, so 2/3 was sold altogether.
A shop starts with 24 items and sells 3/6. It then sells 1/4 of what remains. How many items remain?
Show answer & reasoning
First 12 sell, leaving 12. Taking 1/4 of that remainder removes 3, so 9 remain.
Exit check
A shop starts with 28 items and sells 1/4. It then sells 1/3 of what remains. How many items remain?
Show answer & reasoning
First 7 sell, leaving 21. Taking 1/3 of that remainder removes 7, so 14 remain.
A shop starts with 18 items and sells 4/6. It then sells 1/2 of what remains. How many items remain?
Show answer & reasoning
First 12 sell, leaving 6. Taking 1/2 of that remainder removes 3, so 3 remain.
A shop starts with 32 items and sells 1/8. It then sells 1/2 of what remains. How many items remain?
Show answer & reasoning
First 4 sell, leaving 28. Taking 1/2 of that remainder removes 14, so 14 remain.