Ruiya Math Blog · No. 3 · One Thinking Move
The denominator moved: the whole is not always the same.
A fraction answer can change even when nobody joins or leaves. The question may simply be asking you to compare the same people with a new whole.
A Ruiya-original PSLE-style practice guide for Primary 5, Primary 6 and PSLE
The first useful observation
The final question is not asking about all three rows. Its whole is only the first two rows.
The pupils are the same 12 people after they move. Only the number of seats we compare them with changes.
The challenge
The hall-seat challenge
- • A hall has 3 equal rows of seats.
- • Initially, 5/8 of all seats are occupied.
- • Then 1/5 of the pupils leave.
- • The remaining pupils sit only in the first 2 rows.
- • What fraction of the seats in the first 2 rows is now occupied?
This is a Ruiya-original analogue, not a past-paper question.
Why this strategy fits
Use a picture your brain already understands.
Imagine pouring the same amount of juice from a large jug into a smaller jug. The amount of juice has not changed, but it fills a larger fraction of the smaller jug.
Find the pupils left. At first, 5/8 of the hall is occupied. If 1/5 of those pupils leave, 4/5 remain. So the remaining pupils fill 4/5 × 5/8 = 1/2 of all the seats.
Name the new whole. The first two equal rows contain 2/3 of all seats. This is the denominator we need at the end.
Compare like with like. The remaining pupils are 1/2 of the whole hall, sitting in a space that is 2/3 of the whole hall. So 1/2 ÷ 2/3 = 1/2 × 3/2 = 3/4.
Check your answer
Use 24 seats. At first 15 are occupied; 3 leave, so 12 remain. Two rows have 16 seats. 12/16 = 3/4.
Answer: 3/4 of the seats in the first 2 rows are occupied.
Traps to watch for
The usual wrong turn is often a very reasonable first thought.
Keeping the old denominator
5/8 describes all three rows at the start. It cannot be the final answer because the final whole is only two rows.
Subtracting 1/5 from 5/8
The 1/5 is a fraction of the pupils, not a fraction of all seats. Multiply by 4/5 instead.
Thinking movement creates pupils
Moving pupils changes where they sit, not how many remain.
How an examiner can change it
Keep the thinking move, not the story.
The story may use shelves, buses, teams or survey groups. Watch for wording such as ‘only these boxes’, ‘the remaining group’ or ‘of the first two rows’. It is a signal that the whole has changed.
Try a fresh variation
A theatre has four equal sections. Initially 3/5 of all seats are filled. One third of the audience leaves, then the rest sit only in the first two sections. What fraction of those two sections is filled?
Do not look for the same numbers. Ask which relationship is still doing the work.
Outside the exam
A useful model helps you make a fair decision.
Occupancy reports, class attendance and warehouse capacity all depend on stating the whole clearly. ‘12 people’ means something different in a room for 16 than in a hall for 24.
Try it with Ruiya
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Send Ruiya a hard question and ask: “What is the relationship I should notice before I calculate?”
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