eduKatePunggol · Secondary 1 Mathematics
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A Fraction of the Remainder Has a New Whole. Choose the reading route closest to your question.
Route 1 · Chapters 1–2
Route 2 · Chapters 3–4
Route 3 · Chapters 5–6
Route 4 · Chapters 7–8
Full chapter index · Independent practice · Secondary 1 Mathematics support
If your child takes one third of the original amount when the question says one third of the remainder, label the new whole first. Start with sixty counters and remove half. Thirty remain. Removing one third of that remainder takes ten more, leaving twenty—not ten.
Secondary 1 Mathematics tuition in Punggol can connect the wording to a changing reference amount. A Punggol Secondary 1 Maths tutor should ask, ‘A third of which quantity?’ The denominator describes equal parts of the stated whole; it does not automatically keep referring to the amount at the beginning.
This guide uses original counter and quantity examples, with no claim about a particular school’s assessment. It separates the amount removed from the amount left and includes reverse questions. Small-group 3-pax support should help each learner identify the current whole before applying the next fraction.
Full chapter index
Identify the current whole · Chapters 1–2
Track removals and remainders · Chapters 3–4
Reverse and check the process · Chapters 5–6
Practise and continue · Chapters 7–8
Chapter 1 of 8 · Identify the current whole
1. What changes when the question says remainder?
The remainder is what is left after the earlier action. In the main example, the original whole is sixty, but the next reference whole is thirty. One third of thirty equals ten. The word remainder changes the quantity to which the fraction applies, even though both stages belong to the same problem.
Write a short label at each stage: original sixty, after first removal thirty, second removal ten, final remainder twenty. This prevents the two different reference amounts from disappearing into a single line of arithmetic. A child may know fraction multiplication and still miss the change in the wording.
Chapter 2 of 8 · Identify the current whole
2. Why does subtracting both fractions from one fail here?
Calculating 1 − 1/2 − 1/3 = 1/6 treats both removed fractions as fractions of the original whole. Applied to sixty, it leaves ten. That calculation would suit a different question: remove half of the original, then another third of the original.
In our actual question, the second removal is one third of the remaining half. As a fraction of the original, that removal is 1/3 × 1/2 = 1/6. Total removed is therefore 1/2 + 1/6 = 2/3, leaving one third. One third of sixty is twenty, matching the staged calculation.
| Stage | Amount removed | Amount left |
|---|---|---|
| Original | — | 60 |
| Remove half of original | 30 | 30 |
| Remove a third of the remainder | 10 | 20 |
Chapter 3 of 8 · Track removals and remainders
3. Can we calculate the remaining fraction directly?
Yes. After removing half, the retained fraction is one half of the original. Removing one third of what remains retains two thirds of that amount. The final fraction is 1/2 × 2/3 = 1/3 of the original. This approach tracks the fraction left at each stage.
Keep removed and retained fractions distinct. If a question says ‘uses two fifths of the remainder’, then three fifths of that remainder stays. Multiplying by two fifths would find the second amount used, not the final quantity left. Name the requested output before choosing the multiplier.
Chapter 4 of 8 · Track removals and remainders
4. How does a different example work?
Begin with eighty counters. Remove one quarter of the original, taking twenty and leaving sixty. Then remove two fifths of the remainder, taking twenty-four and leaving thirty-six. The final retained fraction is 3/4 × 3/5 = 9/20, and 9/20 of eighty is thirty-six.
Check by accounting for every counter: twenty removed first, twenty-four removed second and thirty-six remaining total eighty. This check tests the interpretation as well as the arithmetic. A result can look plausible while still assigning the second fraction to the wrong whole.
Chapter 5 of 8 · Reverse and check the process
5. How can a reverse question be solved?
Suppose the same two-stage process—remove half, then remove one third of the remainder—leaves twenty. We already know the final retained fraction is one third of the original. Therefore the original amount was twenty divided by one third, or sixty.
Alternatively work backwards stage by stage. Twenty is two thirds of the first remainder, so that remainder was thirty. Thirty was half of the original, so the original was sixty. Both methods recover the same starting quantity and can be checked by running the actions forwards.
Chapter 6 of 8 · Reverse and check the process
6. Does changing the order always change the final amount?
Not for every problem. If the only actions are removing fractions a and b successively from the current remainder, the final retained fraction is (1 − a)(1 − b). Reversing these two pure fractional removals gives the same final amount because multiplication commutes.
The individual removed amounts can still differ, and adding a fixed-quantity action can make order matter. For sixty counters, removing half then ten leaves twenty, while removing ten then half leaves twenty-five. Read the actual sequence rather than teaching either ‘order never matters’ or ‘order always changes the answer’.
Question 1: start with ninety, remove one third, then half of the remainder. Find what is left. Question 2: start with one hundred, remove one fifth, then one quarter of the remainder. Question 3: the main half-then-third process leaves fifteen; find the original. Question 4: is a third of the remainder automatically a third of the original?
Answers: 1) Sixty remain after the first step, then thirty remain. 2) Eighty remain, then sixty remain. 3) The retained fraction is one third, so the original was forty-five. 4) No; identify the amount remaining at that stage. Ask for the current whole beside each fraction.
Bring the full question and the line where the second fraction was applied. A precise note is ‘keeps using the original whole after the remainder changes’. In a small-group 3-pax lesson, students can compare two questions differing only in ‘of the original’ and ‘of the remainder’.
For parents considering Secondary 1 Mathematics tuition in Punggol, the useful prompt is ‘What does one whole mean at this step?’ Once that reference is clear, let the child finish the calculation and check the total. The aim is to connect the fraction to its actual quantity, not to add more procedures without meaning.
