A shop begins with 500 notebooks.
It sells 30% of them.
Then it donates 20% of the remainder.
How many notebooks are left?
A tempting shortcut is:
30% + 20% = 50%, so half remains.
That gives 250.
It is wrong.
A percentage only has meaning relative to a base quantity. In a percentage-of-a-remainder problem, the base changes after the first action.
First sale:
30% of 500 = 150.
Remainder:
500 − 150 = 350.
Now the second percentage is taken from 350, not from the original 500.
20% of 350 = 70.
Final remainder:
350 − 70 = 280 notebooks.
The second 20% corresponds to 14% of the original quantity, because:
70/500 = 14%.
This is the entire mechanism.
The percentages did not fail.
The base changed.
The first question: 20% of what?
When a problem says “20%”, the number 20 is incomplete by itself.
We need the reference whole.
For example:
- 20% of 500 = 100;
- 20% of 350 = 70;
- 20% of 200 = 40.
The percentage rate is unchanged.
The actual amount changes because the base changes.
In multi-step percentage problems, attach every percentage to its own named base before calculating.
This one habit prevents most shortcut errors.
A 100-unit model makes the changing base visible
Imagine the original quantity as 100 equal units.
If 30% is removed, 70 units remain.
The second action removes 20% of those 70 remaining units.
20% of 70 = 14.
So the final amount is:
70 − 14 = 56 units.
Therefore 56% of the original quantity remains.
For the 500-notebook example:
56% of 500 = 280.
The 100-unit model exposes why adding 30% and 20% is wrong:
the first percentage is measured against 100 original units, while the second is measured against 70 remaining units.
Remainder language changes the reference whole
Watch for words such as:
- remaining;
- left;
- the rest;
- of those still present;
- of the unsold items;
- of the balance;
- of what remained.
These words often signal that the reference quantity has changed.
The learner should pause and write a new base explicitly.
For example:
Original = 500.
After first action = 350.
Second percentage base = 350.
This is a small notation habit with a large effect on accuracy.
Worked example 1: percentage removed, then percentage of remainder removed
A library has 800 old magazines.
25% are recycled.
Then 40% of the remaining magazines are donated.
How many remain?
First removal:
25% of 800 = 200.
Remainder:
800 − 200 = 600.
Second removal:
40% of 600 = 240.
Final remainder:
600 − 240 = 360.
Check using percentages of the original:
25% was removed first.
The second removal was 240/800 = 30% of the original.
Total removed = 55% of original.
Remaining = 45%.
45% of 800 = 360.
The final check works only after both removed amounts have been expressed against the same original base.
Worked example 2: use retained percentages instead of removed percentages
A warehouse starts with 1,200 boxes.
15% are shipped in the morning.
Then 30% of the remainder are shipped in the afternoon.
Instead of finding each removed amount, track what remains.
After the morning shipment:
100% − 15% = 85% remains.
85% of 1,200 = 1,020.
After the afternoon shipment:
100% − 30% = 70% of the remainder stays.
70% of 1,020 = 714.
Answer:
714 boxes remain.
This method is often cleaner because each stage can be described by a retained fraction or multiplier.
The multiplicative-factor view
If 15% is removed, 85% remains.
As a decimal factor:
0.85.
If 30% of the remainder is then removed, 70% of that remainder survives:
0.70.
Overall retained factor:
0.85 × 0.70 = 0.595.
So 59.5% of the original remains.
59.5% of 1,200 = 714.
Successive percentages combine multiplicatively because each new percentage acts on the output of the previous stage.
The factor notation is a useful upper-primary-to-secondary extension.
The underlying idea can still be taught with bars or unitary reasoning before decimal multipliers are introduced formally.
Why 20% off followed by 20% off is not 40% off
Start with 100 units.
First 20% removed:
80 remain.
Second 20% is calculated from 80:
20% of 80 = 16.
Final remainder:
80 − 16 = 64.
Overall reduction:
36%.
Not 40%.
Factor check:
0.8 × 0.8 = 0.64.
64% remains.
Successive percentages can sometimes be reversed in order
If one step removes 30% and another removes 20% of the current remainder, the final retained factor is:
0.70 × 0.80 = 0.56.
If the order is reversed:
0.80 × 0.70 = 0.56.
The same 56% remains.
This happens because multiplication is commutative.
But the amount removed at each individual stage will differ.
And if one stage uses a fixed amount rather than a percentage, order can matter.
Example:
Remove $20, then remove 10% of the remainder.
This is not generally equivalent to removing 10% first and then $20.
The operation types differ.
Worked example 3: percentage added after a remainder
A club has 400 members.
10% leave.
Then new members equal to 25% of the remaining membership join.
After 10% leave:
90% of 400 = 360.
New members:
25% of 360 = 90.
New total:
360 + 90 = 450.
The final total is 112.5% of the original.
Notice that losing 10% and then gaining 25% does not mean a net gain of 15%.
The two percentages use different bases.
A percentage decrease and equal percentage increase do not cancel
Start with 100.
Decrease by 20%:
80 remains.
Increase the new amount by 20%:
20% of 80 = 16.
Final amount:
96.
The original 100 is not restored.
Why?
The decrease was 20% of 100.
The increase was 20% of 80.
Same rate.
Different base.
Worked example 4: fraction first, percentage of the remainder next
A farmer sells 2/5 of a harvest.
Then 25% of the remainder is stored for seed.
What fraction of the original harvest remains for other uses?
After selling 2/5:
3/5 remains.
25% = 1/4.
If 1/4 of the remainder is stored, then 3/4 of the remainder is left for other uses.
Remaining fraction of original:
3/5 × 3/4 = 9/20.
Answer:
9/20 of the original harvest.
As a percentage:
9/20 = 45%.
This example connects fraction-of-a-remainder reasoning with percentage representation.
Worked example 5: three stages
A school fund begins at $10,000.
20% is used for equipment.
10% of the remainder is used for transport.
25% of what then remains is reserved for future maintenance.
Track retained percentages:
- after equipment: 80% remains;
- after transport: 90% of that remains;
- after maintenance reserve: 75% of that remains available for other purposes.
Calculation:
10,000 × 0.80 = 8,000.
8,000 × 0.90 = 7,200.
7,200 × 0.75 = 5,400.
Final amount:
$5,400.
Overall retained percentage:
0.8 × 0.9 × 0.75 = 0.54 = 54%.
The three stated removal percentages sum to 55%, but 45% is not the final remainder.
The final remainder is 54% because the later percentages operate on successively smaller bases.
Reverse problems: reconstructing the original quantity
After 20% of a collection is removed, 25% of the remainder is removed.
360 items remain.
How many items were there originally?
Track retained factors:
80% remains after the first step.
75% of that remains after the second.
Overall retained fraction:
0.8 × 0.75 = 0.6.
So 360 is 60% of the original.
Original:
360 ÷ 0.6 = 600.
Primary learners may prefer the fraction route:
80% = 4/5.
75% = 3/4.
(4/5)×(3/4) = 3/5.
If 3/5 = 360, then 1/5 = 120 and 5/5 = 600.
The fraction representation may make the reverse structure easier to see.
Reverse percentage is not “undo the percentage sign”
If 80% of an original quantity equals 240, the original is not:
240 + 20% of 240.
That would add 48 and give 288.
But 80% of 288 is 230.4, not 240.
The correct question is:
if 80% corresponds to 240, what corresponds to 100%?
1% = 240 ÷ 80 = 3.
100% = 300.
Or:
240 ÷ 0.8 = 300.
Reverse reasoning reconstructs the original base; it does not apply an equal percentage increase to the reduced value.
Bar models for changing percentage bases
For a problem that removes 30% and then 20% of the remainder:
Draw the original as 10 equal bars if convenient.
Removing 30% removes 3 bars.
Seven bars remain.
The second 20% must now be taken from those seven remaining bars.
Because fifths are more convenient for 20%, each remaining bar may need to be subdivided or the remainder may be redrawn as a new whole.
This redraw is not wasted work.
It visually announces that the reference whole has changed.
When the base changes, it is often useful to redraw the remainder as the new 100%.
A two-column state table can be even clearer
For each stage, record:
- current quantity;
- percentage action;
- amount changed;
- new quantity.
Example with 500:
- Stage 0: 500.
- Stage 1: remove 30% of 500 = 150 → 350.
- Stage 2: remove 20% of 350 = 70 → 280.
This state-based representation prevents the learner from accidentally returning to the original base during the second calculation.
Percentage points are a different idea
If a survey result changes from 40% to 55%, that is an increase of 15 percentage points.
The relative percentage increase is:
(15/40)×100% = 37.5%.
This distinction is broader than a typical Primary 6 percentage-of-remainder problem, but it reinforces the same principle:
percentage statements need a named base.
Whole-object contexts can expose inconsistent answers
Suppose a class has 37 pupils.
A problem states that exactly 20% of the class left.
20% of 37 = 7.4 pupils.
That is impossible if pupils are indivisible and the percentage is meant to be exact.
The issue may be that:
- the percentage was rounded;
- the data are hypothetical rather than literal;
- or the givens are inconsistent.
Good mathematical reasoning checks whether the numerical result fits the kind of quantity being counted.
A general model for successive removals
Suppose an original quantity is Q.
Remove p%.
The retained factor is:
1 − p/100.
Then remove q% of the remainder.
The second retained factor is:
1 − q/100.
Final quantity:
Q(1 − p/100)(1 − q/100).
This algebraic form is a generalisation rather than a required primary-school formula.
Its value is explanatory:
it shows exactly why successive percentage changes multiply rather than add.
Common misconception 1: add the percentage rates
30% removed, then 20% of the remainder removed does not mean 50% removed.
Repair: write the base beside each percentage before calculating.
Common misconception 2: calculate every percentage from the original quantity
If the problem says “20% of the remainder”, the original quantity is no longer the base.
Repair: calculate the new state first, then treat it as the new 100%.
Common misconception 3: the remainder percentage is the same as the original percentage
20% of a 70% remainder is 14% of the original.
Repair: convert both amounts to the same base before adding or comparing them.
Common misconception 4: equal percentage decrease and increase cancel
A 20% decrease followed by a 20% increase gives 96% of the original, not 100%.
Repair: state the base for both percentages explicitly.
Common misconception 5: reverse percentage means add back the same percentage
If 80% of an original equals 240, adding 20% of 240 does not reconstruct the original.
Repair: use unitary reasoning or divide by the retained factor.
Common misconception 6: “remainder” is merely a story word
In these problems, “remainder” changes the mathematical reference set.
Repair: translate the story into state labels: original → remainder 1 → remainder 2.
A diagnostic ladder
- Can the learner explain percentage as a quantity relative to a base?
- Can the learner find a percentage of a whole accurately?
- Can the learner find the remainder after a percentage is removed?
- Can the learner identify that the remainder becomes the new base?
- Can the learner solve a two-stage percentage-of-remainder problem without adding the rates?
- Can the learner express the second-stage amount as a percentage of the original?
- Can the learner use fractions or retained percentages when they are more efficient?
- Can the learner solve a reverse problem from the final remainder?
- Can the learner explain why equal percentage decrease and increase do not cancel?
- Can the learner check whole-number constraints and units?
A five-minute home investigation
Start with 100 counters.
Remove 20.
Ask:
What percentage remains?
Answer: 80%.
Now remove 25% of the remaining 80.
25% of 80 = 20.
60 counters remain.
Then ask:
- Why was the second 25% equal to 20 counters rather than 25 counters?
- What percentage of the original did the second removal represent?
- What fraction of the original remains?
- Could the same final result be found using 4/5 × 3/4?
This makes the changing base physically visible.
What parents and teachers should listen for
- “The second percentage is of the remainder, so I need the new base first.”
- “After 30% is removed, 70% remains.”
- “Twenty percent of that 70% is 14% of the original.”
- “I cannot add the rates until I have converted the amounts to the same base.”
- “I used 80% and 75% as retained fractions because it was easier than finding both removed amounts.”
- “To work backwards, I treat the final amount as a known fraction or percentage of the original.”
How this fits current Singapore Primary 6 Mathematics
Singapore’s Primary Mathematics syllabus applicable to Primary 6 from 2026 includes percentage work such as finding the whole given a part and its percentage, and finding percentage increase or decrease. Earlier fraction, decimal and percentage knowledge remains prerequisite.
“Percentage-of-a-remainder problems” is best understood as a multi-step application rather than a separate named official syllabus heading. It combines percentage, fractions, changing reference quantities and problem representation.
The 2026 PSLE Mathematics assessment objectives require learners not only to recall facts and perform procedures but also to interpret information, apply concepts in varied contexts, reason mathematically and select appropriate strategies. Tracking the correct percentage base is exactly this kind of transfer work.
The official MOE and SEAB documents remain the authority for examinable scope. The broader treatment here is intended to make the mathematics underneath multi-step percentage questions explicit.
The deeper lesson: the base is part of the number
People often talk about “20%” as if it were a complete amount.
Mathematically, it is a rate relative to a reference quantity.
Change the reference quantity and the same 20% produces a different actual amount.
Percentage-of-a-remainder problems are therefore not really about difficult arithmetic.
They are about maintaining quantity identity through time.
The safest question in any percentage problem is not “What percentage do I use?” but “What quantity is being treated as 100% at this step?”
Final thought
Shortcuts become dangerous when they erase the base.
Keep the states visible.
Name the current 100%.
Then calculate.
Once a learner develops that habit, percentage-of-a-remainder questions stop feeling like traps and become ordinary proportional reasoning carried through more than one stage.