Before-and-After Problems: Tracking What Changed and What Stayed Fixed

A box contains some counters.

Twenty are added.

Then one quarter of the new total is removed.

After that, 12 counters are transferred into a second box.

At the end, the first box contains 60 counters.

What was true at the beginning?

This kind of question feels difficult because several versions of the same quantity exist inside one short story.

There is the original amount.

The amount after 20 are added.

The amount after one quarter is removed.

The amount after the transfer.

Before-and-after problems are not mainly about doing more operations. They are about keeping the identity of each mathematical state clear while the situation changes.

Once the states are separated, the problem usually becomes quieter.

The quick answer: write the states, then mark the changes and invariants

A reliable first pass has three columns:

State	What changed?	What stayed fixed?
Before	Starting quantities	Identify the original relationships
After Step 1	Addition, removal, transfer or scaling	Look for quantities untouched by the step
After Step 2	Another change	Check whether total, difference or one quantity is preserved
Final	Known end state	Use as a constraint or a place to work backwards from

The central questions are:

What quantity does this number describe now?
What entered the system?
What left the system?
What moved within the system?
What stayed unchanged?
Did the total stay fixed?
Did the difference stay fixed?
Did a fraction or percentage act on the original amount or on a new remainder?

These questions are more useful than asking for a keyword.

A state is a snapshot, not merely a number

Suppose Mei has $100.

She spends $30.

Then she receives $20.

The numbers are easy:

$100 → $70 → $90.

But the deeper habit is to understand that $70 is not just “70”. It is Mei’s money after the first change but before the second.

That identity matters when a later sentence says “half of the remainder”, “25% of this amount” or “the difference then”.

The word then can change the mathematical base.

Three kinds of change must be kept separate

1. Quantity enters or leaves the system

If 15 books are added to a shelf, the shelf total increases by 15.

If 15 books are removed from the room entirely, the total number of books in the room decreases by 15.

2. Quantity moves within a closed system

If A gives 15 marbles to B, A decreases by 15 while B increases by 15.

The combined total does not change.

3. A quantity is rescaled

If 25% of an amount is removed, the new amount is 75% of the previous state.

This is not the same kind of change as subtracting a fixed 25 units.

Fixed change and proportional change behave differently. A good state model makes that difference visible before arithmetic begins.

Worked example 1: addition followed by a fraction of the new amount

A container held some water.

20 litres were added.

Then one quarter of the new amount was removed.

60 litres remained.

We know the final state.

If one quarter was removed, three quarters remained.

So 60 litres represents 3 equal parts.

1 part = 20 litres.

4 parts = 80 litres.

That 80 litres was the state after 20 litres had been added.

Original amount = 80 − 20 = 60 litres.

The answer happens to equal the final amount, but the states are not the same. That coincidence should not erase the reasoning.

A before-and-after diagram can prevent base confusion

Write:

Original → +20 → new whole → remove 1/4 of new whole → 60

The phrase “new whole” is useful because it reminds the learner that the fraction acts after the addition.

Worked example 2: a transfer preserves total but changes difference

A had 74 marbles and B had 46.

A gave 12 marbles to B.

Before the transfer:

total = 74 + 46 = 120;
difference = 74 − 46 = 28.

After the transfer:

A = 62;
B = 58;
total = 120;
difference = 4.

The transfer of 12 changed the difference by 24 because one side fell by 12 while the other rose by 12.

This gives a useful principle:

During an internal transfer, total is conserved but the difference can change by twice the transferred amount.

That principle is powerful in PSLE-style problems because it lets the learner identify which relationship is stable and which is not.

Worked example 3: equal additions preserve difference but usually change ratio

A has 30 counters and B has 50.

Both receive 10 more.

Before:

difference = 20;
ratio = 3:5.

After:

A = 40;
B = 60;
difference = 20;
ratio = 2:3.

The difference stayed fixed.

The ratio changed.

This is one reason before-and-after problems cannot be solved by carrying every relationship forward unchanged.

Worked example 4: equal multiplication preserves ratio but changes difference

A = 12 and B = 18.

Both quantities are doubled.

Before:

ratio = 2:3;
difference = 6.

After:

A = 24;
B = 36;
ratio = 2:3;
difference = 12.

Multiplicative scaling preserved the ratio but scaled the difference.

So the learner should ask not “What stayed the same?” in a vague way, but “Which relationship stayed the same under this kind of transformation?”

This is where invariants become practical

An invariant is something that remains unchanged under a particular transformation.

In school Mathematics, common before-and-after invariants include:

constant total during a transfer inside a closed system;
constant difference when equal fixed amounts are added to or removed from both quantities;
constant ratio when both quantities are scaled by the same non-zero factor;
one unchanged quantity when only the other quantity changes;
fixed geometry when a diagram is rearranged without changing area;
fixed unit rate in a proportional relationship.

The invariant is often the bridge connecting the before-state to the after-state.

Worked example 5: percentage after a changing quantity

A shop had some notebooks.

30 notebooks were delivered.

Then 40% of the notebooks were sold.

90 remained.

If 40% were sold, 60% remained.

90 = 60% of the state after delivery.

State after delivery = 90 ÷ 0.6 = 150.

Original stock = 150 − 30 = 120 notebooks.

The most common error is applying 40% to the original stock before the delivery. The percentage acts on the state that exists at that point in the story.

Worked example 6: a ratio is only one kind of before-and-after structure

A:B = 2:3.

A receives 12 while B remains unchanged.

The new ratio is 4:5.

Because B is unchanged, B can connect the two states.

Scale the ratios so B has the same number of common comparison units:

Before 2:3 becomes 10:15.

After 4:5 becomes 12:15.

A increased by 2 common units, and that increase equals 12.

1 unit = 6.

Original A = 60 and B = 90.

This is a ratio-specific version of the larger before-and-after principle: find the quantity or relationship that survives the transition and use it to calibrate the two states.

Do not mix “before units” and “after units” without an anchor

Ratio units belong to the state in which the ratio is defined.

Two units in a 2:3 ratio are not automatically the same size as four units in a later 4:5 ratio.

The states become comparable only when a real quantity—such as an unchanged B—links them.

This is why representation discipline matters more than drawing more bars.

When should you work backwards?

Working backwards is especially useful when:

the final state is fully known;
the starting state is hidden;
the changes are reversible;
a remaining fraction or percentage is known;
the after-state is more constrained than the before-state.

But before-and-after reasoning is broader than working backwards.

Sometimes the cleanest route is forward.

Sometimes two states are compared through a constant total or difference.

Sometimes both ends are partly unknown and a bar model or equation links them.

“Before and after” describes the structure of the problem. “Working backwards” is one possible route through that structure.

A state-change ledger for difficult questions

For a dense Paper 2 problem, write a compact ledger.

Quantity	Before	Change	After
A	unknown	−12	known or related
B	unknown	+12	known or related
Total	same total	0	same total
Difference	unknown	changes by 24	known or related

This makes hidden consequences visible.

It also reduces working-memory load because the learner no longer has to remember every state mentally.

Common misconception 1: every number belongs to the original state

In a multi-stage problem, a number may belong only to the state after one or two changes.

Repair: label each number with its state before calculating.

Common misconception 2: transfer means the total decreases

A transfer moves quantity within the system. It does not remove quantity from the system.

Repair: calculate the combined total before and after.

Common misconception 3: equal addition preserves ratio

Equal addition preserves difference, not ratio in general.

Repair: test with a simple numerical counterexample such as 30:50 becoming 40:60.

Common misconception 4: the percentage base never changes

“25% of the remainder” acts on the current remainder, not automatically on the original total.

Repair: write “100% = ?” at each percentage state.

Common misconception 5: if something stayed fixed once, it stays fixed throughout the problem

An invariant belongs to a particular transformation.

A total may be fixed during a transfer and then change when new quantity enters later.

Repair: reassess invariants after every stated change.

A diagnostic ladder for before-and-after reasoning

Can the learner identify the original and final states?
Can the learner insert intermediate states in the correct order?
Can the learner distinguish addition, removal, transfer and scaling?
Can the learner say which quantity a fraction or percentage acts on?
Can the learner recognise a constant total?
Can the learner recognise a constant difference?
Can the learner recognise when one quantity stays unchanged?
Can the learner distinguish ratio units in different states?
Can the learner choose whether to work forwards, backwards or through an invariant?
Can the learner reconstruct every state and check the result against the story?

How this fits the current Singapore Mathematics framework

“Before-and-after problems” are best understood as a broad problem-solving family rather than a single named content strand in the current MOE Primary Mathematics syllabus.

The 2021 Primary Mathematics syllabus applies to Primary 6 from 2026 and places mathematical problem solving at the centre, with reasoning, communication, connections, applications and modelling among the important processes. The content needed for before-and-after problems can draw on whole numbers, fractions, percentages, ratios, measurement and algebraic thinking depending on the learner’s level.

For the 2026 PSLE Mathematics examination, SEAB’s assessment objectives include interpreting information, applying concepts in varied contexts, analysing information, making inferences and selecting appropriate strategies. That is why state tracking is useful: it helps a learner decide what relationship the question is actually testing before choosing a calculation.

What parents and tutors should listen for

“This 90 is the amount after the delivery, not the original amount.”
“The transfer changes A and B but keeps the combined total fixed.”
“The difference stays fixed because both quantities received the same amount.”
“The ratio stays fixed only because both quantities were multiplied by the same factor.”
“This 25% acts on the remainder, so I need the current 100%.”
“The after-ratio units are not automatically the same size as the before-ratio units.”

These explanations are evidence of structure, not merely successful arithmetic.

The deeper lesson: change becomes manageable when identity survives

A hard before-and-after question presents a moving world.

Quantities enter, leave, transfer and scale.

The learner’s job is to prevent the mathematical identities from dissolving as that movement occurs.

Which state is this?

Which quantity is this?

What changed?

What stayed fixed?

The arithmetic becomes dependable when every number is attached to the right state and every state is connected by a relationship the learner can explain.