Primary 6 Mathematics Learning Guide | Everything Changed Problems and Units-and-Parts Reasoning

Wait, What? What If Nothing Stayed the Same?

Some changing-ratio problems do not offer an obvious invariant. Both quantities may change, the total may change, and the difference may change. The learner cannot align the ratios by a constant part, constant total or constant difference. These questions are often described informally as everything changed problems.

The solution is not to give up on units. It is to represent each state carefully, connect the before and after quantities through the actual numerical changes, and solve the resulting relationships. This is the core of units-and-parts reasoning when no simple invariant is available.

When nothing obvious stays fixed, the bridge between states is the change itself.

Quick Answer

A reliable routine is:

WRITE THE BEFORE RATIO → ASSIGN BEFORE UNITS → WRITE THE AFTER RATIO → ASSIGN AFTER UNITS → EXPRESS EACH AFTER QUANTITY FROM THE BEFORE STATE AND THE KNOWN CHANGE → FORM TWO CONSISTENT RELATIONSHIPS → SOLVE → CHECK BOTH STATES.

1. Why Constant-Part Methods Fail Here

Suppose A:B = 2:3. Then A gains 10 while B loses 5, and the new ratio becomes 4:5. A changed. B changed. The total changed by +5. The difference also changed.

There is no single quantity to align directly. The changes themselves must connect the two states.

2. Use Separate Unit Systems for Separate States

Let the before ratio A:B = 2:3 be represented by 2x and 3x. Let the after ratio 4:5 be represented by 4y and 5y. The two unit values x and y need not be equal because the total structure changed.

This is the key difference from constant-part problems: before-units and after-units may represent different actual amounts.

3. Build Change Equations

If A gains 10, then 4y = 2x + 10. If B loses 5, then 5y = 3x − 5. These two relationships connect the two unit systems.

Once the equations are formed, solve x and y or use elimination reasoning.

4. Worked Example: Both Quantities Change

A:B = 2:3. A receives 10 items while B gives away 5 items. The new ratio is 4:5. Find A and B at first.

Before: A = 2x, B = 3x. After: A = 4y, B = 5y.

4y = 2x + 10.
5y = 3x − 5.
From the first equation, 2x = 4y − 10.
Multiply by 3/2 conceptually or substitute: 3x = 6y − 15.
But 3x = 5y + 5 from the second relationship.
So 6y − 15 = 5y + 5.
y = 20.
Then 4y = 80 = 2x + 10, so 2x = 70 and x = 35.
Original A = 70; original B = 105.

Check after: A=80 and B=100, ratio 4:5.

5. The Bar-Model View

Draw the before state with 2 equal A-units and 3 equal B-units. Draw the after state separately with 4 equal A-units and 5 equal B-units. Then annotate the known additions or removals between corresponding quantities.

The bars do not have to share one unit size across both states. The arrows between states carry the numerical changes.

6. Use Equations Without Fear

Primary 6 Algebra is designed to help learners represent unknown quantities compactly. When every major quantity changes, algebra can be clearer than forcing a single unit system to do too much.

The equation is simply a sentence describing how the before state became the after state.

7. Everything Changed Versus Constant Part

If one quantity is unchanged, use constant-part alignment because it is usually simpler. If both quantities change but the total stays fixed due to a transfer, use constant-total reasoning. If both quantities gain or lose the same amount, consider constant-difference reasoning.

Only when these simpler invariants are absent should the learner move to a full two-state units-and-parts model.

8. Everything Changed Versus Constant Total

Suppose A gives 10 to B. Both quantities change, but total A+B stays fixed. That is not an everything-changed problem in the strongest sense because there is still a simple invariant total.

The correct classification saves work.

9. Everything Changed Versus Constant Difference

If A and B both increase by the same amount, their difference remains fixed. Age problems often work this way. Again, use the simpler invariant first.

The phrase “everything changed” should not make the learner ignore hidden invariants that still exist.

10. Worked Example: Both Increase by Different Amounts

A:B = 3:4. A increases by 18 and B increases by 10. The new ratio is 4:5. Find their original amounts.

Before: A=3x, B=4x. After: A=4y, B=5y.

4y = 3x + 18.
5y = 4x + 10.
Multiply first relation by 4: 16y = 12x + 72.
Multiply second relation by 3: 15y = 12x + 30.
Subtract: y = 42.
Then 4y = 168 = 3x + 18, so 3x = 150 and x = 50.
Original A=150, B=200.

Check after: 168:210 simplifies to 4:5.

11. Elimination Is a Natural Primary-to-Secondary Bridge

The previous solution made the x-terms equal so they could be eliminated. This is an early form of simultaneous-equation reasoning, expressed through ratio units and arithmetic rather than formal Secondary notation.

Students who understand why the elimination works are well prepared for later algebra.

12. Units and Parts Without Formal Algebra

The same reasoning can be shown in a structured table:

State	A	B
Before	3 before-units	4 before-units
Change	+18	+10
After	4 after-units	5 after-units

The learner then scales the two relationships until one before or after component can be compared consistently. The logic is identical to equation elimination.

13. Worked Example: One Gains, One Loses

A:B = 5:7. A gains 12 while B loses 6. The new ratio is 3:4. Let before amounts be 5x and 7x, after amounts 3y and 4y.

3y = 5x + 12
4y = 7x − 6

The learner can scale these equations to eliminate x or y. The main point is structural: both quantities changed differently, so a separate before-unit and after-unit system is justified.

14. Use the Change Directions to Check Plausibility

If A gains while B loses, A’s share of the pair should usually increase. If the new ratio suggests A became relatively smaller, inspect the model unless the starting sizes and changes make that possible.

Directional reasoning is a valuable check before exact calculation is trusted.

15. The Total Can Still Help Even If It Is Not Constant

If A gains 18 and B gains 10, total increases by 28. This gives an additional relationship between the before and after totals. Sometimes comparing total ratio units across states is an efficient alternative route.

Everything changed does not mean nothing useful can be tracked.

16. Total-Change Route

In the 3:4 to 4:5 example, before total = 7x and after total = 9y. Since total increases by 28, 9y = 7x + 28. Combined with one individual change relation, this can form another solvable system.

Multiple valid routes strengthen verification.

17. Everything Changed and Remainder Branching

Some problems combine changing ratios with fractions of remainders. In that case, first control the changing whole, then build the before-and-after ratio states. Do not collapse both layers into one vague unit diagram.

Question decomposition becomes essential.

18. Everything Changed and Percentage

If A increases by a percentage while B decreases by a different percentage, both quantities change multiplicatively. Let original values be ratio-based unknowns and apply the growth or remaining factors before forming the new ratio.

This is an advanced but natural extension of units-and-parts reasoning.

19. Common Error Families

Error	What it looks like	Repair
Forced invariant	Pretends one ratio term stayed constant when both changed	Classify the problem honestly before aligning
Same-unit assumption	Uses one unit size for unrelated before and after ratios	Use separate x and y units
Change-sign error	Writes + when quantity was removed	Build a before/change/after table
Equation mismatch	Links A-before to B-after	Track each quantity across states separately
Premature complexity	Uses full system when constant total or part exists	Search for simpler invariants first
No verification	Does not check both ratios and all stated changes	Reconstruct the full before-and-after story

20. A First-Weak-Link Diagnostic

Classification: Can the learner confirm that no simple invariant solves the problem?
Two-state modelling: Can before and after ratios be represented separately?
Unit independence: Can before-units and after-units remain distinct?
Change equations: Can each quantity be connected across states?
Elimination: Can the relationships be combined to solve unit values?
Direction: Can increase/decrease signs be controlled?
Verification: Can every change and ratio be reconstructed?
Transfer: Can the method handle money, objects and percentages?

21. Examination Control

First test for constant part, total or difference.
If none fits, use separate before and after unit systems.
Track A across states and B across states independently.
Write every known change with its correct sign.
Use elimination or total-change reasoning to connect the unit systems.
Check both ratios after solving.
Do not force a simpler heuristic beyond its valid conditions.

22. What Parents Can Ask

“Did anything actually stay constant?”
“Can the before and after ratio units have different sizes?”
“How did A change?”
“How did B change?”
“What two relationships connect the states?”
“Can you check every change after finding the values?”

23. What Tutors Should Protect

Problem classification. Do not fabricate invariants.
Separate unit systems. Before and after ratios may require different unit values.
State correspondence. Track the same quantity through time.
Algebraic bridge. Show how units-and-parts becomes equations.
Method economy. Prefer simpler invariant methods when valid.
Prompt reduction. Let learners choose the representation.
Transfer. Mix additive and multiplicative changes.

24. Official Process Connection

The Singapore Primary Mathematics framework emphasises problem solving, proportional reasoning, representation, connections and algebraic thinking. Everything-changed units-and-parts problems integrate those processes when no single invariant gives a direct route.

Official reference: MOE Mathematics Syllabus — Primary One to Six.

25. The Secondary Mathematics Handover

These problems are an excellent bridge into simultaneous equations. Primary ratio units become algebraic variables, while before-and-after change statements become equations connecting them.

Continue the Primary 6 Mathematics Series

The Quiet Return

Everything-changed problems become manageable when the learner accepts that one unit system may no longer be enough. Separate the states, connect them with the stated changes and let the relationships do the work.

The mature Primary 6 habit is to ask: if nothing simple stayed fixed, what equations or unit relationships connect the before state to the after state?

Small Group Tutorials