Primary 6 Mathematics Learning Guide | Constant Total Problems and Internal Transfer

Wait, What? Both Parts Can Change While the Total Stays Exactly the Same

Some Primary 6 ratio problems describe a transfer from one part of a system to another. Money moves from A to B. Water is poured from one container into another. Counters move from one box to another. Both parts change, yet nothing enters or leaves the combined system.

This is a constant total problem. The key invariant is the sum. When a quantity is transferred internally, one part decreases by the transfer amount while the other increases by the same amount. The total does not change.

Internal transfer changes ownership, not total quantity.

Quick Answer

A reliable constant-total routine is:

IDENTIFY THE TRANSFER → CONFIRM NOTHING ENTERS OR LEAVES → WRITE THE BEFORE RATIO → WRITE THE AFTER RATIO → ALIGN TOTAL RATIO UNITS → MAP THE UNIT SHIFT TO THE DOUBLE EFFECT OF THE TRANSFER → FIND ONE UNIT → RECONSTRUCT BOTH STATES → CHECK THE TOTAL.

1. What Makes the Total Constant?

If 12 counters move from Box A to Box B, A loses 12 and B gains 12. The combined total changes by −12 + 12 = 0.

The total is constant only because the movement is internal. If 12 new counters are added to B from outside, the total increases and this method does not apply.

2. The Double Effect of a Transfer

If A transfers 10 to B, the difference A−B changes by 20: A becomes 10 smaller while B becomes 10 larger. This double effect is one of the most useful observations in transfer problems.

It explains why a transfer amount can create a ratio-unit change twice as large as the physical transfer when comparing the two parts.

3. Worked Example: Equalising Two Quantities

A:B = 5:3. A gives 24 to B and they become equal. Find the original quantities.

1. Before difference = 2 ratio units.
2. After transfer, difference becomes 0.
3. A gives 24 and B gains 24, so the difference closes by 48.
4. Therefore 2 units = 48.
5. 1 unit = 24.
6. A initially = 5×24 = 120.
7. B initially = 3×24 = 72.

Check: 120−24 = 96 and 72+24 = 96. Total before and after is 192.

4. Align Total Units When Ratios Change

Suppose A:B changes from 3:5 to 1:1 after an internal transfer. Before total ratio units = 8. After total ratio units = 2. Because the actual total is unchanged, scale the after ratio to 4:4 so both states have 8 total units.

Now the movement of units between A and B can be compared directly.

5. Worked Example: Ratio Changes After Transfer

A:B = 3:5. B transfers 30 to A. The new ratio is 1:1. Find A and B at first.

1. Before total units = 8.
2. After ratio 1:1 has 2 units; scale to 4:4 so total is also 8 units.
3. A increases from 3 units to 4 units.
4. B decreases from 5 units to 4 units.
5. One aligned unit transferred = 30.
6. A initially = 3×30 = 90.
7. B initially = 5×30 = 150.

Check: 90+30 = 120 and 150−30 = 120.

6. Why Total Units Must Match

Before and after ratios can have different-looking totals even when the real-world sum is identical. A 2:3 ratio has 5 units; a 4:6 ratio has 10 units, yet they represent the same relative relationship. Equivalent scaling lets both states use a common total-unit system.

Only after total units are aligned should before-and-after unit shifts be compared.

7. Constant Total Versus Constant Part

In a constant-part problem, one quantity stays fixed while the total usually changes. In a constant-total problem, both parts may change while their sum stays fixed.

Ask: did one part remain untouched, or did something move between the two parts?

8. Constant Total Versus Everything Changed

If A gains 10 from outside and B loses 5 to somewhere else, total changes. That is not a pure constant-total problem. Use a two-state units-and-parts or algebraic model instead.

Do not use an invariant that the story does not guarantee.

9. The Bar-Model View

Draw one long bar representing the fixed combined total. Partition it according to the before ratio. Draw the same-length total bar again for the after ratio. The movement of the internal boundary represents the transfer.

This makes constant total visually unmistakable: the whole bar never changes length.

10. The Algebraic View

If A:B = 3:5, write A=3u and B=5u. If B gives 30 to A and they become equal, then 3u+30 = 5u−30. Therefore 60 = 2u and u=30.

The algebra shows the same double effect: the transfer appears on opposite sides of the equality.

11. Worked Example: New Ratio 5:3

A:B = 2:6. B gives 20 to A. The new ratio becomes 5:3. Find the original total.

1. Before total units = 8.
2. After total units = 8 already.
3. A rises from 2 units to 5 units: +3 units.
4. B falls from 6 units to 3 units: −3 units.
5. The transfer of 20 corresponds to 3 units.
6. 1 unit = 20/3.
7. Original total = 8×20/3 = 160/3.

This result is mathematically valid but may be contextually unsuitable if the quantities must be whole items. If a real counting context requires integers, the problem data would need compatible values. Feasibility matters.

12. Whole-Number Feasibility

When the quantities represent people, books or objects, the reconstructed amounts usually need to be whole numbers. A fractional unit value can signal either an error or a deliberately continuous context such as litres or kilograms.

Always return to the meaning of the quantity.

13. Transfer Direction Is Part of the Model

If A gives to B, A must decrease and B must increase. If your unit alignment predicts the opposite direction, inspect the ratio scaling or the story interpretation.

Direction provides a strong plausibility check.

14. Worked Example: Percentage After Transfer

A and B have money in ratio 7:5. A gives $40 to B. Afterwards, A has 50% of the combined total. Find the original total.

After transfer A and B are equal because 50% of a two-person total means half the total. Thus the final ratio is 1:1. Before difference is 2 ratio units. Transfer of $40 closes the difference by $80, so 2 units=$80, one unit=$40, and original total=12 units=$480.

This connects ratio, percentage and constant-total reasoning.

15. Constant Total and Difference

Internal transfer keeps total fixed but changes difference by twice the transfer. This gives two simultaneous invariants/relationships:

• sum stays constant;
• difference changes by twice the transfer amount.

Either relationship may provide the shorter route depending on the question.

16. Common Error Families

Error	What it looks like	Repair
External-addition confusion	Treats new outside quantity as transfer	Confirm nothing enters or leaves
Total-unit mismatch	Compares before and after ratio units directly	Align total units first
Single-effect thinking	Thinks transfer 20 changes difference by 20	Track −20 on one side and +20 on the other
Wrong direction	Increases the giver	Draw arrows for transfer
Invariant loss	Does not check total equality	Reconstruct sums before and after
Feasibility neglect	Accepts fractional people or objects	Check quantity type

17. A First-Weak-Link Diagnostic

1. Classification: Can internal transfer be distinguished from external change?
2. Invariant: Can the fixed total be stated?
3. Ratio alignment: Can total units be matched?
4. Transfer effect: Can unit shifts be linked to the transfer amount?
5. Difference reasoning: Can the double effect be explained?
6. Reconstruction: Can both states be recovered?
7. Feasibility: Do the final quantities make contextual sense?
8. Verification: Is total preserved exactly?

18. Examination Control

• Circle transfer words such as gives, moves, pours or shifts.
• Confirm the combined total stays fixed.
• Align total ratio units before comparing states.
• Use the double effect when difference is easier than total alignment.
• Track transfer direction explicitly.
• Rebuild the total at the end.
• Check whether answers must be whole quantities.

19. What Parents Can Ask

• “Did anything enter or leave the system?”
• “What stayed the same?”
• “How did the transfer affect both sides?”
• “Why does the difference change by twice the transfer?”
• “Have the before and after total units been aligned?”
• “Can you prove the total is unchanged?”

20. What Tutors Should Protect

• Invariant fidelity. Use constant total only for internal movement.
• Double-effect reasoning. Make the two-sided change explicit.
• Equivalent-ratio control. Align total units.
• Multiple representations. Compare bars, units and algebra.
• Feasibility. Match results to context.
• Prompt reduction. Let learners classify the invariant themselves.
• Transfer. Vary money, objects, liquids and ratio contexts.

21. Official Process Connection

Constant-total transfer problems develop proportional reasoning, representation, connections, algebraic thinking and problem solving within the Singapore Primary Mathematics framework. The “constant total” label is a teaching structure for identifying the invariant.

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

Continue the Primary 6 Mathematics Series

• Primary 6 Mathematics Learning Hub
• Gap-and-Difference Problems and Comparison Change
• Grouping, Number × Value and Equal-Group Reconstruction
• Double-IF Problems and Two-Scenario Reasoning

The Quiet Return

Constant-total problems become simple once the learner recognises that the transfer changes the distribution but not the combined quantity.

The mature Primary 6 habit is to ask: did the quantity leave the system, or did it merely move inside it?