Primary 6 Mathematics Learning Guide | Constant Part Problems and Changing Ratios

Wait, What? Ratios Can Change While One Quantity Stays Exactly the Same

Many Primary 6 ratio problems describe two states: before and after. One quantity changes because something is added, removed, transferred or used, while another quantity stays unchanged. The ratio changes, but the unchanged quantity creates a bridge between the two states.

This is commonly taught as the constant part method. The mathematical job is to identify which quantity remains fixed, scale the two ratios so that the constant part carries the same number of units in both states, then use the change in the other quantity to determine the value of one unit.

When a ratio changes, do not compare the ratios until the unchanged quantity has been aligned.

Quick Answer

A reliable constant-part routine is:

IDENTIFY THE UNCHANGED QUANTITY → WRITE BOTH RATIOS → SCALE THE RATIOS SO THE CONSTANT PART MATCHES → COMPARE THE CHANGING PART → MAP THE UNIT DIFFERENCE TO THE KNOWN CHANGE → FIND ONE UNIT → RECONSTRUCT ALL QUANTITIES → CHECK BOTH STATES.

1. What Is a Constant Part?

A constant part is a quantity that remains unchanged between two ratio states. For example, red:blue = 3:5. After 24 red beads are added, the ratio becomes 5:5. The blue beads did not change, so blue is the constant part.

That unchanged blue quantity allows the before and after ratios to be aligned.

2. Why Ratios Must Be Aligned

Suppose A:B changes from 2:3 to 5:6, and B stays constant. The “3” before and the “6” after refer to the same actual quantity, so they cannot represent equal-sized units yet. Scale the first ratio by 2 to get 4:6. Now B is 6 units in both states.

Only after this alignment can the changing A-units be compared meaningfully.

3. Worked Example: Red Beads Added

Red:Blue = 3:5. After 24 red beads are added, the ratio becomes 5:5. How many blue beads were there at first?

1. Blue stays unchanged at 5 units in both ratios.
2. Red increases from 3 units to 5 units.
3. Increase = 2 units = 24 beads.
4. 1 unit = 12 beads.
5. Blue = 5 units = 60 beads.

Check: Red starts at 36, then 24 are added to make 60, matching Blue’s 60.

4. Worked Example: Ratio Needs Scaling

A:B = 2:3. After 18 is added to A, the ratio becomes 5:6. B is unchanged. Find A at first.

1. B before = 3 units; B after = 6 units.
2. Scale the first ratio by 2: A:B = 4:6.
3. A changes from 4 units to 5 units.
4. 1 unit = 18.
5. A at first = 4 units = 72.
6. B = 6 units = 108.

Check after: 72 + 18 = 90, and 90:108 simplifies to 5:6.

5. The Constant Can Be the First Quantity

If A stays unchanged while B changes, align A across the ratios instead. There is no preferred side. The constant is determined by the story, not by the position of the ratio term.

6. Addition and Removal

The method works whether the changing quantity increases or decreases. If items are removed from A while B stays fixed, align B and compare how many A-units disappeared.

The sign of the change tells whether units were gained or lost.

7. Worked Example: Amount Removed

Pencils:Pens = 7:4. After 18 pencils are removed, the ratio becomes 4:4. Pens stay unchanged. How many pencils were there at first?

1. Pens are 4 units in both states.
2. Pencils fall from 7 units to 4 units.
3. 3 units = 18 pencils.
4. 1 unit = 6 pencils.
5. Original pencils = 7 × 6 = 42.

Check: 42 − 18 = 24 and pens = 24, so final ratio is 1:1.

8. Why You Cannot Compare Raw Ratio Numbers Immediately

If a constant quantity is 3 units in one ratio and 6 units in another, the unit sizes differ. Subtracting the changing numbers before scaling can produce a meaningless unit difference.

Ratio numbers describe relative parts, not fixed physical units across separate states unless they have been aligned.

9. Constant Part and Bar Models

Bar models make the alignment visible. Draw the unchanged quantity with equal total length in both states. Then divide or scale the bars so that its unit count matches. The changing quantity can then be compared above or below it.

This visual method is especially useful before learners become comfortable with algebraic scaling.

10. Constant Part and Algebra

If A:B = 2:3, write A=2u and B=3u. After A increases by 18, the new ratio is 5:6. Since B remains 3u, and in the new ratio B corresponds to 6 parts, each new ratio part equals 0.5u. A after corresponds to 5×0.5u = 2.5u. The increase is 0.5u = 18, so u=36 and A originally = 72.

The algebra is the same alignment logic written symbolically.

11. Constant Part Versus Constant Total

These are different problem families. In constant-part problems, one quantity stays unchanged and the total usually changes. In constant-total problems, an internal transfer occurs so the sum stays fixed while both parts may change.

Choosing the wrong invariant creates the wrong model.

12. Constant Part Versus Constant Difference

In constant-difference problems, both quantities change by the same additive amount, so their difference stays fixed. Age problems are a classic example. In constant-part problems, one quantity itself stays fixed.

Ask exactly what remains unchanged: one amount, the sum, or the difference.

13. Worked Example: Money Ratio

Alice:Ben money = 4:7. After Alice receives $30, the ratio becomes 7:7. Ben’s money is unchanged. How much did Ben have?

1. Ben = 7 units before and after.
2. Alice rises from 4 units to 7 units.
3. 3 units = $30.
4. 1 unit = $10.
5. Ben = 7 units = $70.

14. Worked Example: Different Constant Unit Counts

A:B = 5:8. After 21 is removed from A, A:B becomes 3:6. B stays unchanged.

1. B before = 8 units; B after = 6 units.
2. LCM of 8 and 6 is 24.
3. Scale first ratio by 3: 15:24.
4. Scale second ratio by 4: 12:24.
5. A decreases by 3 aligned units = 21.
6. 1 aligned unit = 7.
7. A initially = 15 × 7 = 105.

Check B = 24×7 = 168. After removal A=84 and 84:168=1:2=3:6.

15. Use the Lowest Useful Common Multiple

When the constant part has different unit counts, use a common multiple to align them. The least common multiple is often efficient, but any common multiple preserves the relationship if used consistently.

16. Constant Part and Percentage

A changing ratio can be converted into fractions or percentages of a total, but remember that the total may itself change when one part is added or removed. The constant part anchors the ratio, not necessarily the total.

This distinction prevents wrong-whole errors.

17. Common Error Families

Error	What it looks like	Repair
Wrong invariant	Assumes total stays fixed when one part was added externally	State exactly what remains unchanged
Raw-ratio subtraction	Compares changing parts before aligning constant units	Scale the constant part first
Incorrect scaling	Changes one side of a ratio only	Multiply both parts by the same factor
Change-direction loss	Adds when the story says removed	Track before and after states
Unit-value mismatch	Uses pre-alignment units as though they equal post-alignment units	Work only with aligned units
No state check	Does not verify both ratios	Reconstruct before and after quantities

18. A First-Weak-Link Diagnostic

1. Invariant: Can the unchanged quantity be identified?
2. Alignment: Can its ratio units be made equal?
3. Comparison: Can the changing-part unit difference be found?
4. Mapping: Can the known numerical change be matched to units?
5. Unit value: Can one aligned unit be found?
6. Reconstruction: Can all original quantities be recovered?
7. Verification: Can both ratios be checked?
8. Classification: Can constant part be distinguished from constant total and difference?

19. Examination Control

• Underline the quantity that did not change.
• Write both ratios before calculating.
• Align the constant part using equivalent ratios.
• Only then compare the changing part.
• Map the unit change to the known amount added or removed.
• Reconstruct both states and simplify the final ratio.
• If no part stayed constant, use another method.

20. What Parents Can Ask

• “Which quantity stayed exactly the same?”
• “Does it have the same number of ratio units in both states?”
• “How can you scale the ratios to make it match?”
• “How many aligned units changed?”
• “What amount does that unit difference represent?”
• “Can you check both the before and after ratios?”

21. What Tutors Should Protect

• Invariant classification. Constant part must be explicit.
• Equivalent-ratio fidelity. Scale both terms together.
• Alignment before comparison. Never subtract unaligned ratio units.
• Multiple representations. Use bars, tables and algebra.
• Verification. Reconstruct both states.
• Prompt reduction. Let learners find the invariant independently.
• Transfer. Use money, objects, people and measurement contexts.

22. Official Process Connection

Changing-ratio problems draw on proportional reasoning, representation, connections and problem solving in the Singapore Primary Mathematics framework. The constant-part method is a teaching structure for making the invariant visible across two ratio states.

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

Continue the Primary 6 Mathematics Series

• Primary 6 Mathematics Learning Hub
• Remainder Branching and Fractions of a Changing Whole
• Assumption Method, Replacement Difference and All-One-Type Reasoning
• Everything Changed Problems and Units-and-Parts Reasoning

The Quiet Return

Constant-part problems are solved by refusing to compare moving ratios too early. First anchor the mathematics to what did not move.

The mature Primary 6 habit is to ask: what stayed fixed, and have I made that constant carry the same units in both states?