PRIMARY 5 MATHEMATICS LEARNING GUIDE · BATCH 9 · GUIDE 34
Excess-and-shortage problems are really two competing distribution plans sharing one hidden total. One plan leaves something over. Another plan does not have enough. The learner’s job is to compare the plans, measure the gap between them and use that difference to recover the missing number of groups or items.
This guide isolates the structure so it does not remain a memorised heuristic. It develops equal distribution, surplus, deficit, plan difference, group count, reverse checks, fractions, money and capacity applications.
Series route: return to the Primary 5 Mathematics Learning Hub. Earlier: Remainder Concept, Fraction of the Remainder & Branching. Continue to One Item Constant, External Unchanged, Repeated Identity & Equal-Fraction Reasoning and Primary 5 Mathematics Examination Guide.
1. One total, two plans
A school has some exercise books.
If each class receives 24 books, 36 books remain.
If each class receives 27 books, the school is 9 books short.
The total number of books is unchanged. Only the distribution plan changes.
2. Compare what changes per group
Plan B gives 3 more books per class than Plan A.
Across all classes, moving from Plan A to Plan B must absorb the 36-book surplus and also cover the 9-book shortage.
Total plan gap = 36 + 9 = 45 books.
Since each class accounts for 3 books of that gap, classes = 45 ÷ 3 = 15.
3. Recover the hidden total
Using Plan A:
Total books = 15 × 24 + 36 = 396.
Check Plan B:
15 × 27 = 405, which is 9 more than 396. Therefore the second plan is indeed short by 9.
4. Excess means actual total exceeds the planned distribution
If 10 bags receive 8 marbles each and 5 marbles remain, total marbles = 10 × 8 + 5 = 85.
The excess is added to the plan total.
5. Shortage means planned distribution exceeds actual total
If 10 bags need 9 marbles each but the supply is 5 short, actual total = 10 × 9 − 5 = 85.
The same actual total can therefore appear as an excess under one plan and a shortage under another.
6. The two equations must describe the same total
If there are n groups:
24n + 36 = 27n − 9.
Therefore 45 = 3n and n = 15.
The familiar heuristic “add excess and shortage, then divide by plan difference” is simply the equation written structurally.
7. Why excess and shortage are added
The first plan sits 36 below the actual total after its group allocations. The second planned allocation sits 9 above the actual total.
The distance between the two planned totals is therefore 36 + 9 = 45.
They lie on opposite sides of the same actual total.
8. If both plans have excess, subtract the excesses
Suppose giving 8 items each leaves 30, while giving 10 each leaves 6.
Both planned totals lie below the actual total. Their difference is 30 − 6 = 24.
Per-group difference = 2, so groups = 24 ÷ 2 = 12.
9. If both plans have shortage, subtract the shortages
Suppose giving 12 each is short by 6, while giving 15 each is short by 42.
The difference between planned totals is 42 − 6 = 36.
Per-group difference = 3, so groups = 12.
10. Decide whether the gap is a sum or a difference
Opposite-side conditions—one excess and one shortage—require adding their magnitudes.
Same-side conditions—both excess or both shortage—require finding the difference between their magnitudes.
A number-line view of planned totals makes this clear.
11. Draw the plans vertically
| Plan | Per group | Adjustment |
|---|---|---|
| A | 24 | +36 excess |
| B | 27 | −9 shortage |
The table exposes three useful quantities: per-group difference, excess/shortage gap and unknown number of groups.
12. Use units to keep the division meaningful
Total gap = 45 books. Difference per class = 3 books/class.
45 books ÷ 3 books/class = 15 classes.
Units confirm the intended quotient.
13. Distribution problems can hide in seating
If 6 students sit at each table, 18 students have no seat. If 8 sit at each table, 2 seats are empty.
Going from 6 to 8 adds 2 seats per table.
The gap between 18 unseated students and 2 empty seats is 20 places.
Tables = 20 ÷ 2 = 10.
Total students = 10 × 6 + 18 = 78.
14. Empty seats are an excess of capacity
“2 seats are empty” means the second plan creates 2 more places than needed. It is therefore an excess on the capacity side.
Translate language into surplus or deficit before calculating.
15. Packaging problems use the same structure
If sweets are packed 12 per bag, 20 remain. If packed 14 per bag, 8 are short of filling all bags equally.
Gap = 20 + 8 = 28. Difference per bag = 2. Bags = 14.
Total sweets = 14 × 12 + 20 = 188.
16. Money allocation works too
A teacher plans equal prizes. At $6 per student, $24 remains. At $8 per student, $10 is short.
Plan gap = 34. Per-student difference = $2. Student count = 17.
Total money = 17 × 6 + 24 = $126.
17. Distribution with equal groups may have a whole-number constraint
If the calculated group count is 12.5 but groups must be whole classes, inspect the data or interpretation. An exact excess-shortage setup with uniform groups should usually return a valid whole-number group count when constructed consistently.
18. Reverse problems can ask for one adjustment
There are 20 groups and two plans differ by 3 items per group. Therefore their planned totals differ by 60 items.
If Plan A leaves 42 extra and Plan B is short, shortage = 60 − 42 = 18.
19. Plan comparison can eliminate the hidden total
The actual total need not be found first. Comparing two equations removes it:
8n + 30 = 10n + 6.
24 = 2n, so n = 12.
The fixed total cancels because it is identical in both plans.
20. Excess-shortage is a special case of invariant reasoning
What remains fixed? The total supply or total number of people.
What changes? Allocation per group and the resulting surplus or deficit.
Once the invariant is identified, the problem becomes a comparison of two states.
21. Do not memorise “add then divide” without classifying the states
If both conditions have excess, adding them is wrong. If both have shortage, adding them is also wrong.
The operation on the adjustments depends on their positions relative to the actual total.
22. Build a number-line interpretation
Think of Plan A total, actual total and Plan B total as points on one line. Excess and shortage are distances between these points.
Opposite sides: distances add. Same side: distances subtract.
This makes the heuristic reconstructable even if forgotten.
23. Compound excess-shortage problems
A club has chairs arranged in rows. With 8 chairs per row, 14 people stand. With 11 chairs per row, 7 chairs are empty.
Capacity gap = 14 + 7 = 21. Difference per row = 3. Rows = 7.
People = 7 × 8 + 14 = 70.
24. Validate with both plans
For 70 people and 7 rows:
8 per row gives 56 seats, leaving 14 standing.
11 per row gives 77 seats, leaving 7 empty.
Both conditions return exactly.
25. Error map
| Visible error | Likely cause | Repair question |
|---|---|---|
| Adds two excesses | Relative positions not identified | Are both plans on the same side of the actual total? |
| Uses excess alone | Second plan not compared | What is the full distance between planned totals? |
| Divides gap by total per-group amount | Difference-per-group missed | How much more does Plan B allocate per group? |
| Cannot interpret empty seats | Capacity language not translated | Does the plan create too much or too little capacity? |
| Finds group count but does not verify | No return path | Do both original conditions reconstruct exactly? |
26. Practice laboratory
- 24 per class leaves 36; 27 per class is short 9. Find classes and total.
- 8 per group leaves 30; 10 per group leaves 6. Find groups.
- 12 per group is short 6; 15 per group is short 42. Find groups.
- 6 students/table leaves 18 standing; 8/table leaves 2 empty seats. Find tables and students.
- 12 sweets/bag leaves 20; 14/bag is short 8. Find bags and sweets.
- $6 per student leaves $24; $8 per student is short $10. Find students and money.
- 20 groups, plan difference 3 each, first plan leaves 42 extra. Find second-plan shortage if the plans lie on opposite sides.
27. Answers
1. Gap 45, per-class 3 → 15 classes; total 396.
2. Excess difference 24, /2 → 12 groups.
3. Shortage difference 36, /3 → 12 groups.
4. Gap 20, /2 → 10 tables; students 78.
5. Gap 28, /2 → 14 bags; sweets 188.
6. Gap $34, /$2 → 17 students; money $126.
7. Planned totals differ by 60. Shortage = 60 − 42 = 18.
28. Full distribution problem
A teacher has some worksheets. If each student receives 5 worksheets, 23 remain. If each student receives 7 worksheets, the teacher is 9 short. How many students and worksheets are there?
Per-student plan difference = 2 worksheets.
Opposite-side gap = 23 + 9 = 32.
Students = 32 ÷ 2 = 16.
Worksheets = 16 × 5 + 23 = 103.
Check: 16 × 7 = 112, exactly 9 more than 103.
29. Final checkpoint
A strong Primary 5 learner recognises excess and shortage as positions around one fixed total, distinguishes opposite-side and same-side cases, finds the per-group plan difference, uses the total gap to recover the number of groups and verifies every answer against both original distribution plans.
Wintour House V1.0 · CivDJ · eduKate Publishing: preserve the hidden total, compare plans rather than guessing totals, turn surplus and deficit into geometric distance, and require both plans to witness the final answer.
