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Excess and Shortage | The PSLE Mathematics Allocation Method

Excess and Shortage is a PSLE Mathematics heuristic for allocation problems where the same stock is distributed in two different ways: one plan leaves too many items, while another plan does not have enough. The difference between those two outcomes reveals the number of groups or recipients.

Students searching for excess and shortage method, PSLE distribution problem sums or Primary 5/6 heuristics often memorise a formula without understanding it. The method is simpler when viewed as two competing requirements for the same fixed stock.

This page is the canonical Excess and Shortage owner under the PSLE Mathematics Heuristics hub.

Quick answer: the core relationship

If one plan is short by some amount and another plan has some amount left over, the gap between the two required totals equals:

shortage + excess.

That same gap also equals:

difference per recipient × number of recipients.

So the method connects outcome-gap to per-recipient gap.

Worked example 1: the classic structure

Problem: If each child receives 6 pencils, there are 5 pencils short. If each child receives 5 pencils, there are 3 pencils left over. How many children are there?

The two plans differ by 1 pencil per child. The total requirement difference is 5 + 3 = 8 pencils. Therefore 1 × number of children = 8, so there are 8 children.

Check the stock: 5 × 8 + 3 = 43 pencils. Under the 6-each plan, 6 × 8 = 48, which is indeed 5 short.

Why shortage and excess are added

The “5 short” plan lies 5 above the available stock. The “3 excess” plan lies 3 below it. The distance between the two required totals is therefore 5 + 3 = 8.

A number line can make this visible: lower requirement → available stock → higher requirement.

Worked example 2: difference of two per-group amounts

Problem: If 7 pupils sit at each table, 4 pupils have no seats. If 8 pupils sit at each table, 3 seats are empty. How many tables are there?

The seating plans differ by 1 place per table. The outcome gap is 4 + 3 = 7 places. Therefore there are 7 tables.

Check: with 7 per table, 49 seats and 4 pupils unseated means 53 pupils. With 8 per table, 56 seats and 3 empty also means 53 pupils.

Worked example 3: per-group gap greater than one

Problem: If 4 oranges are packed in each bag, 10 oranges remain. If 6 oranges are packed in each bag, 8 oranges are short. How many bags are planned?

Per-bag difference = 2 oranges.
Outcome gap = 10 + 8 = 18 oranges.
Number of bags = 18 ÷ 2 = 9.

Available oranges = 4 × 9 + 10 = 46. The 6-per-bag plan needs 54, which is 8 more.

The four-step method

  1. Find the per-group difference.
  2. Find the total outcome gap. Add shortage and excess when the stock lies between the two requirements.
  3. Divide total gap by per-group difference.
  4. Use one scenario to recover the actual total or remaining quantity if asked.

Not every two-scenario problem is excess and shortage

If both scenarios leave excess, subtract the two excesses rather than automatically adding. If both scenarios have shortage, subtract the shortages. The general principle is always difference between outcomes, not “add because this chapter says excess and shortage”.

This is why Gap and Difference is the broader parent idea.

Three outcome patterns

  • One excess, one shortage: outcome gap = excess + shortage.
  • Both excess: outcome gap = larger excess − smaller excess.
  • Both shortage: outcome gap = larger shortage − smaller shortage.

The sign method for advanced learners

Treat excess as positive and shortage as negative relative to available stock. Then the difference between outcomes is simply the numerical distance between the two signed values.

Example: +3 excess and −5 shortage differ by 8. This unifies the cases and reduces memorised formulas.

Common errors

  • adds shortages when both scenarios are shortages
  • uses total per-group amount instead of the difference per group
  • finds the number of recipients but answers with total stock
  • forgets to verify both scenarios
  • uses excess/shortage method when the number of groups differs between scenarios
  • treats ‘left over’ as subtraction without building the two requirement totals

A visual model

Imagine two bars for required totals. The available stock sits somewhere between them when one scenario has excess and the other shortage. The vertical distance between the bars is the sum of the two offsets. That same distance is produced by giving every recipient the extra per-group amount.

Transfer practice

Practice 1

Scenario: 5 each leaves 7; 6 each short by 2.

First calculation: Per-group gap 1; outcome gap 9.

Practice 2

Scenario: 8 each leaves 11; 10 each short by 5.

First calculation: Per-group gap 2; outcome gap 16.

Practice 3

Scenario: 4 each leaves 12; 6 each leaves 2.

First calculation: Both excess: outcome gap 10; per-group gap 2.

Practice 4

Scenario: 7 each short by 15; 9 each short by 3.

First calculation: Both shortage: outcome gap 12; per-group gap 2.

Frequently asked questions

Why do we add excess and shortage?

Because the available stock lies between the two required totals, so the distance from one requirement to the other crosses both offsets.

What if both cases have excess?

Subtract the excess amounts.

What if both cases have shortage?

Subtract the shortage amounts.

What do I divide by?

The difference in amount per recipient or group.

How do I check?

Reconstruct both scenarios using the number of groups found.

Where does this sit in Atlas?

This is the canonical Excess and Shortage owner under PSLE Mathematics Heuristics.

The final excess-shortage rule

Do not memorise “add and divide”. Compare two requirements for the same stock. Find the outcome gap, find the per-group gap, then divide one by the other.

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