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Maths olympiad question · P5–P6★★★★ Expert

Surplus and shortage: olympiad question 9

An expert question on surplus and shortage for P5–P6, with a step-by-step solution, diagrams and the answer. Surplus-and-shortage questions appear in the Hua Xia Cup's P1 to P5 papers; harder ones become questions about distance and time.

The question

A company shares out a year-end bonus among its staff, one of whom is the manager.

The plan: the manager gets two shares, everyone else one, at $200 a share — leaving $110 over.

Then 4 new staff join and must have a share too.

Now each share is $160, the manager still taking two and everyone else one; the last member of staff gets only $150, everyone else in full.

How much money is there altogether?

Show the hints (3)
  1. Hint 1: The manager takes two shares: how does the number of shares compare with the number of people? How many more shares does the second plan have?
  2. Hint 2: The last one gets only $150. To pay him in full, how much more money would it take? Try taking the 4 new shares away first.
  3. Hint 3: Without the new staff, at $160 a share the original shares leave 160 × 4 − (160 − 150) = $630 over; at $200 a share only $110.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.The manager takes two shares, so there is one more share than people. The second plan has 4 more shares (one for each newcomer).
  2. 2.Second plan: the last one gets only $150, $160 − 150 = $10 short of a full share.
  3. 3.In the second plan, take away the newcomers' 4 shares ($160 × 4 = $640) and pay the last one in full ($10 more): what remains is the original shares at $160 each, leaving $640 − 10 = $630 over.
  4. 4.The same shares leave $110 at $200 each and $630 at $160 each. Each share $200 − 160 = $40 more leaves 630 − 110 = $520 less over.
  5. 5.So there were 520 ÷ 40 = 13 shares (12 people).
  6. 6.The money: 200 × 13 + 110 = $2710.
  7. 7.Check: the second plan has 17 shares; at $160 that is $2720, $10 more than $2710, so the last one gets 160 − 10 = $150 ✓
  8. ⚠️Common mistake: counting people instead of shares (forgetting the manager's two), or leaving out the $10 the last one is short.

Answer: 2710 dollars

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