A trader sells grain to three buyers. To the first buyer, the trader gives half the stock plus an extra 3 kg. To the second buyer, the trader gives half of the remaining stock plus an extra 3 kg. To the third buyer, the trader again gives half of the remaining stock plus an extra 3 kg, after which no grain is left. Find the initial quantity of grain.
Answer and solution
Correct answer: 42
Solution
Answer: 42
1. After each sale, remaining grain = previous stock −(half of it+3)=previous stock/2−3.
2. Invert the rule: previous stock=2(remaining stock+3).
3. Before buyer 3: 2(0+3)=6 kg. Before buyer 2: 2(6+3)=18 kg.
4. Before buyer 1: 2(18+3)=42 kg.
5. Forward check: sells 24, then 12, then 6 kg; total 42 and nothing remains.
Why this method works:
Work backwards because the final remainder is known exactly. The forward map takes half the current stock and then subtracts 3; its inverse must add 3 first and then double. Doubling before adding 3 would invert a different operation.
Independent check and common trap:
Forward verification from 42 kg: the first buyer receives 21+3=24, leaving 18; the second gets 9+3=12, leaving 6; the last gets 3+3=6, leaving 0. The three sales add to 42, so no stock is missing.
Apply: Forward remainder=x/2−3, so reverse is x=2(remainder+3). Start from 0: 0→6→18→42.
Check: Forward verification from 42 kg: the first buyer receives 21+3=24, leaving 18; the second gets 9+3=12, leaving 6; the last gets 3+3=6, leaving 0. The three sales add to 42, so no stock is missing.
1. After each sale, remaining grain = previous stock −(half of it+3)=previous stock/2−3.
2. Invert the rule: previous stock=2(remaining stock+3).
3. Before buyer 3: 2(0+3)=6 kg. Before buyer 2: 2(6+3)=18 kg.
4. Before buyer 1: 2(18+3)=42 kg.
5. Forward check: sells 24, then 12, then 6 kg; total 42 and nothing remains.
Why this method works:
Work backwards because the final remainder is known exactly. The forward map takes half the current stock and then subtracts 3; its inverse must add 3 first and then double. Doubling before adding 3 would invert a different operation.
Independent check and common trap:
Forward verification from 42 kg: the first buyer receives 21+3=24, leaving 18; the second gets 9+3=12, leaving 6; the last gets 3+3=6, leaving 0. The three sales add to 42, so no stock is missing.
G Strategy shortcut
Spot: Linear Equations and Word Problems.Apply: Forward remainder=x/2−3, so reverse is x=2(remainder+3). Start from 0: 0→6→18→42.
Check: Forward verification from 42 kg: the first buyer receives 21+3=24, leaving 18; the second gets 9+3=12, leaving 6; the last gets 3+3=6, leaving 0. The three sales add to 42, so no stock is missing.
G Strategy: Spot: Linear Equations and Word Problems. Apply: Forward remainder=x/2−3, so reverse is x=2(remainder+3). Start from 0: 0→6→18→42. Check: Forward verification from 42 kg: the first buyer receives 21+3=24, leaving 18; the second gets 9+3=12, leaving 6; the last gets 3+3=6, leaving 0. The three sales add to 42, so no stock is missing.