KA-0100Working backwards
4 points, difficulty 2 of 3Level 5–6about 75sA number is doubled, then 6 is added, then the result is halved. The answer is 11. What was the original number?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Start at 11 and undo each step.
2The strategy
Reverse both the operations and their order. The halving happened last, so undo it first, then the addition, then the doubling.
3The full solution
Undo the halving: 11 x 2 = 22. Undo the addition: 22 - 6 = 16. Undo the doubling: 16 divided by 2 = 8.
Solution
The reliable way
Run the machine backwards. The last step was halving, so undo it first: 11 x 2 = 22. Before the 6 was added it was 22 - 6 = 16. Before doubling it was 16 divided by 2 = 8. Check forwards: 8 doubled is 16, plus 6 is 22, halved is 11. The transferable idea: reverse the operations and the order together, and always check forwards at the end.
The elegant way
Doubling and then halving cancel, so the machine is really add 6 then halve, making the start 2 x 11 - 6 all halved: 8.
Why this is on the test: Working backwards is worth several marks a paper, and reversing the operations while keeping the original order is the reliable way to get it wrong.