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Maths: Algebra

▶  Watch: how to work backwards5 min

Four steps in about five minutes — one real question, worked step by step, then the traps to watch out for.

In these questions, someone starts with a secret number and does some steps to it. You're told the number it ends on.

Your job is to find the number it started with, from five answers. You know the end, and you want the start, so you work backwards. Sometimes the start is a letter, like n, but it's the same job.

In GL's papers a secret-number question tells you each step in words, in the order it was done. The video above and the worked example below work through the same question, with the options in the same order.

The method

Four steps: list the steps in order, start from the end, flip every step, check it forwards.

  1. Step 1 of 4

    List the steps in order

    Turn the words into a chain of steps, in the order they happen.
  2. Step 2 of 4

    Start from the end

    Begin with the end number you're given, and undo the last step first.
  3. Step 3 of 4

    Flip every step

    Each step flips to its opposite: add and take away swap, and so do multiply and divide.
  4. Step 4 of 4

    Check it forwards

    Run your answer through the steps the way they were done. It must land on the end number.

Worked example

George starts with a secret number. He adds 5 and then divides the result by 2. The result is 7. What was the secret number?

A. 4 B. 7 C. 9 D. 14 E. 19

  1. List the steps in order. George starts with a secret number, so the chain starts with a question mark. He adds 5, and then divides the result by 2. The result is 7, so 7 goes at the end: ? → + 5 → ÷ 2 → 7.
  2. Start from the end. Begin at 7. George divided by 2 last, so that's the first step to undo. Adding 5 comes second.
  3. Flip every step. Divide by 2 flips to multiply by 2, and 7 × 2 = 14. 14 is only halfway back, so keep going. Add 5 flips to take away 5, and 14 − 5 = 9. So the secret number is 9.
  4. Check it forwards. Run 9 through the steps the way George did them: 9 + 5 = 14, and 14 ÷ 2 = 7. It lands on 7. Now try the others. Put in 4, and it lands on 4½, so A is out. Put in 7, and it lands on 6, so B is out. Put in 14, and it lands on 9½, so D is out. Put in 19, and it lands on 12, so E is out. So the answer is C.

Answer: C — 9 (9 + 5 = 14, and 14 ÷ 2 = 7)

Things to watch out for

  1. Undo the last step first. The last thing done is the first thing to undo. In the worked example, A was 4: the steps undone in the wrong order. 7 − 5 = 2, and 2 × 2 = 4. Undo the dividing first, and you get 9.
  2. Every step flips. Not just one of them. In the worked example, E was 19. It flipped the divide but kept the add: 7 × 2 = 14, and 14 + 5 = 19. And D, 14, stopped after one flip.
  3. Answer what's asked. Finding the letter may be only a step. If 8t = 80, what is 5t − 4? Eight lots of t make 80, so t is 10. But 10 is only a step, and it's option A. Now work out 5t − 4, in that order: five 10s are 50, and 50 − 4 = 46.
  4. Check by going forwards. Some wrong answers sit right next to the real one, and only the check catches a near miss. This number machine ends on 71: ? → × 4 → + 11 → 71. 14 looks close, but 14 × 4 = 56, and 56 + 11 = 67. 15 × 4 = 60, and 60 + 11 = 71. Only 15 lands on the end number.

Related

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