How do you solve a math riddle by "working backwards"?

Question

For example, how to you "work backwards" to solve this riddle?

Add 9 to me. Then multiply by 3. If you subtract 16 and then add 7, you get 27. What number am I?

Answer 1

The best way to approach a problem like this is to start at the end and work backwards, using the opposite operations. Here's how this example works.

27 - 7 = 20

20 + 16 = 36

36/3 (36 divided by 3) = 12

12 - 9 = 3

Answer: I am 3.

At each backward step we are using the opposite operation:

27 - 7 = 20 (We subtract 7 from 27 because it said, "add 7, you get 27")

20 + 16 = 36 (Because it said to "subtract 16," we add 16 to the last number we got)

36/3 (36 divided by 3) = 12 (It said to "multiply by 3," so we divide the last number we got by 3)

12 - 9 = 3 (It said "Add 9 to me," so we subtract 9) from the last number we got)

It's sort of like retracing your steps on a trip back home. If I walked 4 blocks east, then turned left and walked 3 blocks north to get to the store from home, on my trip home, I would walk 3 blocks south, then turn right and walk 4 blocks west to get home.

I strongly recommend that you line up each equation vertically, and that you start each equation with the number you reached in the previous equation. This will help you to keep track of your work. It's easy to get confused when you're working backwards, so every bit of organization helps!