HSPT: Number puzzles: work backward from the result
Learn to find a starting number by undoing each step of a number story in reverse order.
How do you find the starting number when a question tells you what was done to it and what came out?
Short answer
Start from the result and undo each step with its inverse, beginning with the last step. Then run your answer forward to check it.
Know first: Inverse operations, Multiplying and dividing whole numbers
Step 1 of 5
Undo the steps in reverse order
Some questions describe what happened to a mystery number and give you the result: A number is multiplied by 4, then 12 is added. The result is 52. To find the starting number, run the story backward.
Two ideas do the work. Each operation has an inverse that undoes it. And the last step done is the first one you undo, like taking off shoes before socks.
| Step in the story | Undo it with |
|---|---|
| Add a number | Subtract it |
| Subtract a number | Add it |
| Multiply by a number | Divide by it |
| Divide by a number | Multiply by it |
Step 2 of 5
Work backward from the result
Worked example: Two steps to undo
A number is multiplied by 4, then 12 is added. The result is 52. What was the starting number?
- A10
- B1
- C40
- D16
- 1
List the steps
First: multiply by 4. Last: add 12.
- 2
Undo the last step
52 − 12 = 40.
- 3
Undo the first step
40 ÷ 4 = 10.
- 4
Check forward
10 × 4 = 40, and 40 + 12 = 52. It works.
- 5
See why the other choices are there
1 undoes the steps in the wrong order: 52 ÷ 4 = 13, then 13 − 12 = 1. 40 stops after one step. 16 adds 12 when it should subtract: (52 + 12) ÷ 4.
Answer
10
Find the wrong step
A student solves: A number is divided by 2, then 9 is subtracted. The result is 5.
- 1
Undoes the last step, subtracting 9, by adding: 5 + 9 = 14.
- 2
Undoes dividing by 2 by dividing again: 14 ÷ 2 = 7.
What went wrong
The inverse of dividing is multiplying. Dividing again makes the number smaller still.
- 3
Answers 7.
The fix
Multiply: 14 × 2 = 28. Check: 28 ÷ 2 = 14, and 14 − 9 = 5.
Step 3 of 5
Run the story backward
Check yourself · Question 1
A number is multiplied by 5, then 10 is added. The result is 85. What was the starting number?
Answer: B
Undo the last step first: 85 − 10 = 75. Then undo multiplying by 5: 75 ÷ 5 = 15. Check: 15 × 5 + 10 = 85.
Trap answer: 7
Wrong order
- Why it looks right
- The story mentions multiplying first, so undoing the multiplication first feels natural.
- Why it's wrong
- Undoing works in reverse. The 10 was added last, so it has to come off first. Check: 7 × 5 + 10 = 45, not 85.
- The right answer
- 15, because 15 × 5 + 10 = 85.
Check yourself · Question 2
A number is divided by 4, then 5 is subtracted. The result is 9. What was the starting number?
Answer: C
Undo the subtraction: 9 + 5 = 14. Undo the division: 14 × 4 = 56. Check: 56 ÷ 4 = 14, and 14 − 5 = 9.
Common mistake
"I undo the steps in the order the question lists them."
- Why it's tempting
- Reading left to right is a strong habit, and the first step is the first thing you see.
- Do this instead
- Undo the last step first. Then plug your answer back into the story to make sure it lands on the result.
Step 4 of 5
What to remember
Remember
- Each step has an inverse: add and subtract undo each other, and so do multiply and divide.
- Undo the last step first and work back to the start.
- Run your answer forward through the story to check it.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
Which step did you undo first in the practice question, and what wrong answer comes from undoing them in the order listed?