Multiplication Tricks
Squaring Numbers Ending in 5 – n × (n + 1), Then 25
Squares of numbers ending in 5 have a pattern that makes them almost instant.
This page explains the rule, shows why it works, and walks through worked examples. Then you can try it yourself and check your understanding with practice questions. Use the previous and next links at the bottom to move through the tricks in order.
The rule
The Rule: Square any two-digit number ending in 5 by multiplying the first digit by itself plus one, then appending 25.
Why it works
Why It Works: Every two-digit number ending in 5 is really “some leading digit, then a 5” — like “4, then 5” for 45. It turns out that when you square any such number, the answer always ends in exactly 25, and the digits in front are simply that leading digit multiplied by one more than itself. It looks like a magic shortcut, but it drops straight out of expanding the square with ordinary algebra.
Formal Proof (for advanced learners)
A number ending in 5 can be written as 10a + 5, where a represents the leading digit(s). Expanding the square: (10a + 5)² = 100a² + 2·(10a)·5 + 25 = 100a² + 100a + 25 = 100·a(a+1) + 25. This is precisely “a times (a + 1), followed by the digits 25.”
Examples
- 25² → 2 × 3 = 6, append 25: 625.
- 45² → 4 × 5 = 20, append 25: 2025.
- 105² → 10 × 11 = 110, append 25: 11025.
Step by step
Example A: 75²
- Take the number in front of the 5: 7.
- Multiply it by the next whole number: 7 × 8 = 56.
- Write 25 after it: 5625.
Example B: 115²
- Take the number in front of the 5: 11.
- Multiply it by the next whole number: 11 × 12 = 132.
- Write 25 after it: 13225.
Try it yourself
Now it is your turn. A new random question appears each time. Type your answer in the box and press Check Answer. If you are right, we show how the trick got you there, step by step. If you are not, you will see the full working two ways, the shortcut and the usual method, so you can spot exactly where your answer differed. Press New Question as often as you like: a few quick rounds is the best way to make the trick feel automatic.
Practice questions
Have a go on paper first, then tap Show answer to check your method.
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Work out 15².
Show answer
225 m². Take the number in front of the 5: 1. Multiply it by the next whole number: 1 × 2 = 2. Write 25 after it: 225.
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Work out 35².
Show answer
1225 m². Take the number in front of the 5: 3. Multiply it by the next whole number: 3 × 4 = 12. Write 25 after it: 1225.
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Work out 55².
Show answer
3025 m². Take the number in front of the 5: 5. Multiply it by the next whole number: 5 × 6 = 30. Write 25 after it: 3025.
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Work out 85².
Show answer
7225 m². Take the number in front of the 5: 8. Multiply it by the next whole number: 8 × 9 = 72. Write 25 after it: 7225.
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A square garden has sides of 65 m. What is its area in square metres?
Show answer
4225 m². Take the number in front of the 5: 6. Multiply it by the next whole number: 6 × 7 = 42. Write 25 after it: 4225.
Tips and watch-outs
- It works for any number ending in 5, however long: 105² = 10 × 11 = 110, then 25 → 11025.
- The answer always ends in 25.