Square any number ending in 5 in three seconds

Try this without a calculator: what is 35²? Most people start the long multiplication, 35 × 35, and lose a minute. There is a much quicker way for every number that ends in 5, and it takes about three seconds once you have seen it.

The trick

Take the digits in front of the 5. For 35 that is 3.
Multiply that number by the next whole number. 3 × 4 = 12.
Write 25 after it. The answer is 1225.

So 35² = 1225. Check it with a long multiplication if you like: 35 × 35 = 1225.

More examples

15²: 1 × 2 = 2, then 25, so 225.
65²: 6 × 7 = 42, then 25, so 4225.
85²: 8 × 9 = 72, then 25, so 7225.
105²: 10 × 11 = 110, then 25, so 11025.

Why it works

Any number that ends in 5 can be written as 10a + 5, where a is the number in front of the 5. For 35, a = 3. Now use the identity (x + y)² = x² + 2xy + y² from Class 8 algebra:

(10a + 5)² = 100a² + 2 × 10a × 5 + 25
= 100a² + 100a + 25
= 100 × a(a + 1) + 25

That is exactly the trick. The part a(a + 1) is "the digits in front, times the next number". Multiplying by 100 pushes it two places to the left, which leaves room for the 25 at the end. Try it with 35: 100 × 3 × 4 + 25 = 1200 + 25 = 1225.

Using it backwards: square roots

The trick also helps you find square roots. Suppose you must find √7225. The number ends in 25, so the root ends in 5. Cover the 25 and look at what is left: 72. Which two consecutive numbers multiply to 72? 8 × 9. So the digits in front of the 5 are 8, and the root is 85. You can check: 85² = 7225.

Does the rule work for 5² itself? Here a = 0, so 0 × 1 = 0, followed by 25, gives 25. Yes.

Try it yourself

Find 45², 75², 95² and 115².
Find √5625 and √2025 using the backward method.
Why does the trick not work for 34² or 36²? (Hint: the first step used 10a + 5.)

Answers: 2025, 5625, 9025 and 13225; √5625 = 75 and √2025 = 45. Squares and square roots are part of Class 8, and you will meet more such patterns in the Class 8 chapters. When you want practice, build a fresh test with the Practice Generator, and share the trick with a friend who thinks maths is only about memorising.