# squaring maths trick how to quickly square a number that ends in 5 e.g. 35 squared.

## First an example.

You need to compute 35 x 35.

Instantly, a child can answer "3 x 4 = 12 so 35 squared is 1,225"

## Now the trick is explained

The number must end in 5, although numbers over 99 are a little harder, and as they get larger it gets harder. Even so, it's a useful trick for small numbers.

Given 45 x 45, you take the 4 and multiply it by (4+1). i.e. 4 * 5 = 20, then you tac on 25 at the end to give 2025.

If the number is 115 squared, then you do 11 x 12 and tac on 25. So 115 squared = 13,225.

## Some more examples

N n squared the trick5 25 0 15 225 2 25 625 6 35 1225 12 45 2025 20 55 3025 30 65 4225 42 75 5625 56 85 7225 72 95 9025 90 105 11025 110 115 13225 132 125 15625 156 135 18225 182 145 21025 210 155 24025 240 165 27225 272 175 30625 306 185 34225 342 195 38025 380 205 42025 420 215 46225 462 225 50625 506 235 55225 552 245 60025 600

## Why does this work?

You can use some algebra to see why this works.

Our number ends in 5 and so can be written:

X5

Where X is all the digits to the left of the 5 so that X can represent 16 in X5 for 165.

We can temporarily forget the actual value of x and try to work out the formula for X5 squared.

Since X is in the 10s column, then we can rewrite X5 as (10x+5).

Let's find (10x+5) squared.

= (10x+5)(10x+5)

= 100xx + 50x + 50x + 25

= 100xx + 100x +25

take out the 100x common to the first two terms...

= 100x(x+1) + 25

QED.

We have our formula. Take the x and multiply by (x+1) and make this 100 times bigger and add 25.

See also:

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## Comments 6 comments

I learned this and many others from a book many, many years ago. I lost it and do not remember the title, unfortunately.

Fifty years ago or more - I have no clue!

I found a formular for the sum, cos1+cos2+cos3.........+cosn. And also for sin, sinh, and cosh. I don't know, have this been found by mathematicians?

6