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 trick

5       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

Pcunix profile image

Pcunix 5 years ago from SE MA

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


Manna in the wild profile image

Manna in the wild 5 years ago from Australia Author

Was it something like 'speed maths' or 'speed arithmetic'? something with 'speed' I think


Pcunix profile image

Pcunix 5 years ago from SE MA

Fifty years ago or more - I have no clue!


Kirui 5 years ago

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?


Manna in the wild profile image

Manna in the wild 5 years ago from Australia Author

Yes those would be trigonometric identities or series expansions.


marcello 2 years ago

I still don't understand your explanation.

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