Awsome Mental Math tricks

Squaring a two-digit number ending in 5

To find the square of a two-digit number ending in 5, take the first digit, multiply by itself plus one, and then put "25" after it.

Example: To find 65², compute 6 * 7 = 42, then write "4225" as the answer.

Example: To find 25², compute 2 * 3 = 6, then write "625" as the answer.

Note: This only works if the second digit is 5!

Multiplying two numbers when a round number is halfway between them

Suppose you are multiplying two moderately large numbers, 84 and 76. Note that the number 80 is halfway between the two.

This means that 84*76 can be rewritten as

(80 + 4)(80 - 4)

which simplifies to 80² - 4², which is easier to figure out: 6400 - 16 = 6384.

Note: This only works if the number halfway between your two original numbers, is a nice round number. Otherwise, squaring the number is almost as much mental work as multiplying the original numbers!

Multiplying large numbers very close to a round number

To multiply 1003 * 1004, for example, rewrite the multiplication mentally as:

(1000 + 3)(1000 + 4)

which expands to:

1000² + (3 + 4)*1000 + 3*4

which you can mentally work out to 1,007,012. In general,

(1000 + a)(1000 + b) = 1,000,000 + (a + b)1000 + a*b

and hopefully you'll be able to recognize a similar strategy when multiplying other large numbers that are "almost round".

Multiplying numbers that are just slightly less than a round number works similarly:

99*98

= (100 - 1)(100 - 2)

= 10000 - 100 - 200 + 2

= 9702

As well as, of course, multiplying a number slightly less than a round number, by a number slightly greater than a round number:

1001*998

= (1000 + 1)(1000 - 2)

= 1000000 + 1000 - 2000 - 2

= 998998

Remember to use the right sign for the last term, the "small" one! If the two large numbers you are multiplying are both slightly greater than a round number, or both slightly less, then the sign on the final small term will be positive; if one of the two large numbers is slightly less than a round number and the other one is slightly greater, than the final small term will be negative.

Also, in all of the examples above, the two numbers being multiplied were close to the same round number. Be prepared for cases where the round numbers might be different, e.g.:

2001*999

= (2000 + 1)(1000 - 1)

= 2000000 + 1000 - 2000 - 1

= 1998999

Powers of 11

Up to the 4th power, the digits in the powers of 11 follow the same pattern as the binomial coefficients:

11² = 121

11³ = 1331

11⁴ = 14641

i.e. if you expand (x + y)⁴ for example, the coefficients:

1x⁴ + 4x³y + 6x²y² + 4xy³ + 1y⁴

exactly match the digits of 11⁴ = 14641.

This makes sense if you consider 11⁴ to be an expansion of (10 + 1)⁴:

(10 + 1)⁴ = 1*10⁴ + 4*10³ + 6*10² + 4*10¹ + 1

Geometric series

These are easy to work out if you remember the formula, and because they're easy to work out, they're likely to show up in mental math:

a + ar + ar2 + ar3 + ... =

a (1 - r)

Example:

10 + 10/3 + 10/9 + 10/27 + ...

= 10/(1 - 1/3)    (by the formula)

= 10 / (2/3)

= 10*(3/2)

= 15

Sum of first n integers

Another easy formula likely to show up:

1 + 2 + 3 + ... + n =

n(n+1) 2

Example: 1 + 2 + 3 + ... + 12 = 12*13/2 = 6*13 = 78

(Note that to compute 12*13/2, rather than multiplying 12*13 and then dividing by 2, we divided 12 by 2 right away to get 6*13. This can make all the difference in mental math!)

Sum of the first n odd integers

The sum of the first n odd integers is n².

Example: The sum of the first 4 odd integers: 1 + 3 + 5 + 7 = 16 = 4²

Sum of the first n square numbers

1 + 22 + 32 + ... + n2 =

n(n+1)(2n+1) 6

Example: The sum of the first 10 square numbers:

The long way: 1 + 4 + 9 + 6 + 25 + 36 + 49 + 64 + 81 + 100 = 385

The faster way: 10*(10+1)(20+1)/6 = 10*11*21/6 = 385

To work out that last expression, by the way:

10*11*21/6

the shortcut is: First look at the numerator, the expression 10*11*21, and look for a number that can be divided by 2, and divide it by 2. In this case, reduce the 10 to a 5:

5*11*21

Then look for a number that can be divided by 3 -- in this case, 21 -- and divide it by 3:

5*11*7

The net effect of this (dividing by 2 and then dividing by 3) is, of course, dividing by 6. So now you only have to work out 5*11*7 = 385, which is easier (although still not easy) to do in your head.