**Mental Calculations - Getting the result fast**

Addition of 5

When adding 5 to a digit greater than 5, it is easier to first subtract 5 and then add 10.

For example,

7 + 5 = 12.

Also 7 - 5 = 2; 2 + 10 = 12.

Subtraction of 5

When subtracting 5 from a number ending with a a digit smaller than 5, it is easier to first add 5 and then subtract 10.

For example,

23 - 5 = 18.

Also 23 + 5 = 28; 28 - 10 = 18.

Division by 5

Similarly, it's often more convenient instead to multiply first by 2 and then divide by 10.

For example,

1375/5 = 2750/10 = 275.

Multiplication by 5

It's often more convenient instead of multiplying by 5 to multiply first by 10 and then divide by 2.

For example,

137×5 = 1370/2 = 685.

Division by 5

Similarly, it's often more convenient instead to multiply first by 2 and then divide by 10.

For example,

1375/5 = 2750/10 = 275.

Division/multiplication by 4

Replace either with a repeated operation by 2.

For example,

124/4 = 62/2 = 31. Also,

124×4 = 248×2 = 496.

Division/multiplication by 25

Use operations with 4 instead.

For example,

37×25 = 3700/4 = 1850/2 = 925.

Division/multiplication by 8

Replace either with a repeated operation by 2.

For example,

124×8 = 248×4 = 496×2 = 992.

Division/multiplication by 125

Use operations with 8 instead.

For example,

37×125 = 37000/8 = 18500/4 = 9250/2 = 4625.

Squaring two digit numbers.

You should memorize the first 25 squares:

12345678910111213141491625364964811001211441691961516171819202122232425225256289324361400441484529576625

**If you forgot an entry**.

Say, you want a square of 13. Do this: add 3 (the last digit) to 13 (the number to be squared) to get16 = 13 + 3. Square the last digit: 3² = 9. *Append*the result to the sum: 169.

As another example, find 14². First, as before, add the last digit (4) to the number itself (14) to get 18 = 14 + 4. Next, again as before, square the last digit: 4² = 16. You'd like to append the result (16) to the sum (18) getting 1816 which is clearly too large, for, say, 14 < 20 so that 14² < 20² = 400.What you have to do is append 6 and carry 1 to the previous digit (8) making 14² = 196.

**Squares of numbers from 26 through 50**.

Let A be such a number. Subtract 25 from A to get x. Subtract x from 25 to get, say, a. ThenA² = a² + 100x. For example, if A = 26, then x = 1and a = 24. Hence

26² = 24² + 100 = 676.

**Squares of numbers from 51 through 99**.

If A is between 50 and 100, then A = 50 + x. Compute a = 50 - x. Then A² = a² + 200x. For example,

63² = 37² + 200×13 = 1369 + 2600 = 3969.

Any Square.

Assume you want to find 87². Find a simple number nearby - a number whose square could be found relatively easy. In the case of 87 we take 90. To obtain 90, we need to add 3 to 87; so now let's subtract 3 from 87. We are getting 84. Finally,

87² = 90×84 + 3² = 7200 + 360 + 9 = 7569.

Squares Can Be Computed Squentially

In case A is a successor of a number with a known square, you find A⊃ by adding to the latter itself and then A. For example, A = 111 is a successor of a = 110whose square is 12100. Added to this 110 and then 111 to get A²:

111²= 110² + 110 + 111 = 12100 + 221 = 12321.

Squares of numbers that end with 5.

A number that ends in 5 has the form A = 10a + 5, where a has one digit less than A. To find the square A² of A, append 25 to the product a×(a + 1) of a with its successor. For example, compute 115².115 = 11×10 + 5, so that a = 11. First compute11×(11 + 1) = 11×12 = 132 (since 3 = 1 + 2). Next, append 25 to the right of 132 to get 13225!

Product of 10a + b and 10a + c where b + c = 10.

Similar to the squaring of numbers that end with 5:

For example, compute 113×117, where a = 11, b = 3, and c = 7. First compute 11×(11 + 1) = 11×12 = 132 (since3 = 1 + 2). Next, append 21 (= 3×7) to the right of 132 to get 13221!

Product of two one-digit numbers greater than 5.

This is a rule that helps remember a big part of the multiplication table. Assume you forgot the product 7×9. Do this. First find the excess of each of the multiples over 5: it's 2 for 7 (7 - 5 = 2) and 4 for 9 (9 - 5 = 4). Add them up to get 6 = 2 + 4. Now find the complements of these two numbers to 5: it's 3 for 2 (5 - 2 = 3) and 1 for 4(5 - 4 = 1). Remember their product 3 = 3×1. Lastly, combine thus obtained two numbers (6 and 3) as63 = 6×10 + 3.

Product of two 2-digit numbers.

The simplest case is when two numbers are not too far apart and their difference is even, for example, let one be 24 and the other 28. Find their average:(24 + 28)/2 = 26 and half the difference (28 - 24)/2 = 2.Subtract the squares:

28×24 = 26² - 2² = 676 - 4 = 672.

The ancient Babylonian used a similar approach. They calculated the sum and the difference of the two numbers, subtracted their squares and divided the result by four. For example,

33×32= (65² - 1²)/4 = (4225 - 1)/4 = 4224/4 = 1056.

Product of numbers close to 100.

Say, you have to multiply 94 and 98. Take their differences to 100: 100 - 94 = 6 and 100 - 98 = 2. Note that 94 - 2 = 98 - 6 so that for the next step it is not important which one you use, but you'll need the result: 92. These will be the first two digits of the product. The last two are just 2×6 = 12. Therefore, 94×98 = 9212.

Multiplying by 11.

To multiply a 2-digit number by 11, take the sum of its digits. If it's a single digit number, just write it between the two digits. If the sum is 10 or more, do not forget to carry 1 over.

For example, 34×11 = 374 since 3 + 4 = 7. 47×11 = 517since 4 + 7 = 11.

Faster subtraction.

Subtraction is often faster in two steps instead of one.

For example,

427 - 38 = (427 - 27) - (38 - 27) = 400 - 11 = 389.

A generic advice might be given as "First remove what's easy, next whatever remains". Another example:

1049 - 187 = 1000 - (187 - 49) = 900 - 38 = 862.

Faster addition.

Addition is often faster in two steps instead of one.

For example,

487 + 38 = (487 + 13) + (38 - 13) = 500 + 25 = 525.

A generic advice might be given as "First add what's easy, next whatever remains". Another example:

1049 + 187 = 1100 + (187 - 51) = 1200 + 36 = 1236.

Faster addition, #2.

It's often faster to add a digit at a time starting with higher digits. For example,

583 + 645= 583 + 600 + 40 + 5 = 1183 + 40 + 5 = 1223 + 5 = 1228.

Multipliply, then subtract.

When multiplying by 9, multiply by 10 instead, and then subtract the other number. For example,

23×9 = 230 - 23 = 207.

The same applies to other numbers near those for which multiplication is simplified:

23×51= 23×50 + 23= 2300/2 + 23= 1150 + 23= 1173. 87×48= 87×50 - 87×2= 8700/2 - 160 - 14= 4350 - 160 - 14= 4190 - 14= 4176.

Multiplication by 9, 99, 999, etc.

There is another way to multiply fast by 9 that has an analogue for multiplication by 99, 999 and all such numbers. Let's start with the multiplication by 9.

To multiply a one digit number *a* by 9, first subtract 1 and form *b* = *a* - 1. Next, subtract *b* from 9: *c* = 9 - *b*. Then just write *b* and *c* next to each other:

9*a* = *b**c*.

For example, find 6×9 (so that *a* = 6.) First subtract:5 = 6 - 1. Subract the second time: 4 = 9 - 5. Lastly, form the product 6×9 = 54.

Similarly, for a 2-digit *a*:

*b**c*= 100*b* + *c* = 100(*a* - 1) + (99 - (*a* - 1)) = 100*a* - 100 + 100 - *a* = 99*a*.

Do try the same derivation for a three digit number. As an example,

543×999= 1000×542 + (999 - 542) = 542457.

Adding a Long List of Numbers

How fast can you calculate the sum

97 + 86 + 83 + 95 + 85 + 70 + 84 + 72 + 77 + 81 + 70 + 85 + 84 + 76 + 92 + 66?