MATHEMATICS  TRICS

Fast Arithmetic Tips

Mental Calculations - Getting the result fast

1. 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.
2. 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.
3. 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.
   More examples and explanation
4. 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.
   More examples and explanation
5. 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.
   More examples and explanation
6. 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.
7. Division/multiplication by 25
   Use operations with 4 instead.
   For example,
   37×25 = 3700/4 = 1850/2 = 925.
   More examples and explanation
8. Division/multiplication by 8
   Replace either with a repeated operation by 2.
   For example,
   124×8 = 248×4 = 496×2 = 992.
9. Division/multiplication by 125
   Use operations with 8 instead.
   For example,
   37×125 = 37000/8 = 18500/4 = 9250/2 = 4625.
10. Squaring two digit numbers.
    
    1. You should memorize the first 25 squares:

       1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 4 9 16 25 36 49 64 81 100 121 144 169 196 15 16 17 18 19 20 21 22 23 24 25 225 256 289 324 361 400 441 484 529 576 625
    2. 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 get18 = 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.
       More examples and explanation
    3. 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 = 1 and a = 24. Hence
       26² = 24² + 100 = 676.
       More examples and explanation
    4. 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.
       More examples and explanation
11. 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.
    More examples and explanation
12. 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 = 110 whose square is 12100. Added to this 110 and then 111 to get A²:

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

    More examples and explanation
13. 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 compute 11×(11 + 1) = 11×12 = 132 (since 3 = 1 + 2). Next, append 25 to the right of 132 to get 13225!
    More examples and explanation
14. 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(since 3 = 1 + 2). Next, append 21 (= 3×7) to the right of 132 to get 13221!
    More examples and explanation
15. Product of two one-digit numbers greater than 5.

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