Tag Archives: math short cut

MSC 15: Squaring Numbers Ending in 5

Squaring is multiplying a number by itself.

It is very easy to square two-digit numbers ending in five by using the Vedic Math sutra or word formula “By one more than the one before.”

The square of a two-digit number ending in 5 is composed of 2 parts.

  • The first or left hand part is obtained by multiplying the ten’s digit “By one more than the one before” or by the next number.
  • The second part is always 5 x 5 or 25.

Example 15.1:     Find 652

  • The first part is the ten’s digit 6 multiplied by the next number 7, (By one more than the one before) so  6 x 7 =  42.
  • The second part is always 25 ( 5 x 5) .
  • 652= 42|25 or 4,225

Even larger numbers can be squared.

Example 15.2:     Find 9952

          9952  =  99 x 100|25

          9952  = 990,025

Proof P15: Let t = the ten’s digit; 10t + 5 = two digit number ending in 5(10t + 5)2 = (10t)2 + 2(10t)(5) + 52= 100t2  + 100t + 25= 100(t2 + t) + 25= 100(t)(t + 1)+ 25

We can extend the technique to decimals and fractions.

Example 15.3:     3.52 = 3 x 4 | .5 x .5  = 12.25

Example 15.4:     (7 ½)2 = 7 x 8 | ½  x ½  =  56 1/4

Exercise 15:     Find the square of the following numbers

  1. )   45
  2. )   95
  3. )   195
  4. )   10,005
  5. )   10.5
  6. )   7.5
  7. )   6 ½
  8. )   9 ½
  9. )   100.5
  10. )   0.025

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

MSC 12 – Dividing by 4 and 8

We can easily divide a number by 4 by halving the number twice and by 8, by halving thrice.  We shall, however try dividing by 4 and 8 using the one line solution introduced in MSC 6 – Division by Two.

Example 12.1:    586 ÷ 4 =

When dividing a whole number by 4, there are only 4 possible remainders: 0, which means that the number is evenly divisible by 4; and 1, 2 and 3 which are equivalent to 0.25, 0.50 and 0.75 respectively. In this example, the dividend is an even number so the remainder can only be 0 or 2.

             1                         1   4                     1   4   6 
          4) 5 18  6           4) 5 18 26             4) 5 18 26  ( r 2

a) Dividing 5 by 4 gives a quotient of 1 and a remainder of 1 which we place in front of the next digit of the dividend,

b) Dividing the next dividend 18 by 4 results in a quotient of 4 and a remainder of 2, which we then place in front of the next digit

c) Finally, we divide 26 by 4 to give 6 and a remainder of Thus the final quotient is 146 2/4 or 146.5.

 Example 12.2:     743  ÷ 4 =

               1   8   5                                                
           4) 7 34  23 (r 3

Here the dividend is an odd number, so we expect a remainder of 1 or 3.

  1. Dividing 7 by 4 gives 1 remainder
  2. 34 divided by 4 gives 8 remainder23 divided by 4 gives 5 remainder
  3. The quotient is 185 3/4 or 185.75. 

 The following are the values of the remainders when dividing by 8.

Remainder                         decimal equivalent
        1                                              0.125
        2                                              0.25
        3                                              0.375
        4                                              0.5
        5                                              0.625
        6                                              0.75
        7                                              0.875

Example 12.3:          983÷ 8 =

                    1   2  2                                  
                8) 9 18  23 (7

Using the one line solution, we have

  1. 9 divided by 8 is 1 remainder 1
  2. 18 divided 8 is 2 remainder 2
  3. 23 divided by 2 is 2 remainder 7

So we arrive at an answer of 122 remainder 7. This is equivalent to 122 7/8 or 122.875

Example 12.4:         6352 ÷ 8 =

                 7   9  4                                               
           8) 63 75 32 (0

Using the one line method, we got the quotient 794. For the careful observer, the dividend, 6,352 is very near 6,400 or exactly 48 less.  6,400÷ 8 = 800 and 48÷ 8 = 6 so 6352 ÷ 8 is simply 800 – 6 = 794.

 Exercise 12:

  1. )  456 ÷ 4 =
  2. )  527 ÷ 4 =
  3. )  983 ÷ 4 =
  4. )  2,538 ÷ 4 =
  5. )  6,789 ÷ 4 =
  6. )  745 ÷ 8 =
  7. )  1,278 ÷ 8 =
  8. )  3,684 ÷ 8 =
  9. )  6,454 ÷ 8 =
  10. )  10,522 ÷ 8 =

 

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!