Multiplication

Mathematics is easy – Some Tips – Part2

Consider two numbers to be multiplied, which are less than 100. Say, 97 and 98. Write both numbers one below the other. Deduct 100 from these numbers and write the difference on right side of these numbers with their sign, like
97 -3 (97 – 100 = -3)
98 -2 (98 – 100 = -2)
Now add diagonally opposite number (97-2 or 98-3 = 95). This is the first part of your answer. Multiply numbers on the right (-3 x -2 = +06). These two digits are the second part of the answer.
97 -3
98 -2
95 -5 (95 – 100 = -5)
94 -6 (94 – 100 = -6)
8930 is the answer (89 = 95 – 6 or 94 – 5, and 30 = -5 x -6)
Observe some more problems.
88 -12 78 -22 96 -4
93 -7 97 -3 98 -2
8184 7566 9408
After some practice, one need not write all these steps. An answer can be written directly.

In the above examples, both numbers are below 100. Let us consider numbers above 100.
Same procedure is followed as above. Earlier while adding diagonals, result was reducing due to negative sign, now it is increasing due to positive sign.
108 +8 (108 -100 = +8)
101 +1 (101 – 100 = +1)
10908 (101 +8 =108+1=109 and 8×1 =08)
Some more examples
103 +3 102 +2 104 +4
107 +7 120 +20 102 +2
11021 12240 10608

Before going to next section, let us revise some properties of numbers. Consider a number 2318.
2318 = 2 x 1000 + 3 x 100 + 1 x 10 + 8 x 1 = 2000+300+10+8
Or 2318 = 23 x 100 + 18 = 2300+18
Now let us see one of the numbers more than hundred and another less than hundred. Here one of the numbers differing from hundred will be positive and the other one negative. Their product will be negative. This negative number should be deducted from hundreds of left hand number. Let us see the example:
95 -5 Unit digit numbers, when multiplied becomes negative
112 +12 It should be deducted from Right hand number
107 / -60 → 107 =95 +12 = 112 – 5 / -60 = 12 x (-5)
10640 → 10700 -60 = 10640

78 -22 98 -2 96 -4
106 +6 102 +2 103 +3
8268 9996 9888
84/-132 100 /-04 99 / -12
8400 – 132 = 8268 9900 – 04 = 9896 9900 – 12 = 9888

So far we have considered numbers near hundred. Let us see numbers near fifty:
First deduct given numbers from 50. Then add diagonal numbers. Then multiply this number by ½ or 0.5. Next step is as usual, multiplying unit digit numbers.
58 +8 (58 – 50 = 8)
52 +2 (52 – 50 = 2)
(60X0.5)(16)
Since we have taken base number to deduct is fifty instead of 100, we have to multiply right hand side number by 1/2. Because 50 is half of 100.

48 -2 47 -3
44 -6 53 +3
2112 2491
42×0.5 =21 (50 x 0.5) x 100 – 9 = 2491

Consider two numbers (10A + B) and (10A + C), where B+C is = 10.
(10A+B) X (10A +C) = 10(A^2+A) +BC.
98X92 = (9X9+9) +8X2 = 9016
(or, 9X (9+1) + 8X2 = 9016 )
84 X 86 = (8X8 +8) + 24 = 7224
(or, (8X9)+ 4X6= 7224)

Consider two other numbers: (10B + A) X (10C+A), where B+C = 10, here the result will be
100(BXC+A) + A^2
63 X 43 = (6X4 +3) + 3X3 = 2709
In other words, while multiplying two numbers, whose unit or tens digit is common and the sum of other two digits is ten, then multiply tens digit numbers and add common number and directly multiply unit digit number.
63 x67 here tenth digit is common =6, unit digit sum is ten 7+3 = 10
Multiply tenth digit number and add common number -> 6×6+6= 42
Multiply unit digit number 7×3 =21
36×76 here tenth digits add up to 10(3 and 7) and unit digit number is common (6)
Answer in this case is (3×7 + 6) + 6×6 -> 2736

Consider another case,
If tenth digits differ by one and unit digits add up to 10 as in 68×52 or 74×66.
Take the larger number. Square the tenth digit number and subtract square of unit digit number.
68×52 = 6^2 – 8^2 = 3600 – 64= 3536
66×74 = 7x7x100-4×4 = 4900 -16 =4884.
Let us consider a general case, where unit digits add up to 10 and tenth digits differ by ‘r’.
(10a + b) x (10c +d) where (b + d) = 10 and (a – c) = r.
= 100(a*c + c) +10* r*d + (b*d) = 100(a*c + c) +10* (a-c)*d +b*d
98*42 = (9*4+4)*100+10*(9-4)*2+8*2
=4000+100+16 = 4116
67*23 = 1541 -> (6*2+2)*100 =1400, (6-2)*3*10+21 = 120+21=141 ->1400+141 = 1541

Tenth/ unit digit is same and other digit (unit/ tenth digit) add up to 5, then:
100 times the product of tenth digits and half of common digit + product of unit digits
64*61 = (36+3)*100 + 04 = 3904
72*73 = 100*(7*7+3.5) + 06 =5256
28*38 = 100*(2*3+4) + 64 = 1064
49*19 = 100*(4*1+4.5) + 81 = 931

One of the digits is common, then while multiplying
100 times (product of tenth digits plus + sum of uncommon digits * common digit/10)+ product of unit digits
(10A+B)*(10A+C) = 100*{(A*A +A*(B+C)/10)} + BC ->
68*64 = 100*{(6*6) +6(8+4)/10) + 8*4 =(36+7.2)*100 + 32 = 4352
(10A +B)* (10C + B) = 100*{(A*C)+B(A+C)/10)} +B*B ->
37 *67 = 100{(3*6) +7*(3+6)/10)} + 7 * 7 = (18+6.3)*100 +49 = 2479