3. Mathematical rules for multiplication
In this section we will look at the mathematical rules and formulas that are useful for mental calculation.
Commutative and associative laws
Like addition, multiplication is also commutative and associative.
With regard to commutativity, we say that whatever two numerical values denoted by a and b, we have
a x b = b x a
We say with respect to associativity that whatever three numerical values denoted by a, b and c, we have
(a x b) x c = a x (b x c)
These rules can be used in mental arithmetic as follows:
To multiply a number by 4, multiply it by 2 and then multiply the result by 2
To multiply a number by 6, multiply it by 3 then multiply the result by 2
To multiply a number by 8, multiply it by 2 then multiply the result by 2
To multiply a number by 12, multiply it by 2 then multiply the result by 2 then multiply the result by 3
To multiply a number by 20, multiply it by 2 then multiply the result by 10
To multiply a number by 40, multiply it by 2 then multiply the result by 2 then multiply the result by 10
Examples :
57 x 4 = 57 x 2 x 2 = (57 x 2) x 2 = 104 x 2 = 208
123 x 12 = 123 x 3 x 2 x 2 = 369 x 2 x 2 = 738 x 2 = 1476
Example of the application of the commutative law :
35 x 45 x 4 = 35 x 45 x 2 x 2 = 35 x 2 x 45 x 2 = 70 x 90 = 6300
Other cases of stepwise calculation
To multiply by 5, multiply by 10 then divide by 2. To divide by 5, multiply by 2 then divide by 10.
Remember that 0.5 = ½. To multiply a number by 0.5, simply divide by 2.
In general, to multiply by a number that has as unit 5, multiply by the double of that number and divide by 2. And to divide by a number having in unit 5, divide by the double of that number and multiply by 2.
Example :
260 / 65= (260 / 130) x 2 = 2 x 2 = 4
Example :
12 x 45 = (12 x 90) / 2 = 1080 / 2 = 540
But for the last example, it is better to do :
12 x 45 = 6 x 2 x 45 = 6 x 90 = 540
To multiply by 25, multiply by 100 and then divide by 4 or divide twice by 2.
Remember that 0.25 = ¼. To multiply a number by 0.25, simply divide it by 4 or divide it twice by 2.
To multiply by 50, multiply by 100 then divide by 2.
Multiplying by 9
9 = 10 – 1, so to multiply a number by 9, simply multiply it by 10, and subtract it from the result.
For example :
64 x 9 = (64 x 10) – 64 = 640 – 64 =(640 + 36) – (64 + 36) = 676 – 100 = 576
Simplifying and Factoring
For any three numerical values denoted by a, b and c, we have
a x (b + c) = (a x b) + (a x c)
(b + c) x a = (b x a) + (c x a)
For example
3 x (4 + 5) = 3 x 4 + 3 x 5
In case of subtraction
a x (b - c) = (a x b) - (a x c)
(b - c) x a = (b x a) - (c x a)
With more values:
a x (b + c + d) = (a x b) + (a x c) + (a x d)
Cases of division:
(b + c) / a = (b / a) + (c / a)
(b - c) / a = (b / a) - (c / a)
Products of two sums:
(a + b) x (c + d) = (a x c) + (a x d) + (b x c) + (b x d)
(a - b) x (c + d) = (a x c) + (a x d) - (b x c) - (b x d)
(a + b) x (c - d) = (a x c) - (a x d) + (b x c) - (b x d)
(a - b) x (c - d) = (a x c) - (a x d) - (b x c) + (b x d)
Remarkable identities :
(a + b)2 = a2 + 2 x a x b + b2
(a - b)2 = a2 - 2 x a x b + b2
(a + b) x (a – b) = a2 - b2
These equalities will be used in the next chapter for the 1st method of mental calculation of multiplication.