Imagine you’re trying to square a 2-digit number in your head. Sounds tricky. Actually, it’s quite simple once you learn the formula, and with practice can be done very quickly.
The most convenient way to consider a 2-digit number is 10a + b, where ‘a’ is the first digit and ‘b’ is the second.
So if we were trying to find the square of a 2-digit number, we would write the formula as:
(10a + b)² or (10a + b)(10a + b)
Multiplying out the brackets gives us:
100a² + 20ab + b²
And if we factor out 10 from the first two terms, we get:
10(10a² + 2ab) + b²
This might look complicated, but once you get to grips with what each letter represents, it’s actually very straightforward.
Suppose we were trying to find the square of 79.
Using 10a + b as the notation for 2-digit numbers, we can imagine the number 79 as:
10(7) + 9
Thus, a = 7 and b = 9
Plugging this information back into our formula gives us:
10(10(7)² + 2(7)(9)) + 9²
Again, this appears confusing, but in actual fact we are simply calculating as follows:
7² = 49
49 x 10 = 490
2 x 7 x 9 = 126
490 + 126 = 616
616 x 10 = 6160
9² = 81
6160 + 81 = 6241
The more you practice using this formula, the more you get used to the order in which to calculate, and the quicker you can complete the sum.
We can actually use the same method with a slightly altered formula to calculate the product of any two 2-digit numbers.
If we imagine our first 2-digit number as 10a + b and our second as 10c + d, the product of the two terms would give us:
(10a + b)(10c +d)
Like the previous method, we multiply out the brackets, giving us:
100ac + 10ad + 10bc + bd
And again, we factor out 10 from all terms with a coefficient that is divisible by 10, giving us our final formula:
10(10ac + ad + bc) + bd
Using 84 x 59 as our example, we allocate as follows:
a = 8, b = 4, c = 5, d =9
And as for the order of calculations:
10 x 8 x 5 = 400
8 x 9 = 72
400 + 72 = 472
4 x 5 = 20
472 + 20 = 492
492 x 10 = 4920
4 x 9 = 36
4920 + 36 = 4956
If you’ve managed to get through all of this information and still feel like you understand what’s going on, give it a try for yourself!
Give yourself a sum to do in your head using this method, then check the answer on a calculator.
The more you practice, the more fluent it will become.