In the first week, we learned how to multiply two numbers in the teens together instantly, as if you memorized the answer. To the average person, this is the only way possible to get the answer so quickly! Now, you can take that up a notch, and multiply together the biggest two digit numbers. Say someone says 97 x 94. You can say, "That's easy! It's 9118!"
We will take it step by step here. For the first two digits, you take the bigger number and see how far away it is from 100. In this case, it is three away. Then, you subtract that from the other number. 94 - 3 = 91. There's your first two digits.
For the last two digits, take how far both numbers are from 100. 97 is 3 away and 94 is 6 away. The answer is precisely 3 x 6 = 18. Put them together and you have 9118.
Let's try another one, 95 x 89. 95 is 5 away from 100, and 89 - 5 = 84. Then, 89 is 11 away from 100, so 11 x 5 = 55. Then, we put them together to get 8455.
You are probably wondering why this works, for teens or nineties. Let's start with the teens. Pretend that the problem is (z + a)(z + b) with z being 10. We will leave it as z for the moment.
(z + a)(z + b) = z(z + a + b) + ab
If you factor it out, the z(z + a + b) becomes z^2 + za + zb.
(z + a)(z + b) = z^2 + za + zb + ab
If you FOIL out the (z + a)(z + b), you get:
z^2 + zb + za + ab = z^2 + za + zb + ab
This shows that they are equal. If you think about it, this is what we are doing. Take 17 x 16.
(10 + 7)(10 + 6) = 10(10 + 7 + 6) + (7)(6)
This is what you are actually doing. What about for 89 x 95.
(100 - 5)(100 - 11) = 100(100 - 5 - 11) + (-5)(-11)
I got commented about the fact that 20 wouldn't work for the teen method I described in the first post. Like 20 x 18 wouldn't work. However, this formula directs us to do it as so.
(10 + 10)(10 + 8) = 10(10 + 10 + 8) + (10)(8)
This will give us the 360 as promised. If you can do 2 x 1 multiplication problems in your head, you might want to try things like 32 x 37 with 30 as your z, or 68 x 66 with 70 as your z. If you get really good at that, you could even try doing 448 x 442 with 400 as your z, which makes you add 48 x 42, using 40 or 50 as your z. This would be difficult, but you would get 198016 as an answer.
Problem of the Week Solutions (from July):
b = 16
z = 1
n = 42
p = 116
odds = 58%
s = 53.1 degrees
t = 36.9 degrees
a = 1
b = 8
c = 0
h = -4
k = -16
area = 50.3 sq. cm