Question

What is `17 * .67`?

Answer 1

`17*0.67 = 11.39`

Explanation:

Notice that `6*17 = 102`, so one way of multiplying by `17` is to:

• Multiply by `100` (i.e. shift left two places or add two `0`'s).
• Add twice the number you started with.
• Divide by `6`.

So starting with `0.67`

• Multiply by `100` to get `67`
• Add twice `0.67` to get `68.34`
• Divide by `6` to get `11.39`

`color(white)()`
Another method

Note that `17 = 16+1 = 2^4+1`

So another method of multiplying by `17` is to double the given number `4` times, then add the original number...

• Double `0.67` to get `1.34`
• Double `1.34` to get `2.68`
• Double `2.68` to get `5.36`
• Double `5.36` to get `10.72`
• Add `0.67` to get `11.39`

`color(white)()`
Know your squares method

If you have memorised the squares of numbers up to `42^2`, then this method may be for you.

First let's get rid of the decimal point temporarily and multiply `17*67`. We can divide by `100` at the end to get our final result.

Note that:

`(a+b)^2 = a^2+2ab+b^2 = a^2-2ab+b^2+4ab = (a-b)^2+4ab`

Hence:

`ab = ((a+b)/2)^2-((a-b)/2)^2`

What this tells us is that we can multiply two numbers `a` and `b` by subtracting the square of half their difference from the square of their average. This works best if both numbers are odd or both numbers are even.

So:

`67*17 = ((67+17)/2)^2 - ((67-17)/2)^2`

`color(white)(67*17) = (84/2)^2-(50/2)^2`

`color(white)(67*17) = 42^2-25^2`

`color(white)(67*17) = 1764-625`

`color(white)(67*17) = 1139`

Hence:

`17*0.67 = (67*17)/100 = 1139/100 = 11.39`