Question

By doubling each dimension, the area of a parallelogram increased from 36 square centimeters to 144 square centimeters. How do you find the percent increase in area?

Answer 1

Once you start to recognise the connections between numbers you will be able to do these much quicker than I have shown.

Explanation:

`color(blue)("Using shortcuts")`

`(144-36)/36xx100= 300%`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Using first principles with full explanation")`

We are only interested in the change in area. As both these values are given then anything else is a 'red herring'.

Assuming the increase is to be measured against the original area

`("change in area")/("original area")->(144-36)/36` as a fraction, giving:

`108/36`increase as a fraction

Notice that for 108 that `1+0+8=9` which is divisible by 3 so 108 is also divisible by 3

Notice that for 36 that `3+6=9` which is also divisible by 3

So to simplify we have:

`(108-:3)/(36-:3) = 36/12`

`(36-:3)/(12-:3)=(12-:4)/(4-:4)=3/1=3`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let the unknown value be `x`

Then we are looking to end up with `x/100`. So by ratio

`3/1-=x/100`

To change 1 into 100 multiply by 100. What you do to the top you do to the bottom.

`(3xx100)/(1xx100) = 300/100=x/100`

So we have the percentage `300/100` which may be written as `300%`