Question

How do you simplify `\frac { \frac { 8} { 15} } { \frac { 5} { 12} }`?

Answer 1

`32/25`

Explanation:

Keep the first fraction as it is, flip the second fraction and then multiply.

Essentially: `(a/b)/(c/d)= a/b*d/c`

`(8/15)/(5/12)=8/15*12/5=96/75`

We can simplify the fraction seeing that both numbers are divisible by `3` to which you'll get:

`32/25`

Answer 2

`32/25`

Explanation:

`color(blue)("Shortcut method - with explanation")`

Turn the divisor upside down and multiply giving:

`8/15xx12/5 " "=" "(8xx12)/(15xx5)" "=" "12/15xx8/5`

This ability to swap round in multiplication is called the property of being 'distributive'.

3 is a common factor of 12 and 15 so we can cancel out.

`12/15xx8/5" "->" "(12-:3)/(15-:3)xx8/5 " "->" "4/5xx8/5`

It would normally look like this:

`(cancel(12)^4)/(cancel(15)^5)xx8/5`
....................................................................
Or as presented originally we have:

`8/15xx12/5" "->" "8/(cancel(15)^5) xx(cancel(12)^4)/5`
..............................................................
5 is a prime number and does not divide exactly into 8 thus we are stuck with: `4/5xx8/5` giving:

`32/25` there are no common factors so it can not be simplified further.

Answer 3

`32/25`

Explanation:

We can do this by remembering that a fraction is way of showing division:

`3 div 5 = 3/5`

`color(red)(8/15)/(color(blue)(5/12))` is a way of showing that two fractions are being divided.

Write it in a simpler way first:

`color(red)(8/15)/(color(blue)(5/12))= color(red)(8/15) divcolor(blue)(5/12)`

This should now be a familiar calculation.

To divide by a fraction, multiply by the reciprocal:

`" "8/15 div 5/12`

`= 8/15 xx12/5`

`= 8/cancel15_5 xxcancel12^4/5" "larr`cancel by `3`

`= 32/25`

The short cut for this is as follows:

`(a/b)/(c/d) = (axxd)/(bxxc)" "` and then simplify.