Question

What is `21\frac { 2} { 3} - 13\frac { 1} { 3}`?

Answer 1

`8 1/3`

Explanation:

We can work with the whole numbers and the fractions separately.

In this case, this is the best option, because the fractions already have the same denominator, so this calculation is easier than others you will come across.

`21 2/3 - 13 1/3` means `21 +2/3 - 13- 1/3`

We can even rewrite it as `color(red)((21-13)) + color(blue)((2/3-1/3))`

So you can see why we can work with the whole numbers, then with the fractions..

`color(red)(21) 2/3 -color(red)(13) 1/3" "larr` subtract the whole numbers. `21-13=8`

=`color(red)(8) color(blue)(2/3 - 1/3)" "larr` now subtract the fractions `2/3-1/3 = 1/3`

=`8 1/3`

Answer 2

`21 2/3-13 1/3" "=" " 8 1/3`

Explanation:

`color(red)("When you get used to these you will miss out many steps")`

A fraction consists of `("count")/("size indicator of what you are counting")`
`color(white)(.)`

`" "("count")/("size indicator")->("numerator")/("denominator")`

You can not directly add or subtract counts unless the size indicators are the same.

By the way: `1"; "2"; "3"; "4.... -> 1/1 "; "2/1"; "3/1"; "4/1"...."`

So even whole numbers have a count part and a size indicator part. People tend not to write the size indicator part.

This is why you can directly add or subtract whole numbers!
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Converting so that we can do a direct subtraction")`

`color(brown)("This is solved the long way round to make sure everything is covered")` `color(brown)("and to demonstrate a useful method")`

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

Write as `" "[21 + 2/3]" "-" "[13+1/3]`

`" "color(green)([ (21color(red)(xx1)) + 2/3]" "-" "[(13color(red)(xx1))+1/3])`

`" "color(green)([ (21color(red)(xx3/3)) + 2/3]" "-" "[(13color(red)(xx3/3))+1/3])`

`" "color(green)([ 63/3 + 2/3]" "-" "[39/3+1/3])`

`" "65/3" "-" "40/3`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`" "65-40 = 25` so we have `25/3`

But `25` is the same as `24+1` so:

Write as: `24/3+1/3" "->" "(cancel(24)^8)/(cancel(3)^1)+1/3`

`" "8 1/3`