Find `1/6+2/3`?

Least Common Denominator of the two denominators `6` and `3` is `6`

Hence `1/6+2/3`

= `1/6+(2xx2)/(2xx3)`

= `1/6+4/6`

= `5/6`

`color(blue)("Important fact demonstrated by example")`

`color(green)("A fraction consists of:"`

`color(green)(("count")/("size indicator of what you are counting")`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(brown)("Consider the example of "2+3=5)`

This may sound obvious but you are adding the counts of 2 and 3.
What is not so obvious is that they are both of the same unit size.

`color(green)("You can not directly add or subtract values unless the unit size is the same")`

Think again about `2+3=5` The unit size is how many of what you are counting it takes to make a whole of something. In this case it takes 1. So if we chose we could write `2+3=5` as `2/1+3/1=5/1` which is not normally done.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Consider the example of "2/4+3/4=5/4)`

The unit size is such that it takes 4 of what you are counting to make a whole. They are all the same unit size so you can directly add the counts of `2+3`.

`color(green)("Adding the counts does not change the unit size.")`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Answering the question")`

`1/6" and "2/3` are not of the same unit size. So we `color(red)(ul("can not say"))` the count is 1+2=3

So we need to make both unit sizes the same

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

multiply `2/3` by 1 but in the form of `1=2/2` giving:

`1/6+(2/3xx1)" "->" "1/6+(2/3xx2/2)`

`1/6+4/6`

We can now directly add the counts: `1+4 = 5`

So we have: `(1+4)/6= 5/6`

If you cannot spot the lowest common multiple, one method that works every time is to cross multiply

`1/6 + 2/3`

Multiply the first fraction (top and bottom) by 3

Multiply the second fraction (top and bottom) by 6

`[3xx1]/[3xx6] +[6xx2]/[6xx3]`

`3/18 +12/18`

we now have the fractions over the same denominator so simply add the numerators

`15/18`

cancel down by dividing by 3

`5/6`

Sometimes it is longer to do it this way but it works every time and you do not waste time looking for the LCM.