Question

How do you add `3/4` and `5/9`?

Answer 1

See the entire explanation below:

Explanation:

To add fractions with different denominators we need to first convert each fraction to have a common denominator.

To do this we can't change the value of either fraction and therefore need to multiply it by some form of `1` or `a/a`.

In the example you provided, `3/4 + 5/9`, we need to get both fractions over a common denominator. In this case the common denominator can be `4 xx 9 = 36`. So, we need to multiply the first fraction by `1` or `9/9` and we need to multiply the second fraction by `1` also, but this time `4/4`.

`3/4 + 5/9` becomes:

`(9/9 xx 3/4) + (4/4 xx 5/9)`

`27/36 + 20/36`

Now that the two fractions are over a common denominator we can add the numerators.

`(27 + 20)/36`

`47/36`

Hope this helps.

Answer 2

Explanation given using your example

Explanation:

A fraction's structure is `("count")/("size indicator")->("numerator")/("denominator")`

The word 'count' speaks for itself.

The word 'size indicator' is a number indicating how many of what you are counting it takes to make a whole 1 of something.

It takes 2 of `1/2` to make 1
It takes 3 of `1/3` to make 1
it takes 45 of `1/45` to make 1 and so on.

By the way; not normally done but you may write whole numbers the same way. That is, for example, 8 may be written as `8/1` as it takes 1 of what you are counting to make 1

`color(brown)("You can not "ul("directly"color(white)(.))"add or subtract the 'counts' in a fraction")`
`color(brown)("unless the size indicators are the same.")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Demonstration using your example")`

`3/4+5/9`

It does not matter what you make the bottom numbers as long as they are the same.

Multiply by 1 and you do not change the value. However, 1 comes in many forms so you can change the way a number looks without changing its value.

`color(green)([3/4color(red)(xx1)] +[5/9color(red)(xx1)] " "=" "[3/4color(red)(xx9/9)] +[5/9color(red)(xx4/4)] )`

`" "=color(green)([27/36]color(white)(.)+color(white)(..)[20/36])`

Now that the size indicators (denominators) are the same you may directly add the counts giving:
`" "color(green)(=(27+20)/36)`

`" "color(green)(=47/36)`

But 47 can be written as 36 + 11 so we can split this as:

`" "color(green)(=36/36+11/36)`

But `36/36=1` giving:

`" "color(green)(=1color(white)(.) 11/36)`