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

Feb 12, 2017

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 $\frac{a}{a}$.

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

$\frac{3}{4} + \frac{5}{9}$ becomes:

$\left(\frac{9}{9} \times \frac{3}{4}\right) + \left(\frac{4}{4} \times \frac{5}{9}\right)$

$\frac{27}{36} + \frac{20}{36}$

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

$\frac{27 + 20}{36}$

$\frac{47}{36}$

Hope this helps.

Feb 12, 2017

#### Explanation:

A fraction's structure is $\left(\text{count")/("size indicator")->("numerator")/("denominator}\right)$

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 $\frac{1}{2}$ to make 1
It takes 3 of $\frac{1}{3}$ to make 1
it takes 45 of $\frac{1}{45}$ 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 $\frac{8}{1}$ as it takes 1 of what you are counting to make 1

$\textcolor{b r o w n}{\text{You can not "ul("directly"color(white)(.))"add or subtract the 'counts' in a fraction}}$
$\textcolor{b r o w n}{\text{unless the size indicators are the same.}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Demonstration using your example}}$

$\frac{3}{4} + \frac{5}{9}$

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.

$\textcolor{g r e e n}{\left[\frac{3}{4} \textcolor{red}{\times 1}\right] + \left[\frac{5}{9} \textcolor{red}{\times 1}\right] \text{ "=" } \left[\frac{3}{4} \textcolor{red}{\times \frac{9}{9}}\right] + \left[\frac{5}{9} \textcolor{red}{\times \frac{4}{4}}\right]}$

$\text{ } = \textcolor{g r e e n}{\left[\frac{27}{36}\right] \textcolor{w h i t e}{.} + \textcolor{w h i t e}{. .} \left[\frac{20}{36}\right]}$

Now that the size indicators (denominators) are the same you may directly add the counts giving:
$\text{ } \textcolor{g r e e n}{= \frac{27 + 20}{36}}$

$\text{ } \textcolor{g r e e n}{= \frac{47}{36}}$

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

$\text{ } \textcolor{g r e e n}{= \frac{36}{36} + \frac{11}{36}}$

But $\frac{36}{36} = 1$ giving:

$\text{ } \textcolor{g r e e n}{= 1 \textcolor{w h i t e}{.} \frac{11}{36}}$