How do you simplify (r + 9)/4 + (r - 3)/2?

Jul 23, 2016

$\frac{3}{4} \left(r + 1\right)$

Explanation:

In a fraction the top number is the count of what have got. The bottom number is the size indicator of what you are counting.

So $\frac{3}{2}$ is stating that you have a count of three of something and that it takes 2 of what you are counting to make a whole of something.

color(brown)("You can not directly add or subtract the counts unless the size indicator is the same".

$\left(\text{count")/("size indicator") ->("numerator")/("denominator}\right)$

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Given:$\text{ } \frac{r + 9}{4} + \frac{r - 3}{2}$

To be able to directly add the 'counts' make the size indicator of $\frac{r - 3}{2}$ the same as that for $\frac{r + 9}{4}$

Consider: $\frac{r - 3}{2}$

Multiply by 1 but in the form of $1 = \frac{2}{2}$ giving:

$\frac{r + 9}{4} + \left[\frac{r - 3}{2} \times \frac{2}{2}\right]$

$= \frac{r + 9}{4} + \frac{2 r - 6}{4}$

Write as:

$\frac{r + 9 + 2 r - 6}{4}$

$= \frac{3 r + 3}{4}$
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Factor out the 3 giving:

$\frac{3 \left(r + 1\right)}{4} \text{ "=" } \frac{3}{4} \left(r + 1\right)$