# Find 1/6+2/3?

Sep 7, 2016

$\frac{1}{6} + \frac{2}{3} = \frac{5}{6}$

#### Explanation:

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

Hence $\frac{1}{6} + \frac{2}{3}$

= $\frac{1}{6} + \frac{2 \times 2}{2 \times 3}$

= $\frac{1}{6} + \frac{4}{6}$

= $\frac{5}{6}$

Sep 8, 2016

Very detailed explanation

$\frac{5}{6}$

#### Explanation:

$\textcolor{b l u e}{\text{Important fact demonstrated by example}}$

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

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

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

$\textcolor{b r o w n}{\text{Consider the example of } 2 + 3 = 5}$

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.

$\textcolor{g r e e n}{\text{You can not directly add or subtract values unless the unit size is the same}}$

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 $\frac{2}{1} + \frac{3}{1} = \frac{5}{1}$ which is not normally done.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Consider the example of } \frac{2}{4} + \frac{3}{4} = \frac{5}{4}}$

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 $2 + 3$.

$\textcolor{g r e e n}{\text{Adding the counts does not change the unit size.}}$

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

$\textcolor{b l u e}{\text{Answering the question}}$

$\frac{1}{6} \text{ and } \frac{2}{3}$ are not of the same unit size. So we $\textcolor{red}{\underline{\text{can not say}}}$ 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 $\frac{2}{3}$ by 1 but in the form of $1 = \frac{2}{2}$ giving:

$\frac{1}{6} + \left(\frac{2}{3} \times 1\right) \text{ "->" } \frac{1}{6} + \left(\frac{2}{3} \times \frac{2}{2}\right)$

$\frac{1}{6} + \frac{4}{6}$

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

So we have: $\frac{1 + 4}{6} = \frac{5}{6}$

Jun 20, 2018

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

$\frac{1}{6} + \frac{2}{3}$

Multiply the first fraction (top and bottom) by 3

Multiply the second fraction (top and bottom) by 6

$\frac{3 \times 1}{3 \times 6} + \frac{6 \times 2}{6 \times 3}$

$\frac{3}{18} + \frac{12}{18}$

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

$\frac{15}{18}$

cancel down by dividing by 3

$\frac{5}{6}$

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