# How do you solve x/2 + 2/3 = 5/6?

May 23, 2016

$x = \frac{1}{3}$

#### Explanation:

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

$\frac{x}{2} = \frac{5}{6} - \frac{2}{3}$

The L.C.M of the denominators of the fractions of the R.H.S $= 6$

$\frac{x}{2} = \frac{5}{6} - \frac{2 \cdot 2}{3 \cdot 2}$

$\frac{x}{2} = \frac{5}{6} - \frac{4}{6}$

$\frac{x}{2} = \frac{1}{6}$

$x = \left(\frac{1}{6}\right) \cdot 2$

$x = \frac{1}{3}$

May 23, 2016

Alternative approach

$x = \frac{1}{3}$

#### Explanation:

Notice that all the denominators are factors of 6

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To change the way a number looks without changing its value multiply by 1 but where 1 is in another form. For example

Changing $\frac{x}{2}$ into 6 "th"^("s") multiply by 1 but in the form of $1 = \frac{3}{3} \text{ } \to \frac{x}{2} \times \frac{3}{3} = \frac{x \times 3}{2 \times 3} = \frac{3 x}{6}$

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Given:$\text{ } \frac{x}{2} + \frac{2}{3} = \frac{5}{6}$

Write as:$\text{ } \left(\frac{x}{2} \times \frac{3}{3}\right) + \left(\frac{2}{3} \times \frac{2}{2}\right) = \frac{5}{6}$

$\implies \frac{3 x}{6} + \frac{4}{6} = \frac{5}{6}$

As everything is divided by 6 the equation is also true if we totally ignored the 6. Alternatively if you are not 'happy' about doing that this will get you to the same point:

Multiply everything by 6 giving:

$3 x + 4 = 5$

Subtract 4 from both sides:

$3 x = 1$

Divide both sides by 3

$x = \frac{1}{3}$