# Two-thirds of a number is five-sixths. How do you find the number?

Jun 29, 2017

The number is $\frac{15}{12}$

#### Explanation:

Let the unknown value be $x$

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

Multiply both sides by $\textcolor{red}{\frac{3}{2}}$

color(green)(2/3color(red)(xx3/2)xx x=5/6color(red)(xx3/2)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider just the left hand side

Using the principle (by example) that $2 \times 6 = 6 \times 2$ we have:

$\frac{2 \times 3}{3 \times 2} \times x$

$\frac{3 \times 2}{3 \times 2} \times x$

$\frac{3}{3} \times \frac{2}{2} \times x$

$1 \times 1 \times x \text{ "->" } x$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Back to the question

So $x = \frac{5}{6} \times \frac{3}{2}$

$x = \frac{15}{12}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Check

left hand side $\to \text{ } \frac{{\cancel{2}}^{1}}{{\cancel{3}}^{1}} \times \frac{{\cancel{15}}^{5}}{{\cancel{12}}^{6}} = \frac{5}{6}$ as per the right hand side.

Jun 29, 2017

$\frac{5}{4}$

#### Explanation:

$\text{let x represent the number}$

$\text{then " 2/3" of } x = \frac{5}{6}$

$\text{replace of by multiplication}$

$\Rightarrow \frac{2}{3} \times x = \frac{5}{6}$

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

$\textcolor{b l u e}{\text{cross multiplying}}$

$\Rightarrow 2 x \times 6 = 5 \times 3$

$\Rightarrow 12 x = 15$

$\text{divide both sides by 12}$

$\frac{\cancel{12} x}{\cancel{12}} = \frac{15}{12}$

$\Rightarrow x = \frac{15}{12} = \frac{5}{4} \leftarrow \text{ in simplest form}$