# What number should be deducted by the numerator and the denominator of the fraction 7/13 to obtain the fraction 1/3?

Apr 11, 2018

8/39

#### Explanation:

Suppose the value that will be deducted from $\frac{7}{13}$ is $x$ to form $\frac{1}{3}$

So,

$\frac{7}{13} - x = \frac{1}{3}$

Solve the equation

$x = \frac{7}{13} - \frac{1}{3}$

$x = \frac{\left(7 \times 3\right) - \left(13 \times 1\right)}{39}$

$x = \frac{21 - 13}{39}$

$x = \frac{8}{39}$

Apr 11, 2018

Just to show that you get the same answer if you approach it a very slightly different way

Deduct $\frac{8}{39}$

#### Explanation:

Let the unknown value be represented by $x$

Complying with the wording of the question gives:

$\textcolor{g r e e n}{\frac{7}{13} \textcolor{red}{- x} = \frac{1}{3}} \text{ } \ldots \ldots . E q u a t i o n \left(1\right)$

But what happens if we change the sign from subtract to add ?

$\textcolor{g r e e n}{\frac{7}{13} \textcolor{red}{+ x} = \frac{1}{3}} \text{ } \ldots . . E q u a t i o n \left(2\right)$

Subtract $\frac{7}{13}$ from both sides

$\textcolor{g r e e n}{\textcolor{red}{+ x} = \frac{1}{3} - \frac{7}{13}}$

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

color(green)(x=color(white)(.) [1/3color(red)(xx1)]color(white)(".")-color(white)(".")[7/13color(red)(xx1)]

color(green)(x= [1/3color(red)(xx13/13)]-[7/13color(red)(xx3/3)]

$\textcolor{g r e e n}{x = \textcolor{w h i t e}{\text{ddd")13/39color(white)("ddd")-color(white)("ddd}} \frac{21}{39}}$

color(green)(x=-8/39
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Substitute into $E q u a t i o n \left(2\right)$

$\textcolor{g r e e n}{\frac{7}{13} \textcolor{red}{+ \left(x\right)} = \frac{1}{3}}$

$\textcolor{g r e e n}{\frac{7}{13} \textcolor{red}{+ \left(- \frac{8}{39}\right)} = \frac{1}{3}}$

Two signs that are not the same give a minus. So $+ \left(- \frac{8}{39}\right)$ becomes just $- \frac{8}{39}$

$\textcolor{g r e e n}{\frac{7}{13} \textcolor{red}{- \frac{8}{39}} = \frac{1}{3}} \leftarrow \text{ Format as required by the question}$

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So it works out correctly whichever way you chose as long as you 'fully' follow the rules of mathematics.