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When the numerator of a fraction is increased by $4$, the fraction increases by $2/3$. What is the denominator of the fraction?

I tried,

Let the numerator of the fraction be $x$ and the denominator be $y$.

Accordingly, $$\frac{x+4}y=\frac xy+\frac 23$$

I am not able to find the second equation.

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Again, you've got a fine start:

You wrote: $$\frac{x+4}y=\color{red}{\frac xy}+\color{blue}{\frac 23}\tag{1}$$

But note that $$\frac{x+4}{y} = \color{red}{\frac xy} + \color{blue}{\frac 4y}\tag{2}$$

From $(1),(2),$ it must follow that $$\color{blue}{\frac 4y = \frac 23 } \iff 2y = 4\cdot 3 = 12 \iff y = \frac{12}{2} = 6$$

So the denominator, $y$ is $6$.

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  • $\begingroup$ just curious, How should we go about calculating x, incase it was asked? $\endgroup$ – MonK Jul 28 '14 at 13:19
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    $\begingroup$ The value of $x$ is not determined by the given information. Now that you know $y = 6$, the statement of the problem says that $\frac{x+4}6=\frac x6 + \frac23$, which is true for all $x$. $\endgroup$ – David K Jul 28 '14 at 13:26
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If you add 4 to the numerator, the value of your fraction will be increased by $\frac4y$, where y is your denominator.

So $\frac4y=\frac23$ and $y=6$

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In general you are correct: to solve for two unknowns, you would usually need two equations. But in this case you are lucky, and the one equation gives a solution for the one unknown you are asked to find.

$\dfrac{x+4}{y}=\dfrac{x}{y}+\dfrac{2}{3}$ so multiplying both sides by $3y$ gives $3x+12 = 3x+2y$ making the denominator $y=6$.

It gives no specific solution for $x$, for example: $\dfrac{5+4}{6}=\dfrac{5}{6}+\dfrac{2}{3}$ and $\dfrac{7+4}{6}=\dfrac{7}{6}+\dfrac{2}{3}$.

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Given,

$$\frac{n+4}{d}=\frac{n}{d}+\frac{2}{3}$$

So,

$$\frac{n}{d}+\frac{4}{d}=\frac{n}{d}+\frac{2}{3}$$

Or,

$$\frac{4}{d}=\frac{2}{3}$$

That, gives us $d=6$

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We have $\frac{x+4}y={\frac xy}+{\frac 23}=\frac{3x+2y}{3y}$ but: $$\frac{x+4}{y}=\frac{\color{red}3(x+4)}{\color{red}3y}=\frac{3x+12}{3y}$$ So $$\frac{3x+12}{3y}=\frac{3x+2y}{3y}$$ So if $y\neq 0$ then $3x+12=3x+2y$.

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$$x=\frac{a}{b}$$

$$x+\frac{2}{3}=\frac{a+4}{b} \Rightarrow \frac{a}{b}+\frac{2}{3}=\frac{a+4}{b} \Rightarrow a+\frac{2}{3}b=a+4 \Rightarrow b=\frac{12}{2}=6$$

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Hint $ $ linearity: $\ \ell(x) = x/d\,\Rightarrow\, \ell(x\!+\!x')=\ell(x)\color{#c00}{+\ell(x')},\,$ so $\,\ell(x)\,$ increases by $\,\color{#c00}{\ell(x')} = x'/d$

Remark $\ $ Here linearity follows from the distributive law $\ d^{-1}(x+x') = d^{-1}x + d^{-1}x'$

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