Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 7 down vote accepted

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 $6$.

share|cite|improve this answer
just curious, How should we go about calculating x, incase it was asked? – MonK Jul 28 '14 at 13:19
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 $x$. – David K Jul 28 '14 at 13:26

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 $y=6$

share|cite|improve this answer

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 $3x+12=3x+2y$.

share|cite|improve this answer

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 $\dfrac{7+4}{6}=\dfrac{7}{6}+\dfrac{2}{3}$.

share|cite|improve this answer







That, gives us $d=6$

share|cite|improve this answer


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

share|cite|improve this answer

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 $\ d^{-1}(x+x') = d^{-1}x + d^{-1}x'$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.