The product of two fractions is $\frac{1}{9}$. The larger fraction divided by the smaller fraction is $4$. What is the sum of the two fractions?

Question

Question : The product of two fractions is $\frac{1}{9}$. The larger fraction divided by the smaller fraction is $4$. What is the sum of the two fractions?

My Try : $\frac{a}{b}\cdot\frac{c}{d}=\frac{1}{9}$

I will assume that $\frac{a}{b}>\frac{c}{d}$

I'm trying to find $\frac{a}{b}+\frac{c}{d}=?$

I keep getting stuck. Any help would be great!

Answer 1

There's a thing called computational algebra? Isn't all algebra computational?

And as for this problem,

$$ \frac{a}{b} \cdot \frac{c}{d} = 1/9.$$ $$ \frac{ad}{bc} = 4.$$

$ad=4bc$; $a=\dfrac{4bc}{d}$

$\dfrac{4c}{d} \cdot \dfrac{c}{d} = 1/9$

$\dfrac{c}{d} = \sqrt{1/36} = 1/6$

$bc=\dfrac{ad}{4}; c=\dfrac{ad}{4b}$

$\dfrac{a}{b} \cdot \dfrac{a}{4b} = 1/9; \dfrac{a}{b}=\sqrt{4/9}=2/3$

$\text{Sum} = 2/3 + 1/6 = 5/6$.

$%I'm new to MathJax so I don't know how to put that in a nice mathematical format.$

Answer 2

$ab = \frac{1}{9}, a/b=4, a>b$.

We can obtain $a = \frac{2}{3}, b = \frac{1}{6}$ or $a = -\frac{2}{3}, b = -\frac{1}{6}$.

$a>b$, therefore, $a+ b = \frac{5}{6}$.

Answer 3

Just call them $x,y$. Then $xy=\frac 19,x/y=4$

Answer 4

Why muddy the water by stipulating that the quantities be fractions?
Given two quantities, $A$ and $C$, where $AC=\frac19$ and $\frac{A}{C}=4$. So, $A=4C$. So, $AC=4C^2=\frac19$. So, $C^2=\frac1{36}$, and $C=\frac16$. So, $AC=\frac{A}6=\frac19$. So, $A=\frac69=\frac23$. Summing up, $A=\frac23$ and $C=\frac16$, if the OP is satisfied that the magnitude of $A$ be greater than the magnitude of $C$; otherwise, if the OP insists on a number-line interpretation of “greater than,” then $C=-\frac23$ and $A=-\frac16$.
..........BINGO!