Question

Divide the sum of $\dfrac{65}{12}$ and $\dfrac{12}{7}$ by their difference.
A. $\dfrac{599}{311}$
B. $\dfrac{680}{216}$
C. $\dfrac{642}{133}$
D. $\dfrac{501}{301}$

Answer 1

Hint:
We need to have the same denominators in order to add or subtract the fractions.
We can choose a common multiple of 12 and 7 to become a common denominator.
In order to divide a number by a fraction, simply multiply the number by the fraction's reciprocal.
Since, the sum and the difference will have the same denominators, on dividing, our answer will be the numerator of the sum upon the numerator of the difference.


Complete step-by-step answer:
Let us convert both the fractions $\dfrac{65}{12}$ and $\dfrac{12}{7}$ , into fractions with the same denominators.
The least common multiple of 12 and 7 is 12 × 7 = 84, therefore the fractions are:
 $\dfrac{65}{12}=\dfrac{65\times 7}{12\times 7}=\dfrac{455}{84}$
And, $\dfrac{12}{7}=\dfrac{12\times 12}{7\times 12}=\dfrac{144}{84}$ .
∴ The sum of the two fractions = $\dfrac{455}{84}+\dfrac{144}{84}=\dfrac{599}{84}$ .
And the difference of the two fractions = $\dfrac{455}{84}-\dfrac{144}{84}=\dfrac{311}{84}$ .
When we divide the sum by the difference, we can write: $\left( \dfrac{599}{84} \right)\div \left( \dfrac{311}{84} \right)$ .
In order to divide by a fraction, simply multiply by the fraction's reciprocal.
∴ $\left( \dfrac{599}{84} \right)\div \left( \dfrac{311}{84} \right)=\left( \dfrac{599}{{84}} \right)\times \left( \dfrac{{84}}{311} \right)=\dfrac{599}{311}$ .
The answer is A. $\dfrac{599}{311}$ .

Note: For any two fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$ , we have:
$\dfrac{\left( \dfrac{a}{b}+\dfrac{c}{d} \right)}{\left( \dfrac{a}{b}-\dfrac{c}{d} \right)}=\dfrac{\left( \dfrac{ad+bc}{bd} \right)}{\left( \dfrac{ad-bc}{bd} \right)}=\left( \dfrac{ad+bc}{{bd}} \right)\times \left( \dfrac{{bd}}{ad-bc} \right)=\dfrac{ad+bc}{ad-bc}$ .
∴ $\dfrac{\left( \dfrac{65}{12}+\dfrac{12}{7} \right)}{\left( \dfrac{65}{12}-\dfrac{12}{7} \right)}=\dfrac{65\times 7+12\times 12}{65\times 7-12\times 12}=\dfrac{455+144}{455-144}=\dfrac{599}{311}$ .
It is not necessary to change the denominator to LCM only. We can choose any other common number as well which is quick to observe and easier for calculation. However, practice has shown that the LCM is the best choice.