Question

Divide the sum of ${\displaystyle \frac{-13}{5}}$ and ${\displaystyle \frac{12}{7}}$ by the product of ${\displaystyle \frac{-31}{7}}$ and ${\displaystyle \frac{-1}{2}}.$

Answer 1

Hint: We solve this question by using the basic concepts of fractions such as adding, multiplying, dividing and reciprocating of fractions. We first consider the sum of the two fractions ${\displaystyle \frac{-13}{5}}$ and ${\displaystyle \frac{12}{7}}$ . We then obtain the result of the product of the two terms ${\displaystyle \frac{-31}{7}}$ and ${\displaystyle \frac{-1}{2}}.$ We divide the result of the first set of fractions by the result of the second set of fractions.

Complete step-by-step solution:
In order to answer this question, let us consider the first set of fractions and obtain their sum. This can be represented as,
$\Rightarrow {\displaystyle \frac{-13}{5}}+{\displaystyle \frac{12}{7}}$
To add the above two fractions, we need to consider the denominators of the two fractions and find their LCM. We are required to find the LCM of least common multiple of 5 and 7 which is given as
$\Rightarrow LCM(5,7)=5\times 7=35$
Since the above two numbers are prime numbers, their LCM is simply given by their product.
We are required to convert the given denominators in the above expression to 35 and in order to do so, we multiply the numerator and denominator of the first term by 7 and multiply the numerator and denominator of the second term by 5. This can be given as,
$\Rightarrow {\displaystyle \frac{-13\times 7}{5\times 7}}+{\displaystyle \frac{12\times 5}{7\times 5}}$
Taking a product of the respective terms we simplify it as,
$\Rightarrow {\displaystyle \frac{-91}{35}}+{\displaystyle \frac{60}{35}}$
Since the denominator is the same now, we take the denominator common and add the two numerators.
$\Rightarrow {\displaystyle \frac{-91+60}{35}}$
Adding the two numerators, which is nothing but the subtraction of the two since they are of opposite signs, we get an improper fraction as follows,
$\Rightarrow {\displaystyle \frac{-31}{35}}$
Next, we shall find the product of the two terms ${\displaystyle \frac{-31}{7}}$ and ${\displaystyle \frac{-1}{2}}.$ This is given as,
$\Rightarrow {\displaystyle \frac{-31}{7}}\times {\displaystyle \frac{-1}{2}}$
Since there are no common factors to cancel out, we just take the respective product of the numerators and denominators respectively.
$\Rightarrow {\displaystyle \frac{31}{14}}$
The above result is obtained by multiplying the two negative numbers in the numerator which is gives us a positive number and we multiply 7 and 2 to get the denominator.
Now, we need to divide the first result ${\displaystyle \frac{-31}{35}}$ by the second result ${\displaystyle \frac{31}{14}}.$ This can be shown as,
$\Rightarrow {\displaystyle \frac{{\displaystyle \frac{-31}{35}}}{{\displaystyle \frac{31}{14}}}}$
We know that the division of two fractions can be converted to the product of the two fractions by considering the reciprocal of the denominator. This is represented as,
$\Rightarrow {\displaystyle \frac{-31}{35}}\times {\displaystyle \frac{14}{31}}$
We can see the same terms in the numerator and denominator which is 31. So, we cancel them. Next, we see a common factor of 7 between the numerator term 14 and denominator term 35 and cancel it out. This gives us,
$\Rightarrow -{\displaystyle \frac{2}{5}}$
Hence, the result obtained is $-{\displaystyle \frac{2}{5}}$ or -0.4.

Note: We need to know the basic calculation techniques for simplifying and manipulating fractions. We need to be careful while converting the division of two fractions to their multiplication because we should always remember to reciprocate the denominator when we do so. Or else, it could lead to a wrong result.