Question

What should be added to twice the rational number ${\displaystyle \frac{-7}{3}}$ to get ${\displaystyle \frac{3}{7}}$?

Answer 1

Hint: Assume that the number that should be added is x. Now, first find the value of twice the rational number ${\displaystyle \frac{-7}{3}}$ by multiplying the numerator with 2. Now, take the sum of the obtained rational number with x and equate it with ${\displaystyle \frac{3}{7}}$ to form a linear equation in x. Solve this equation for the value of x to get the answer.

Complete step by step solution:
Here we have been asked to find the number that must be added to twice the rational number ${\displaystyle \frac{-7}{3}}$ to get ${\displaystyle \frac{3}{7}}$ as the resultant fraction.
Now, let us assume that the number that should be added is x. The meaning of twice of ${\displaystyle \frac{-7}{3}}$ means we need to multiply the numerator of this rational number with 2, so we get,
$\Rightarrow 2\times \left({\displaystyle \frac{-7}{3}}\right)={\displaystyle \frac{-14}{3}}$
So we need to add ${\displaystyle \frac{-14}{3}}$ with x to get ${\displaystyle \frac{3}{7}}$, therefore the expression will be given as:
$\Rightarrow x+\left({\displaystyle \frac{-14}{3}}\right)={\displaystyle \frac{3}{7}}$
Clearly this is a linear equation in x so we need to solve for the value of x. Leaving x in the L.H.S and taking all other terms to the R.H.S we get,
$\Rightarrow x={\displaystyle \frac{3}{7}}-\left({\displaystyle \frac{-14}{3}}\right)$
We know that $(-1)\times (-1)=1$ so we get,
$\Rightarrow x={\displaystyle \frac{3}{7}}+{\displaystyle \frac{14}{3}}$
Taking the L.C.M of 7 and 3 which is 21 we get,
$\begin{array}{rl}& \Rightarrow x={\displaystyle \frac{(3\times 3)+(14\times 7)}{21}}\\ & \Rightarrow x={\displaystyle \frac{9+98}{21}}\\ & \therefore x={\displaystyle \frac{107}{21}}\end{array}$
Hence the required number that must be added is ${\displaystyle \frac{107}{21}}$.

Note: You must remember how to add and subtract two rational numbers. To find the value of twice the rational number always multiply the numerator with 2 and not the denominator. Note that to add fractions we take the L.C.M of their denominators and not numerators. Here both 3 and 7 were prime numbers and we know that the L.C.M of two prime numbers is simply their product.