Question

By what number should ${\displaystyle \frac{2}{3}}$ be divided to get ${\displaystyle \frac{-4}{5}}$?

Answer 1

Hint: Assume the required number by which ${\displaystyle \frac{2}{3}}$ must be divided as x. Now, to divide ${\displaystyle \frac{2}{3}}$ by x, take the reciprocal of x and multiply it with ${\displaystyle \frac{2}{3}}$. Equate the obtained product with ${\displaystyle \frac{-4}{5}}$ and form a linear equation in x. Cross multiply the terms and solve for the value of x by cancelling the common factors to get the answer.

Complete step by step solution:
Here we have been asked to find the number by which ${\displaystyle \frac{2}{3}}$ will be divided so that the obtained quotient will be equal to ${\displaystyle \frac{-4}{5}}$. Let us assume the required number as x, so we have the relation,
$\Rightarrow {\displaystyle \frac{2}{3}}÷x={\displaystyle \frac{-4}{5}}$
Now, when we have to divide a number ‘a’ with another number ‘b’ then what we do is, we first take the reciprocal of the number ‘b’ and then multiply it with the number ‘a’. To take the reciprocal of a number we divide 1 by that number. For example: - reciprocal of b will be $${\displaystyle \frac{1}{b}}$$.
Let us come to the question. Taking the reciprocal of x we get $${\displaystyle \frac{1}{x}}$$. Now, we have to multiply ${\displaystyle \frac{2}{3}}$ with $${\displaystyle \frac{1}{x}}$$, so we get,
$\begin{array}{rl}& \Rightarrow {\displaystyle \frac{2}{3}}\times {\displaystyle \frac{1}{x}}={\displaystyle \frac{-4}{5}}\\ & \Rightarrow {\displaystyle \frac{2}{3x}}={\displaystyle \frac{-4}{5}}\end{array}$
By cross multiplication we get,
$\begin{array}{rl}& \Rightarrow 10=-12x\\ & \Rightarrow 12x=-10\end{array}$
Solving the above linear equation for the value of x by dividing both the sides by 12 and cancelling the common factors we get,
$\begin{array}{rl}& \Rightarrow x=-{\displaystyle \frac{10}{12}}\\ & \therefore x=-{\displaystyle \frac{5}{6}}\end{array}$
Hence, the above value of x is our answer.

Note: Note that x can never be 0 because the reciprocal of 0 is undefined. Remember the definitions of certain terms like reciprocal of a number and the process to calculate it. Division is the inverse process of multiplication and that is why when we have to divide a number by another number then we multiply the reciprocal of the divisor with the dividend.