Question

If x and y vary inversely as each other and x = 3 when y = 8, find y when x = 4.

Answer 1

Hint: We know that two quantities that are inversely proportional, we can write it as $x\propto \dfrac{1}{y}$. We can then replace the proportionality sign with equal to sign by inserting a proportionality constant k. Thus, we have $xy=k$. Now, we can find the value of k by putting the values $x=3$ and $y=8$. Hence, we can find the value of $y$ when $x=4$.

Complete step by step answer:
We know that when two quantities a and b vary inversely as each other or are said to be inversely related, when $a\propto \dfrac{1}{b}$.
Hence, for this question, we can write $x\propto \dfrac{1}{y}$.
We know that we can replace the proportional sign with an equal to by bringing in a proportional constant into the equation.
So, if k is the proportionality constant, then we can also write
$x=k\cdot \dfrac{1}{y}$
Let us multiply both sides of the above equation by y. Thus, we get
$xy=k...\left( i \right)$
We are given in the question that when $y=8$ then $x=3$. So, using these two values in equation (i), we can write
$3\times 8=k$
And so, the value of our proportional constant is
$k=24$
So, equation (i) now becomes,
$xy=24$
We need to find the value of $y$ when $x=4$. So, putting the value of $x$ in the above equation, we get
$4y=24$
We can now divide both sides of the above equation by 4. Thus, we get
$\dfrac{4y}{4}=\dfrac{24}{4}$
On further simplifying, we have
$y=6$
Hence, we can say that, when $x=4$, then $y=6$.

Note: We know that when two quantities are directly proportional, then we write it as $a\propto b$, and when the two quantities are inversely proportional, we write it as $a\propto \dfrac{1}{b}$. We must also remember to insert the proportionality constant, when proportional sign is replaced by equal to sign.