Question

How do you find “k” if y varies inversely as x and if y=24 when x=3.

Answer 1

Hint: In this question, given that y varies inversely as x. To find the equation for this we will first see what type of relations we have given. For that, we have to learn about inverse and direct relation and using these we will find the relation suitable for the given problem. The relation shown by the sign $\propto$ is called proportionality. To form an equation we need to change the proportion $\propto$ sign to equal = sign. So, we will multiply by constant. We will use the inversely as the relation $A\propto {\displaystyle \frac{1}{B}}$. We can write this as $A={\displaystyle \frac{k}{B}}$. Now, simply substitute the values of x and y to find the value of k.

Complete step-by-step solution:
In this question, the inverse linear variation proportion is given.
We can write the inverse proportion as,
$\Rightarrow A\propto {\displaystyle \frac{1}{B}}$
Let us write the above equation in its equation form.
$\Rightarrow A={\displaystyle \frac{k}{B}}$
In this question, we have given two things like x and y.
Here, given that y varies inversely as x,
In the above equation, the value of A is y and the value of B is x.
Therefore,
$\Rightarrow y={\displaystyle \frac{k}{x}}$
Now, in question given that if y=24 when x=3.
Substitute the values in the above equation.
$\Rightarrow 24={\displaystyle \frac{k}{3}}$
Let us multiply both sides by 3.
$\Rightarrow 24\times 3={\displaystyle \frac{k}{3}}\times 3$
Therefore,
$\Rightarrow k=72$

The value of k is equal to 72.

Note: Direct relation: In direct relation, two things are connecting directly means they both will behave the same way if there is a decrement in one thing then the other will decrease too, and if one is increasing then the other will increase too. If we have two things as A and B, then direct relation is denoted by $A\propto B$. So, the equation becomes $A=kB$.
Inverse proportion: This type of relation is indirectly where A is moving opposite of B. If A increases, B will decrease, and if A decreases, B will increase. It is represented as $A\propto {\displaystyle \frac{1}{B}}$ , so the equation becomes $A={\displaystyle \frac{k}{B}}$.