# Ex.13.2 Q8 Direct and Inverse Proportions Solution - NCERT Maths Class 8

## Question

A factory requires \(42\) machines to produce a given number of articles in \(63\) days. How many machines would be required to produce the same number of articles in \(54\) days?

## Text Solution

**Reasoning:**

Two numbers \(x\) and \(y\) are said to vary in inverse proportion if

\[\begin{align}xy = {\rm{ }}k,{\rm{ }}x{\rm{ }} = {\rm{ }}\frac{1}{y}k\end{align}\]

Where \(k\) is a constant.

\[{x_1}{y_1} = {x_2}{y_2}\]

**What is Known:**

\(42\) machines to produce a given number of articles in \(63\) days.

**What is Unknown:**

Machines required for producing same no. of articles in \(54\) days.

**Steps:**

If the number of days decreases the machine required will increase. So, it is an inverse proportion.

\[\begin{align}{{\rm{x}}_1}{{\rm{y}}_1} &= {{\rm{x}}_2}{{\rm{y}}_2}\\42 \times 63 &= 54 \times {{\rm{y}}_2}\\{y_2} &= \frac{{42 \times 63}}{{54}}\\{{\rm{y}}_2} &= 49\end{align}\]

\(49\) machines will be required to produce the same number of articles in \(54\) days.