# Ex.13.2 Q10 Direct and Inverse Proportions Solution - NCERT Maths Class 8

## Question

Two persons could fit new windows in a house in \(3\) days.

(i) One of the persons fell ill before the work started. How long would the job take now?

(ii) How many persons would be needed to fit the windows in one day?

## Text Solution

(i) One of the persons fell ill before the work started. How long would the job take now?

**What is Known?**

\(2\) Persons can fit the windows in \(3\) days.

**What is Unknown?**

One man \(\left( {2 - 1} \right)\) can fit the window in how many days?

**Reasoning:**

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

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

Where* \(k\)* is a constant.

\[\begin{align}{x_1}{y_1} = {x_2}{y_2}\end{align}\]

**Steps:**

If the no. of men working decreases, the number of days will increase.

\[\begin{align}{x_1}{y_1}& = {x_2}{y_2}\\2 \times 3& = 1 \times {y_2}\\{y_2} &= \frac{{2 \times 3}}{1}\\{y_2} &= 6\end{align}\]

Hence the job will be completed in \(6\) days.

(ii) How many persons would be needed to fit the windows in one day?

**What is Known?**

\(2\) Persons can fit the windows in \(3\) days.

**What is Unknown?**

The persons need to fix the window in \(1\) day.

**Steps:**

If the number of days decreases, then the persons needed will be increase. Hence it is inverse proportion.

\[\begin{align}{x_1}{y_1} &= {x_2}{y_2}\\3 \times 2 &= 1 \times {y_2}\\{y_2} &= \frac{{3 \times 2}}{1}\\{y_2} &= 6\end{align}\]

\(6\) persons will need to fix the window in one day.