Unitary Method — Assertion Reason Type Questions

Assertion (A): If 56 men can do a piece of work in 42 days then 48 men will take 48 days to complete the work.

Reason (R): Less men at work means more days are taken to complete the same work. Conversely, more number of men will take less number of days to complete the same work.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Given:

56 men can do the work in 42 days.

Let 48 men complete the same work in x days.

Since less men take more days to complete the same work, the number of men and the number of days are in inverse variation.

⇒ 56 × 42 = 48 × x

$\Rightarrow x = \dfrac{56 \times 42}{48}\\[1em] = \dfrac{2352}{48}\\[1em] = 49 \text{ days }$

Since 48 men will take 49 days and not 48 days,

Thus, Assertion (A) is false.

We know that,

Less men at work means more days are taken to complete the same work, and more men at work means less days are taken to complete the same work.

Thus, Reason (R) is true.

A is false and R is true.

Hence, option 2 is the correct option.

Assertion (A): 6 pipes can fill the tank in 64 minutes. Then number of pipes required to fill the tank in 96 minutes is 4.

Reason (R): If two quantities are related in such a way that the increase in one quantity causes a corresponding decrease in the other quantity and vice versa, then such a variation is called an inverse variation.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Given:

6 pipes can fill the tank in 64 minutes.

Let x pipes fill the same tank in 96 minutes.

Since more pipes take less time to fill the same tank, the number of pipes and the time taken are in inverse variation.

⇒ 6 × 64 = x × 96

$\Rightarrow x = \dfrac{6 \times 64}{96}\\[1em] = \dfrac{384}{96}\\[1em] = 4 \text{ pipes }$

Since 4 pipes are required to fill the tank in 96 minutes,

Thus, Assertion (A) is true.

We know that,

When an increase in one quantity causes a corresponding decrease in the other quantity, and vice versa, such a variation is called an inverse variation.

Thus, Reason (R) is true.

Both A and R are true.

Hence, option 3 is the correct option.

Assertion (A): A and B together can do a piece of work in 20 days, while B alone can finish it in 60 days, then A's 1 day's work is $\dfrac{1}{40}$.

Reason (R): The number of days required to complete a work and one day's work are reciprocal to each other.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Given:

A and B together can do the work in 20 days.

⇒ (A + B)'s 1 day's work = $\dfrac{1}{20}$

B alone can finish the work in 60 days.

⇒ B's 1 day's work = $\dfrac{1}{60}$

A's 1 day's work = (A + B)'s 1 day's work − B's 1 day's work

$= \dfrac{1}{20} - \dfrac{1}{60}\\[1em] = \dfrac{3 - 1}{60}\\[1em] = \dfrac{2}{60}\\[1em] = \dfrac{1}{30}$

Since A's 1 day's work is $\dfrac{1}{30}$ and not $\dfrac{1}{40}$,

Thus, Assertion (A) is false.

We know that,

If a person completes a work in n days, then his 1 day's work = $\dfrac{1}{n}$, i.e. the number of days required to complete a work and the one day's work are reciprocal to each other.

Thus, Reason (R) is true.

A is false and R is true.

Hence, option 2 is the correct option.

Assertion (A): X can do a piece of work in 3 days, while Y can do it in 6 days. They will complete the work in 2 days if they work together.

Reason (R): Time required to do a certain work =

$\dfrac{\text{Work to be done}}{\text{Work done in unit time}}$.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Given:

X can do the work in 3 days.

⇒ X's 1 day's work = $\dfrac{1}{3}$

Y can do the work in 6 days.

⇒ Y's 1 day's work = $\dfrac{1}{6}$

(X + Y)'s 1 day's work

$= \dfrac{1}{3} + \dfrac{1}{6}\\[1em] = \dfrac{2 + 1}{6}\\[1em] = \dfrac{3}{6}\\[1em] = \dfrac{1}{2}$

Time taken by X and Y together to complete the work

$= \dfrac{\text{Work to be done}}{\text{Work done in 1 day}}\\[1em] = \dfrac{1}{\dfrac{1}{2}}\\[1em] = 2 \text{ days }$

Since X and Y together complete the work in 2 days,

Thus, Assertion (A) is true.

We know that,

Time required to do a certain work = $\dfrac{\text{Work to be done}}{\text{Work done in unit time}}$.

Thus, Reason (R) is true.

Both A and R are true.

Hence, option 3 is the correct option.