Unitary Method — Multiple Choice Questions

In the computer lab of class 7, 8 computers are required for 10 students. The number of computers required for 40 students is:

1. 32

2. 16

3. 5

4. 50

Answer

This is a case of direct variation (more students, more computers).

Computers required for 1 student = $\dfrac{8}{10}$

Computers required for 40 students = $40 \times \dfrac{8}{10} = \dfrac{320}{10}$ = 32

Hence, option 1 is the correct option.

6 identical pens cost ₹ 90, the cost of 15 such pens is:

1. ₹ 225

2. ₹ 36

3. ₹ 180

4. ₹ 150

Answer

Cost of 6 pens = ₹ 90

Cost of 1 pen = $\dfrac{90}{6}$ = ₹ 15

Cost of 15 pens = 15 × 15 = ₹ 225

Hence, option 1 is the correct option.

6 men can do a piece of work in 90 days. In how many days will 15 men complete the same work?

1. 225

2. 36

3. 180

4. 150

Answer

This is a case of inverse variation (more men, fewer days).

1 man will complete the work in 6 × 90 days.

15 men will complete the work in $\dfrac{6 \times 90}{15} = \dfrac{540}{15}$ = 36 days.

Hence, option 2 is the correct option.

A certain distance is covered in 20 minutes at an average speed of 24 km per hour. What should be the average speed, if the same distance is to be covered in 15 minutes?

1. 32 m per second

2. 32 m per hour

3. 32 km per second

4. 32 km per hour

Answer

Speed = 24 km per hour

Time = 20 minutes = $\dfrac{20}{60}$ hours (60 minutes = 1 hour)

Distance = Speed × Time = $24 \times \dfrac{20}{60}$ = 8 km

Now, 15 minutes $= \dfrac{15}{60} \text { h }= \dfrac{1}{4}$ h:

Speed = $\dfrac{\text{Distance}}{\text{Time}} = \dfrac{8}{\dfrac{1}{4}} = 8 \times 4$ = 32 km per hour

Hence, option 4 is the correct option.

20 workers complete a certain piece of work in 28 days. The number of days required to complete half of this work by 8 workers is:

1. 35

2. 70

3. 105

4. 140

Answer

Total work = 20 × 28 = 560 worker-days

Half of the work = $\dfrac{560}{2}$ = 280 worker-days

Number of days for 8 workers = $\dfrac{280}{8}$ = 35 days

Hence, option 1 is the correct option.

A alone can do a piece of work in 6 days and B alone can do the same work in 12 days. How long will A and B take to complete this working together?

1. 18 days

2. 4 days

3. 9 days

4. 8 days

Answer

A's 1 day's work = $\dfrac{1}{6}$

B's 1 day's work = $\dfrac{1}{12}$

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

$= \dfrac{1}{6} + \dfrac{1}{12} \\[1em] \text { L.C.M. of 6 and 12 = 12 }\\[1em] = \dfrac{2}{12} + \dfrac{1}{12} \\[1em] = \dfrac{3}{12} \\[1em] = \dfrac{1}{4}$

Time taken together = 4 days

Hence, option 2 is the correct option.

An empty water tank can be filled completely by a pipe in 6 hours and a completely filled water tank can be emptied by another pipe in 9 hours. If both the pipes are opened together, the empty tank will be filled in:

1. 15 hours

2. 3 hours

3. 12 hours

4. 18 hours

Answer

Filling pipe's 1 hour's work = $\dfrac{1}{6}$

Emptying pipe's 1 hour's work = $\dfrac{1}{9}$

Work done in 1 hour when both pipes are open

$= \dfrac{1}{6} - \dfrac{1}{9} \\[1em] \text { L.C.M. of 6 and 9 = 18 }\\[1em] = \dfrac{3}{18} - \dfrac{2}{18} \\[1em] = \dfrac{3 - 2}{18} \\[1em] = \dfrac{1}{18}$

Time taken to fill the tank = 18 hours

Hence, option 4 is the correct option.

If A : B = $\dfrac{1}{2} : \dfrac{1}{3}$ and A − B = 50; then B is equal to:

1. 25

2. 50

3. 100

4. 150

Answer

A : B = $\dfrac{1}{2} : \dfrac{1}{3}$

Multiply both terms by 6

= $\dfrac{1}{2} \times 6:\dfrac{1}{3} \times 6$ = 3 : 2

A : B = 3 : 2

Let A = 3x, B = 2x

Given, A - B = 50

⇒ 3x − 2x = 50

⇒ x = 50

B = 2x = 2 × 50 = 100

Hence, option 3 is the correct option.

If A : B : C = 5 : 2 : 3 and A + B − C = 200, then B is equal to:

1. 50

2. 250

3. 100

4. 150

Answer

A : B : C = 5 : 2 : 3

Let A = 5x, B = 2x, C = 3x

Given, A + B − C = 200

⇒ 5x + 2x − 3x = 200

⇒ 4x = 200

⇒ x = $\dfrac{200}{4}$ = 50

B = 2x = 2 × 50 = 100

Hence, option 3 is the correct option.

If 5 : 30x = 15 : 6, then x is equal to:

1. 15

2. $\dfrac{1}{15}$

3. 30

4. $\dfrac{1}{30}$

Answer

In a proportion, product of extremes = product of means:

⇒ 5 × 6 = 30x × 15

⇒ 30 = 450x

⇒ x = $\dfrac{30}{450} = \dfrac{1}{15}$

Hence, option 2 is the correct option.