# PE 1 - Time & Work | Arithmetic - Time & Work

**PE 1 - Time & Work | Arithmetic - Time & Work**

X, Y and Z can together complete a piece of work in 20 days. If Y does half what X and Z together do in 1 day, in how many days can Y alone do the complete work?

- (a)
70 days

- (b)
40 days

- (c)
60 days

- (d)
66 (2/3) days

Answer: Option C

**Explanation** :

Let the time taken by each of them be X, Y and Z respectively.

∴ $\frac{1}{X}+\frac{1}{Y}+\frac{1}{Z}=\frac{1}{20}$ …(1)

Also, Y does half the work done by X and Z in a day, hence

$\frac{1}{Y}=\frac{1}{2}\left(\frac{1}{X}+\frac{1}{Z}\right)$ or

$\frac{1}{X}+\frac{1}{Z}=\frac{2}{Y}$ …(2)

From (1) and (2)

$\frac{2}{Y}+\frac{1}{Y}=\frac{1}{20}$

Y = 60 days

Hence, option (c).

Workspace:

**PE 1 - Time & Work | Arithmetic - Time & Work**

A can do a job alone in 6 days. B can do 3 times that job in 10 days and C can do 5 times that job in 20 days. If all of them work together, they will finish one job in :

- (a)
$\frac{50}{31}$ days

- (b)
$\frac{60}{43}$ days

- (c)
2 days

- (d)
2.5 days

Answer: Option B

**Explanation** :

To complete the same piece of work, A takes 6 days, B takes 10/3 days, C takes 20/5 = 4 days

∴ $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$ = $\frac{1}{6}+\frac{3}{10}+\frac{1}{4}$ = $\frac{10+18+15}{60}=\frac{43}{60}$

They will finish the job in 60/43 days.

Hence, option (b).

Workspace:

**PE 1 - Time & Work | Arithmetic - Time & Work**

A can do a piece of work in 20 days and B alone can do it in 30 days. A and B undertook to do it for Rs. 600. With the help of C they finished it in 10 days. How much is paid to C?

- (a)
80

- (b)
50

- (c)
60

- (d)
100

Answer: Option D

**Explanation** :

Given, $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{10}$

⇒ $\frac{1}{20}+\frac{1}{30}+\frac{1}{C}=\frac{1}{10}$

⇒ $\frac{1}{C}=\frac{1}{10}-\frac{1}{12}=\frac{1}{60}$

C alone will take 60 days.

Since all three work for same number of days, the payment will be divided in the ratio of their efficiencies.

Rs. 300 is divided in the ratio A : B : C = $\frac{1}{20}:\frac{1}{30}:\frac{1}{60}$ = 3 : 2 : 1.

∴ C gets =$\frac{1}{6}$ × 600 = 100.

Hence, option (d).

Workspace:

**PE 1 - Time & Work | Arithmetic - Time & Work**

Rudra can do a piece of work in 9 days, Pratap in $2\frac{4}{7}$ days and Singh in 18 days. All begin to do it together but Rudra leaves after 1 day and Pratap 2 days before the completion of work. How long would the work last?

- (a)
$2\frac{5}{9}$ days

- (b)
10 days

- (c)
12 days

- (d)
None of these

Answer: Option D

**Explanation** :

Rudra can do a piece of work in 9 days, Pratap in $\frac{18}{7}$ days and Singh in 18 days.

Let the work be completed in ‘n’ days.

∴ Rudra works for 1 day, Pratap for (n – 2) days and Singh for n days.

∴ $\frac{1}{9}+\frac{(n-2)}{18/7}+\frac{\left(n\right)}{18}$ = 1

⇒ $\frac{8n-12}{18}$ = 1

n = $\frac{30}{8}$ days or $3\frac{3}{4}$ days

Hence, option (d).

Workspace:

**PE 1 - Time & Work | Arithmetic - Time & Work**

X does 1/3^{rd} of a job in 8 days. Y completes rest of the job in 32 days. In how many days could X & Y together have completed the job.

- (a)
24 days

- (b)
32 days

- (c)
9 days

- (d)
16 days

Answer: Option D

**Explanation** :

If X can complete 1/3^{rd} of the work in 8 days, X can complete the whole work in 3 × 8 = 24 days.

Y completes 2/3^{rd} of the work in 32 days.

∴ Y takes 32 × 3/2 = 48 days to complete the whole work.

Now, we calculate time taken by both of them together to complete the work.

∴ $\frac{1}{X}+\frac{1}{Y}$ = $\frac{1}{24}+\frac{1}{48}$ = $\frac{3}{48}$ = $\frac{1}{16}$

⇒ X and Y together will take 16 days to complete the entire work.

Hence, option (d).

Workspace:

**PE 1 - Time & Work | Arithmetic - Time & Work**

Two workers A and B working together completed a job in 5 days. If A worked twice as efficiently as he actually did and B worked 1/3^{rd} as efficiently as he actually did, the work would have completed in 3 days. Find time for B to complete the job alone.

- (a)
20 days

- (b)
15 days

- (c)
25 days

- (d)
None of these

Answer: Option C

**Explanation** :

Let them take A and B days respectively to complete the work alone.

∴ $\frac{1}{A}+\frac{1}{B}=\frac{1}{5}$ …(1)

Also, $\frac{1}{A\u22152}+\frac{1}{3B}=\frac{1}{3}$

⇒ $\frac{2}{A}+\frac{1}{3B}=\frac{1}{3}$ …(2)

(1) × 2 - (2), we get

$\frac{2}{B}-\frac{1}{3B}=\frac{2}{5}-\frac{1}{3}$

∴ B = 25 days.

Hence, option (c).

Workspace:

**PE 1 - Time & Work | Arithmetic - Time & Work**

A alone would take 8 days more to complete the job than if both A and B worked together. If B worked alone, he took 4.5 days more to complete the job than A and B worked together. What time would they take if both A and B worked together?

- (a)
6 days

- (b)
7 days

- (c)
5 days

- (d)
None of these

Answer: Option A

**Explanation** :

Let the time A and B take while working together be ‘x’ days.

As per question $\frac{1}{\mathrm{x}+8}+\frac{1}{\mathrm{x}+\frac{9}{2}}=\frac{1}{\mathrm{x}}$

or, $\frac{\mathrm{x}+25\u22152}{\left(\mathrm{x}+8\right)\left(\mathrm{x}+\frac{9}{2}\right)}=\frac{1}{\mathrm{x}}$

$2{\mathrm{x}}^{2}+\frac{25}{2}\mathrm{x}$ = ${\mathrm{x}}^{2}+\frac{25}{2}\mathrm{x}+36$

x^{2} = 36

⇒ x = 6 days

**Remember**:

Solution for an equation of the form $\frac{1}{x+a}$ + $\frac{1}{x+b}$ = $\frac{1}{x}$

is x = $\sqrt{a\times b}$

Hence, option (a).

Workspace:

**PE 1 - Time & Work | Arithmetic - Time & Work**

A man, a woman and a boy do a job in 6, 8 and 24 days respectively. How many boys must assist 2 men and 1 woman to do the job in 2 days?

- (a)
3

- (b)
2

- (c)
1

- (d)
None of these

Answer: Option C

**Explanation** :

Let the work to be done = 24 units.

∴ Efficiency of a man = 24/6 = 4 units/day

∴ Efficiency of a woman = 24/8 = 3 units/day

∴ Efficiency of a boy = 24/24 = 1 units/day

Let x boys assist 2 men and 1 woman to complete the work in 2 days.

∴ (2m + 1w + xb) × 2 = 24

⇒ (2 × 4 + 1 × 3 + x × 1) = 12

⇒ x = 1

Hence, option (c).

Workspace:

**PE 1 - Time & Work | Arithmetic - Time & Work**

A, B, C together can finish a work in 20 days. A and C together can do thrice as much as B does. Also A and B can do twice as much as C does. Find the time taken by A to finish the work :

- (a)
36

- (b)
48

- (c)
24

- (d)
None of these

Answer: Option B

**Explanation** :

Let the time taken by them is A, B and C days respectively.

∴ $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{20}$ …(1)

$\frac{1}{A}+\frac{1}{C}=\frac{3}{B}$ …(2)

$\frac{1}{\mathrm{A}}+\frac{1}{\mathrm{B}}=\frac{2}{\mathrm{C}}$ …(3)

Adding 1/B both sides in (2) we get,

$\frac{1}{20}=\frac{4}{B}$

⇒ B = 80 days.

Similarly, adding 1/C both sides in (3) we get,

$\frac{1}{20}=\frac{3}{C}$

⇒ C = 60 days.

From (1) we get

$\frac{1}{A}+\frac{1}{80}+\frac{1}{60}=\frac{1}{20}$

⇒ A = 48 days

Hence, option (b).

Workspace:

**PE 1 - Time & Work | Arithmetic - Time & Work**

Three pipes A, B and C can fill a cistern in 15, 20 and 30 min respectively. They were all turned on at the same time. After 5 minutes the first two pipes were turned off. In what time will the cistern be filled?

- (a)
7.5 mins

- (b)
12.5 mins

- (c)
8.5 mins

- (d)
None of these

Answer: Option B

**Explanation** :

Let the work to be done = 60 units.

∴ Efficiency of pipe A = 60/15 = 4 units/min

∴ Efficiency of pipe B = 60/20 = 3 units/min

∴ Efficiency of pipe C = 60/30 = 2 units/min

Work done in 5 minutes = (4 + 3 + 2) × 5 = 45 units.

Work left = 60 – 45 = 15 units.

∴ time taken by Pipe C to complete the remaining work = 15/2 = 7.5 minutes.

∴ Cistern is filled in (5 + 7.5) = 12.5 minutes.

Hence, option (b).

Workspace:

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