Exercise: 11-B
Multiple Choice Questions
Q1: If a work can be completed by A in 30 days and by B in 60 days, then the number of days taken by them to finish the work, working together, is:
i. Calculate individual work rates
Step 1: A’s 1 day work = \(\frac{1}{30}\)
Step 2: B’s 1 day work = \(\frac{1}{60}\)
ii. Calculate combined work rate
Step 3: Combined 1 day work = \(\frac{1}{30} + \frac{1}{60} = \frac{2}{60} + \frac{1}{60} = \frac{3}{60} = \frac{1}{20}\)
iii. Calculate time taken working together
Step 4: Time taken = \(\frac{1}{\frac{1}{20}} = 20\) days
Answer: a. 20
Q2: A can do a job in 6 days while A and B can do it together in 2 days. How many days will be taken by B to do the job alone?
i. Calculate work rates of A and (A + B)
Step 1: A’s 1 day work = \(\frac{1}{6}\)
Step 2: (A + B)’s 1 day work = \(\frac{1}{2}\)
ii. Calculate B’s 1 day work
Step 3: B’s 1 day work = (A + B)’s 1 day work – A’s 1 day work
\[
= \frac{1}{2} – \frac{1}{6} = \frac{3 – 1}{6} = \frac{2}{6} = \frac{1}{3}
\]iii. Calculate time taken by B alone
Step 4: Time taken by B = \(\frac{1}{\frac{1}{3}} = 3\) days
Answer: a. 3
Q3: A, B and C together can finish a piece of work in 4 days. A alone can do it in 12 days and B alone in 18 days. How many days will be taken by C to do it alone?
i. Calculate work rates of A, B and (A + B + C)
Step 1: A’s 1 day work = \(\frac{1}{12}\)
Step 2: B’s 1 day work = \(\frac{1}{18}\)
Step 3: (A + B + C)’s 1 day work = \(\frac{1}{4}\)
ii. Calculate C’s 1 day work
Step 4: C’s 1 day work = (A + B + C)’s 1 day work – (A’s 1 day work + B’s 1 day work)
\[
= \frac{1}{4} – \left(\frac{1}{12} + \frac{1}{18}\right)
\]Step 5: Find LCM of 12 and 18 which is 36:
\[
\frac{1}{12} = \frac{3}{36}, \quad \frac{1}{18} = \frac{2}{36} \\
\frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} \\
\text{C’s 1 day work} = \frac{1}{4} – \frac{5}{36} = \frac{9}{36} – \frac{5}{36} = \frac{4}{36} = \frac{1}{9}
\]iii. Calculate days taken by C alone
Step 6: Days taken by C alone = \(\frac{1}{\frac{1}{9}} = 9\) days
Answer: a. 9
Q4: A and B can do a piece of work in 18 days; Band C in 24 days; C and A in 36 days. In how many days can they do it all working together?
i. Calculate the work rates of A+B, B+C, and C+A
Step 1: (A + B)’s 1 day work = \(\frac{1}{18}\)
Step 2: (B + C)’s 1 day work = \(\frac{1}{24}\)
Step 3: (C + A)’s 1 day work = \(\frac{1}{36}\)
ii. Add the above work rates
Step 4: Sum = \(\frac{1}{18} + \frac{1}{24} + \frac{1}{36}\)
Step 5: Find LCM of 18, 24, and 36 = 72
\[
\frac{1}{18} = \frac{4}{72}, \quad \frac{1}{24} = \frac{3}{72}, \quad \frac{1}{36} = \frac{2}{72}
\]
Sum = \(\frac{4}{72} + \frac{3}{72} + \frac{2}{72} = \frac{9}{72} = \frac{1}{8}\)
iii. Use the relation to find combined work rate of A, B, and C
Step 6: Since each pair’s sum is counted twice in the sum, then:
\[
2 \times (A + B + C)’s\ 1\ day\ work = \frac{1}{8} \\ \\
\Rightarrow (A + B + C)’s\ 1\ day\ work = \frac{1}{16}
\]iv. Calculate the number of days to finish the work together
Step 7: Time taken = \(\frac{1}{\frac{1}{16}} = 16\) days
Answer: c. 16
Q5: A can finish a work in 12 days and B can do it in 15 days. After A had worked for 3 days, B also joined A to finish the remaining work. In how many days the remaining work will be finished?
i. Calculate work done by A in 3 days
Step 1: A’s 1 day work = \(\frac{1}{12}\)
Work done by A in 3 days = \(3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4}\)
ii. Calculate remaining work
Step 2: Remaining work = \(1 – \frac{1}{4} = \frac{3}{4}\)
iii. Calculate combined 1 day work of A and B
Step 3: B’s 1 day work = \(\frac{1}{15}\)
Combined 1 day work = \(\frac{1}{12} + \frac{1}{15} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20}\)
iv. Calculate time taken to finish the remaining work
Step 4: Time = \(\frac{\text{Remaining work}}{\text{Combined 1 day work}} = \frac{\frac{3}{4}}{\frac{3}{20}} = \frac{3}{4} \times \frac{20}{3} = 5\) days
Answer: b. 5
Q6: A can do a piece of work in 20 days and B can do it in 12 days. B worked at it for 9 days. In how many days A alone can finish the remaining work?
i. Calculate B’s work done in 9 days
Step 1: B’s 1 day work = \(\frac{1}{12}\)
Work done by B in 9 days = \(9 \times \frac{1}{12} = \frac{9}{12} = \frac{3}{4}\)
ii. Calculate remaining work
Step 2: Remaining work = \(1 – \frac{3}{4} = \frac{1}{4}\)
iii. Calculate time taken by A to finish remaining work
Step 3: A’s 1 day work = \(\frac{1}{20}\)
Time taken by A to finish remaining work = \(\frac{\text{Remaining work}}{\text{A’s 1 day work}} = \frac{\frac{1}{4}}{\frac{1}{20}} = \frac{1}{4} \times 20 = 5\) days
Answer: b. 5
Q7: A works twice as fast as B. If B can complete a piece of work independently in 12 days, then the number of days taken by A and B together to finish the work is
i. Define the work rates of A and B
Step 1: B’s 1 day work = \(\frac{1}{12}\)
Since A works twice as fast as B, A’s 1 day work = \(2 \times \frac{1}{12} = \frac{1}{6}\)
ii. Calculate combined 1 day work of A and B
Step 2: Combined 1 day work = \(\frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}\)
iii. Calculate time taken to finish the work together
Step 3: Time taken = \(\frac{1}{\frac{1}{4}} = 4\) days
Answer: a. 4
Q8: A can do a piece of work in 80 days. He works at it for 10 days and then B alone finishes the work in 42 days. The number of days taken by the two together to finish the work, is
i. Calculate A’s and B’s one day work
Step 1: A can do the work in 80 days
⇒ A’s 1 day work = \(\frac{1}{80}\)
Step 2: A works for 10 days
⇒ Work done by A = \(10 \times \frac{1}{80} = \frac{10}{80} = \frac{1}{8}\)
Step 3: Remaining work = \(1 – \frac{1}{8} = \frac{7}{8}\)
This remaining work is done by B in 42 days.
⇒ B’s 1 day work = \(\frac{7}{8 \times 42} = \frac{7}{336} = \frac{1}{48}\)
ii. Calculate combined one day work of A and B
Step 4: A’s 1 day work = \(\frac{1}{80}\), B’s 1 day work = \(\frac{1}{48}\)
LCM of 80 and 48 = 240
\[
\frac{1}{80} = \frac{3}{240}, \quad \frac{1}{48} = \frac{5}{240}
\]
Combined 1 day work = \(\frac{3 + 5}{240} = \frac{8}{240} = \frac{1}{30}\)
iii. Find total time taken if both work together
Step 5: Time = \(\frac{1}{\frac{1}{30}} = 30\) days
Answer: The two together would finish the work in 30 days. (Option c)
Q9: Pipe A can fill a tank in 45 hours and pipe B can empty it in 36 hours. If both the pipes are opened in the empty tank. in how many hours will it be full?
i. Calculate filling and emptying rates of the pipes
Step 1: A’s 1 hour filling rate = \(\frac{1}{45}\)
Step 2: B’s 1 hour emptying rate = \(\frac{1}{36}\)
ii. Calculate net filling rate when both pipes are open
Step 3: Net filling rate = A’s filling rate – B’s emptying rate
\[
= \frac{1}{45} – \frac{1}{36}
\]Step 4: Find LCM of 45 and 36:
\(45 = 3^2 \times 5\), \(36 = 2^2 \times 3^2\)
LCM = \(2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180\)
\[
\frac{1}{45} = \frac{4}{180}, \quad \frac{1}{36} = \frac{5}{180} \\
\text{Net filling rate} = \frac{4}{180} – \frac{5}{180} = -\frac{1}{180}
\]iii. Interpret the result and conclude
Step 5: Net filling rate is negative, which means pipe B empties the tank faster than A fills it.
So, the tank will never be full if both pipes are opened simultaneously.
Answer: The tank will not be filled as pipe B empties it faster than pipe A fills it.
Q10: Pipe A can fill a cistern in 4 hours while pipe B can empty it in 6 hours. lf both of them are opened at the same time when the certern is empty, the number of hours taken by them to fill it, is
i. Calculate filling and emptying rates of pipes A and B
Step 1: A’s 1 hour filling rate = \(\frac{1}{4}\)
Step 2: B’s 1 hour emptying rate = \(\frac{1}{6}\)
ii. Calculate net filling rate when both pipes are open
Step 3: Net filling rate = A’s filling rate – B’s emptying rate
\[
= \frac{1}{4} – \frac{1}{6} = \frac{3 – 2}{12} = \frac{1}{12}
\]iii. Calculate time taken to fill the cistern
Step 4: Time taken = \(\frac{1}{\text{Net filling rate}} = \frac{1}{\frac{1}{12}} = 12\) hours
Answer: d. 12
Q11: A cistern which has a leak in the bottom is filled in 15 hours. Had there been no leek it could have been filled in 12 hours. If the cistern is full, the leak can empty it in
i. Calculate filling rate without leak and effective filling rate with leak
Step 1: Filling rate without leak = \(\frac{1}{12}\) per hour
Step 2: Effective filling rate with leak = \(\frac{1}{15}\) per hour
ii. Calculate leak’s emptying rate
Step 3: Leak’s emptying rate = Filling rate without leak – Effective filling rate with leak
\[
= \frac{1}{12} – \frac{1}{15} = \frac{5 – 4}{60} = \frac{1}{60}
\]iii. Calculate time taken by leak to empty the full cistern
Step 4: Time taken by leak = \(\frac{1}{\frac{1}{60}} = 60\) hours
Answer: d. 60 hours
Q12: A cistern can be filled by pipes A and B in 4 hours and 6 hours respectively. When full, the cistern can be emptied by a pipe C in 8 hours. If all the pipes be turned on at the same time, the cistern will be full in
i. Calculate filling rates of pipes A and B and emptying rate of pipe C
Step 1: A’s 1 hour filling rate = \(\frac{1}{4}\)
Step 2: B’s 1 hour filling rate = \(\frac{1}{6}\)
Step 3: C’s 1 hour emptying rate = \(\frac{1}{8}\)
ii. Calculate net filling rate when all pipes are open
Step 4: Net filling rate = A’s rate + B’s rate – C’s rate
\[
= \frac{1}{4} + \frac{1}{6} – \frac{1}{8}
\]Step 5: Find LCM of 4, 6, and 8 which is 24
\[
\frac{1}{4} = \frac{6}{24}, \quad \frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24} \\
\text{Net filling rate} = \frac{6}{24} + \frac{4}{24} – \frac{3}{24} = \frac{7}{24}
\]iii. Calculate time taken to fill the cistern
Step 6: Time taken = \(\frac{1}{\text{Net filling rate}} = \frac{1}{\frac{7}{24}} = \frac{24}{7} \approx 3.4286\) hours
Convert decimal to minutes:
\(0.4286 \times 60 = 25.7\) minutes (approximately 26 minutes)
Answer: b. 3 hours 26 minutes.
