Exercise (Data Sufficiency Questions)

Directions (Questions 1–11): Each of the questions given below consists of a statement and/or a question followed by two statements labelled I and II. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question.

Q1. Will \(Q\) take more than 8 hours to complete the job alone?
I. P works faster than \(Q\).
II. \(P\) and \(Q\) can together finish the job in 5 hours. (J.M.E.T., 2005)

Solution:

From II, we can conclude that if \(P\) and \(Q\) worked with equal efficiency, each of them alone would do the job in 10 hours. But according to I, \(Q\) is slower than \(P\). So \(Q\) alone would take more than 10 hours to complete the job. Thus, both I and II together are necessary to get the answer.
\(\therefore\) Correct answer is \((e)\).

Q2. In how many days can Mohan alone complete the work?

I. Mohan and Prakash together can complete the work in 17 days.
II. Rakesh works double as fast as Mohan and can alone complete the work in 10 days. (Bank P.O., 2006)

Solution:

From II, it is clear that Mohan alone takes double the time as taken by Rakesh alone to do the work i.e., 20 days. I is insufficient.
Thus, II alone gives the answer.
\(\therefore\) Correct answer is \((b)\).

Q3. In how many days can B alone complete the work? (Bank P.O., 2009)
I. B and \(C\) together can complete the work in 8 days.
II. A and B together can complete the work in 12 days.

Solution:

I. gives, \((B+C)\) ‘s 1 day’s work \(=\frac{1}{8} \dots(i)\)
II. gives, \((\mathrm{A}+\mathrm{B})\) ‘s 1 day’s work \(=\frac{1}{12} \dots(ii)\)
We cannot find B’s 1 day’s work using (i) and (ii). Thus, both I and II together are not sufficient.
\(\therefore\) Correct answer is \((d)\).

Q4. How long will Machine \(Y\), working alone, take to produce \(x\) candles? (M.B.A., 2002)
I. Machine \(X\) produces \(x\) candles in 5 minutes.
II. Machine \(X\) and Machine \(Y\) working at the same time produce \(x\) candles in 2 minutes.

Solution:

I. gives, Machine \(\mathrm{X}\) produces \(\frac{x}{5}\) candles in \(1 \mathrm{~min}\).
II. gives, Machines \(X\) and \(Y\) produce \(\frac{x}{2}\) candles in \(1 \mathrm{~min}\).
From I and II, Y produces \(\left(\frac{x}{2}-\frac{x}{5}\right)=\frac{3 x}{10}\) candles in \(1 \mathrm{~min}\). \(\frac{3 x}{10}\) candles are produced by \(Y\) in \(1 \mathrm{~min}\).
\(x\) candles will be produced by \(\mathrm{Y}\) in \(\left(\frac{10}{3 x} \times x\right) \mathrm{min}=\frac{10}{3} \mathrm{~min}\)
Thus, I and II both are necessary to get the answer.
\(\therefore \quad\) Correct answer is \((e)\).

Q5. B alone can complete a work in 12 days. How many days will \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) together take to complete the Work? (SNAP, 2005)
I. A and B together can complete the work in 3 days.
II. B and \(C\) together can complete the work in 6 days.

Solution:

Given: B’s 1 day’s work \(=\frac{1}{12}\).
I. gives, \((\mathrm{A}+\mathrm{B})\) ‘s 1 day’s work \(=\frac{1}{3}\)
\(
\Rightarrow \text { A’s } 1 \text { day’s work }=\left(\frac{1}{3}-\frac{1}{12}\right)=\frac{3}{12}=\frac{1}{4} \text {. }\)
II. gives, \((B+C)\) ‘s 1 day’s work \(=\frac{1}{6}\)
\(
\begin{aligned}
& \Rightarrow \text { C’s } 1 \text { day’s work }=\left(\frac{1}{6}-\frac{1}{12}\right)=\frac{1}{12} . \\
& \therefore \quad(A+B+C) \text { ‘s } 1 \text { day’s work }=\left(\frac{1}{4}+\frac{1}{12}+\frac{1}{12}\right)=\frac{5}{12} .
\end{aligned}
\)
Hence, they all finish the work in \(\frac{12}{5}=2 \frac{2}{5}\) days.
Thus, I and II both are necessary to get the answer.
\(\therefore \quad\) Correct answer is (e).

Q6. Is it cheaper to employ \(X\) to do a certain job than to employ \(Y\)?
I. \(X\) is paid \(20 \%\) more per hour than \(Y\), but \(Y\) takes 2 hours longer to complete the job.
II. \(X\) is paid \(₹ 80\) per hour.

Solution:

Suppose \(X\) takes \(x\) hours and \(Y\) takes \((x+2)\) hours to complete the job.
II. \(X\) is paid \(₹ 80\) per hour.
Total payment to \(\mathrm{X}=₹(80 x)\).
I. \(X=120 \%\) of \(Y=\frac{120}{100} Y=\frac{6}{5} Y \Rightarrow Y=\frac{5}{6} X\).
\(\therefore \quad \mathrm{Y}\) is paid \(₹\left(\frac{5}{6} \times 80\right)\) per hour
\(\Rightarrow \quad Y\) is paid \(₹\left[\frac{200}{3}(x+2)\right]\).
We cannot compare (80x) and \(\frac{200}{3}(x+2)\).
\(\therefore\) Correct answer is \((d)\).

Q7. A and B together can complete a task in 7 days. B alone can do it in 20 days. What part of the work was carried out by A?
I. A completed the job alone after A and B worked together for 5 days.
II. Part of the work done by A could have been done by \(B\) and \(C\) together in 6 days.

Solution:

B’s 1 day’s work \(=\frac{1}{20} \cdot(\mathrm{A}+\mathrm{B})^{\prime}\) s 1 day’s work \(=\frac{1}{7}\).
I. \((A+B)\) ‘s 5 days’ work \(=\frac{5}{7}\).
Remaining work \(=\left(1-\frac{5}{7}\right)=\frac{2}{7}\).
\(\therefore \quad \frac{2}{7}\) work was carried by A.
II. is irrelevant.
\(\therefore \quad\) Correct answer is \((a)\).

Q8. Who is the slowest among the three workers \(P, Q\), and \(\mathrm{R}?\) (M.A.T., 2009)

I. \(P\) and \(Q\) together fence a garden of perimeter \(800 \mathrm{~m}\) in 11 hours.
II. \(P, Q\) and \(R\) together can fence a garden of perimeter \(800 \mathrm{~m}\) in 5 hours.

Solution:

Clearly, using I and II, we can find only R’s 1 hour’s work while the same cannot be found for \(P\) and \(Q\).
Hence, the speeds of \(P, Q\), and \(R\) cannot be compared. Thus, correct answer is \((d)\).

Q9. How many women can complete a piece of work in 15 days? (Bank P.O., 2009)
I. 12 women can complete the same piece of work in 20 days.
II. 10 men can complete the same piece of work in 12 days.

Solution:

I. gives, 1 woman’s 1 day’s work \(=\frac{1}{20 \times 12}=\frac{1}{240}\). \(\therefore 1\) woman’s 15 days’ work \(=\left(\frac{1}{240} \times 15\right)=\frac{1}{16}\).
So, 16 women can complete the work in 15 days. Thus, I alone gives the answer. While II is irrelevant. \(\therefore\) Correct answer is \((a)\).

Q10. In how many days 10 men will finish the work while working together? (Bank P.O., 2008)
I. Only 12 women can finish the work in 16 days.
II. 4 men and 6 women can finish the work in 16 days.

Solution:

I. gives, 1 woman’s 1 day’s work \(=\frac{1}{16 \times 12}=\frac{1}{192}\).
II. gives, \((4 \mathrm{M}+6 \mathrm{~W})=12 \mathrm{~W} \Rightarrow 4 \mathrm{M}=6 \mathrm{~W} \Rightarrow \mathrm{M}=\frac{3}{2} \mathrm{~W}\).
So, 1 man’s 1 day’s work \(=\left(\frac{3}{2} \times \frac{1}{192}\right)=\frac{1}{128}\).
10 men’s 1 day’s work \(=\left(\frac{1}{128} \times 10\right)=\frac{5}{64}\).
Hence, 10 men together take \(\frac{64}{5}\) i.e., \(12 \frac{4}{5}\) days to finish the work.
Thus, both I and II are necessary to answer the question. \(\therefore\) Correct answer is (e).

Q11. How many women can complete the work in 10 days? (Bank P.O., 2009)
I. Work done by one woman in one day is \(75 \%\) of the work done by one man in one day.
II. Work done by one woman in one day is \(150 \%\) of the work done by one child in one day.

Solution:

Both I and II tell us about the comparative efficiencies of a man, a woman, and a child. From the given information, the answer cannot be obtained.
\(\therefore\) Correct answer is \((d)\).