Directions (Questions 1–8): Each of the following questions consists of a question followed by three statements I, II and III. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question.
Directions (Questions 9-10): Each of these questions is followed by three statements. You have to study the question and all three statements given to decide whether any information provided in the statement(s) is/are redundant and can be dispensed with while answering the given question.
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In how many days can A and B working together complete a job?
I. A alone can complete the job in 30 days.
II. B alone can complete the job in 40 days.
III. B takes 10 days more than A to complete the job.
I. A can complete the job in 30 days.
\(
\therefore \text { A’s } 1 \text { day’s work }=\frac{1}{30} \text {. }
\)
II. B can complete the job in 40 days.
\(
\therefore \text { B’s } 1 \text { day’s work }=\frac{1}{40} \text {. }
\)
III. B takes 10 days more than A to complete the job. I and II gives, \((A+B)\) ‘s 1 day’s work \(=\left(\frac{1}{30}+\frac{1}{40}\right)=\frac{7}{120}\).
\(\therefore\) I and III also give the same answer.
II and III also give the same answer.
\(\therefore\) Correct answer is \((d)\).
In how many days \(\mathrm{A}\) alone can complete a work?
I. \(A\) and \(B\) can complete the work in 8 days.
II. B takes twice the time taken by \(\mathrm{A}\) in completing the work.
III. A and B together take \(\frac{1}{3}\) of the time taken by B alone in completing the work. (Bank P.O., 2004)
I. \((\mathrm{A}+\mathrm{B})\) ‘s 1 day’s work \(=\frac{1}{8}\).
II. Suppose \(A\) takes \(x\) days to complete the work. Then, \(B\) takes \(2 x\) days to complete it.
\(
\therefore \frac{1}{x}+\frac{1}{2 x}=\frac{1}{8} \Rightarrow \frac{3}{2 x}=\frac{1}{8} \Leftrightarrow x=\frac{3 \times 8}{2}=12 \text {. }
\)
So, A alone takes 12 days to complete the work.
III. B alone takes \((3 \times 8)=24\) days to complete the work.
\(
\therefore \text { A’s } 1 \text { day’s work }=\frac{1}{8}-\frac{1}{24}=\frac{2}{24}=\frac{1}{12} \text {. }
\)
So, A alone takes 12 days to complete the work. Thus, (I and II) or (I and III) give the answer.
\(\therefore\) Correct answer is (e).
In how many days can the work be completed by A and B together? (M.A.T., 2005)
I. A alone can complete the work in 8 days.
II. If \(\mathrm{A}\) alone works for 5 days and \(\mathrm{B}\) alone works for 6 days, the work gets completed.
III. B alone can complete the work in 16 days.
I. A can complete the job in 8 days.
So, A’s 1 day’s work \(=\frac{1}{8}\).
II. A works for 5 days, B works for 6 days and the work is completed.
III. B can complete the job in 16 days.
So, B’s 1 day’s work \(=\frac{1}{16}\).
I and III : \((\mathrm{A}+\mathrm{B})\) ‘s 1 day’s work \(=\left(\frac{1}{8}+\frac{1}{16}\right)=\frac{3}{16}\).
\(\therefore\) Both can finish the work in \(\frac{16}{3}\) days.
II and III : Suppose A takes \(x\) days to finish the work.
Then, \(\frac{5}{x}+\frac{6}{16}=1 \Rightarrow \frac{5}{x}=\left(1-\frac{3}{8}\right)=\frac{5}{8} \Rightarrow x=8\).
\(\therefore \quad(\mathrm{A}+\mathrm{B})\) ‘s 1 day’s work \(=\left(\frac{1}{8}+\frac{1}{16}\right)=\frac{3}{16}\).
\(\therefore \quad\) Both can finish it in \(\frac{16}{3}\) days.
I and II : A’s 1 day’s work \(=\frac{1}{8}\). Suppose B takes \(x\) days to finish the work.
Then from \(I,\left(5 \times \frac{1}{8}+6 \times \frac{1}{x}=1\right) \Leftrightarrow \frac{6}{x}=\left(1-\frac{5}{8}\right)\) \(=\frac{3}{8} \Rightarrow x=\left(\frac{8 \times 6}{3}\right)=16\).
\(\therefore \quad(\mathrm{A}+\mathrm{B})^{\prime}\) s 1 day’s work \(=\left(\frac{1}{8}+\frac{1}{16}\right)=\frac{3}{16}\).
\(\therefore\) Both can finish it in \(\frac{16}{3}\) days.
Hence, the correct answer is \((c)\).
In how many days will B alone complete the work?
I. A and B together can complete the work in 8 days.
II. B and C together can complete the work in 10 days.
III. \(A\) and \(C\) together can complete the work in 12 days.
I. \((\mathrm{A}+\mathrm{B})^{\prime}\) s 1 day’s work \(=\frac{1}{8} \dots(i)\).
II. \((\mathrm{B}+\mathrm{C})^{\prime}\) s 1 day’s work \(=\frac{1}{10} \dots(ii)\).
III. \((\mathrm{A}+\mathrm{C})^{\prime}\) s 1 day’s work \(=\frac{1}{12} \dots(iii)\).
Adding (i), (ii) and (iii), we get:
\(
\begin{aligned}
& 2(\mathrm{~A}+\mathrm{B}+\mathrm{C})^{\prime} \text { s } 1 \text { day’s work }=\frac{1}{8}+\frac{1}{10}+\frac{1}{12}=\frac{37}{120} \\
& \Rightarrow(\mathrm{A}+\mathrm{B}+\mathrm{C})^{\prime} \text { s } 1 \text { day’s work }=\frac{37}{240} . \\
& \therefore \text { B’s } 1 \text { day’s work }=\left(\frac{37}{240}-\frac{1}{12}\right)=\frac{17}{240} .
\end{aligned}
\)
Hence, B alone can complete the work in \(\frac{240}{17}\) i.e., \(14 \frac{2}{17}\) days.
Thus, I, II and III together give the answer.
\(\therefore\) Correct answer is (c).
How many workers are required for completing the construction work in 10 days?
I. \(20 \%\) of the work can be completed by 8 workers in 8 days.
II. 20 workers can complete the work in 16 days.
III. One-eighth of the work can be completed by 8 workers in 5 days.
I. \(\frac{20}{100}\) work can be completed by \((8 \times 8)\) workers in 1 day.
\(\Rightarrow\) Whole work can be completed by \((8 \times 8 \times 5)\) workers in 1 day
\(=\frac{8 \times 8 \times 5}{10}\) worke rs in 10 days \(=32\) workers in 10 days.
II. \((20 \times 16)\) workers can finish it in 1 day
\(\Rightarrow \frac{(20 \times 16)}{10}\) workers can finish it in 10 days
\(\Rightarrow 32\) workers can finish it in 10 days.
III. \(\frac{1}{8}\) work can be completed by \((8 \times 5)\) workers in 1 day
\(\Rightarrow\) Whole work can be completed by \((8 \times 5 \times 8)\) workers in 1 day
\(=\frac{8 \times 5 \times 8}{10}\) workers in 10 days \(=32\) workers in 10 days.
\(\therefore \quad\) Any one of the three gives the answer.
\(\therefore \quad\) Correct answer is (e).
In how many days can 16 men and 8 women together complete the piece of work? (Bank P.O. 2006)
I. 8 men complete the piece of work in 10 days.
II. 16 women complete the piece of work in 10 days.
III. 5 women take 32 days to complete the piece of work.
I. 1 man’s 1 day’s work \(=\frac{1}{10 \times 8}=\frac{1}{80}\).
II. 1 woman’s 1 day’s work \(=\frac{1}{10 \times 16}=\frac{1}{160}\).
III. 1 woman’s 1 day’s work \(=\frac{1}{32 \times 5}=\frac{1}{160}\).
Since II and III give the same information, either of them may be used.
\((16\) men +8 women)’s 1 day’s work
\(
=\left(\frac{1}{80} \times 16+\frac{1}{160} \times 8\right)=\frac{1}{5}+\frac{1}{20}=\frac{5}{20}=\frac{1}{4} .
\)
\(\therefore 16\) men and 8 women together can complete the work: in 4 days.
Thus, I and either II or III give the answer.
\(\therefore\) Correct answer is \((d)\).
In how many days can the work be done by 9 men and 15 women?
I. 6 men and 5 women can complete the work in 6 days.
II. 3 men and 4 women can complete the work in 10 days.
III. 18 men and 15 women can complete the work in 2 days.
Clearly, any two of the three will give two equations in \(x\) and \(y\), which can be solved simultaneously.
\(\therefore\) Correct answer is (c).
\(\left.\begin{array}{l}\text { For example I and II together give } \\ \qquad\left(6 x+5 y=\frac{1}{6}, 3 x+4 y=\frac{1}{10}\right)\end{array}\right]\).
In how many days can 10 women finish a work? (N.M.A.T. 2005; R.B.I., 2002)
I. 10 men can complete the work in 6 days.
II. 10 men and 10 women together can complete the work in \(3 \frac{3}{7}\) days.
III. If 10 men work for 3 days and thereafter 10 women replace them, the remaining work is completed in 4 days.
I. \((10 \times 6)\) men can complete the work in 1 day
\(
\Rightarrow 1 \text { man’s } 1 \text { day’s work }=\frac{1}{60} \text {. }
\)
II. \(\left(10 \times \frac{24}{7}\right)\) men \(+\left(10 \times \frac{24}{7}\right)\) women can complete the work in 1 day.
\(
\begin{aligned}
\Rightarrow & \left(\frac{240}{7}\right) \text { men’s } 1 \text { day’s work }+\left(\frac{240}{7}\right) \text { women’s } 1 \text { day’s } \\
& \quad \text { work }=1 \\
\Rightarrow & \left(\frac{240}{7} \times \frac{1}{60}\right)+\left(\frac{240}{7}\right) \text { women’s } 1 \text { day’s work }=1 . \\
\Rightarrow & \left(\frac{240}{7}\right) \text { women’s } 1 \text { day’s work }=\left(1-\frac{4}{7}\right)=\frac{3}{7} \\
\Rightarrow & 10 \text { women’s } 1 \text { day’s work }=\left(\frac{3}{7} \times \frac{7}{240} \times 10\right)=\frac{1}{8} .
\end{aligned}
\)
So, 10 women can finish the work in 8 days.
III. (10 men’s work for 3 days) + (10 women’s work for 4 days) \(=1\)
\(
\begin{aligned}
& \Rightarrow(10 \times 3) \text { men’s } 1 \text { day’s work }+(10 \times 4) \text { women’s } 1 \text { day’s } \\
& \text { work }=1
\end{aligned}
\)
\(\Rightarrow 30\) men’s 1 day’s work +40 women’s 1 day’s work \(=1\). Thus, I and III will give us the answer. And, II and III will give us the answer.
\(\therefore \quad\) Correct answer is \((a)\).
In how many days can the work be completed by \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) together?
I. A and B together can complete the work in 6 days.
II. B and C together can complete the work in \(3 \frac{3}{4}\) days.
III. A and \(C\) together can complete the work in \(3 \frac{1}{3}\) days.
I. \((A+B)\) ‘s 1 day’s work \(=\frac{1}{6}\).
II. \((\mathrm{B}+\mathrm{C})\) ‘s 1 day’s work \(=\frac{4}{15}\).
III. \((\mathrm{A}+\mathrm{C})\) ‘s 1 day’s work \(=\frac{3}{10}\).
Adding, we get \(2(A+B+C)\) ‘s 1 day’s work
\(
\begin{aligned}
& =\left(\frac{1}{6}+\frac{4}{15}+\frac{3}{10}\right)=\frac{22}{30} \\
& \Rightarrow(A+B+C) \text { ‘s } 1 \text { day’s work }=\left(\frac{1}{2} \times \frac{22}{30}\right)=\frac{11}{30} .
\end{aligned}
\)
Thus, \(A, B\) and \(C\) together can finish the work in \(\frac{30}{11}\) days.
Hence I, II and III are necessary to answer the question.
\(\therefore \quad\) Correct answer is (e).
8 men and 14 women are working together in a field. After working for 3 days, 5 men and 8 women leave the work. How many more days will be required to complete the work?
I. 19 men and 12 women together can complete the work in 18 days.
II. 16 men can complete two-third s of the work in 16 days.
III. In a day, the work done by three men is equal to the work done by four women. (M.A.T., 2006)
Clearly, I only gives the answer.
Similarly, II only gives the answer.
And, III only gives the answer.
\(\therefore \quad\) Correct answer is \((d)\).
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