Directions (Questions 1 to 12): Each of the questions below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements is/are sufficient to answer the question. 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; and
Give answer (e) if the data in both Statements I and II together are necessary to answer the question.
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How much time did \(X\) take to reach the destination?
I. The ratio between the speeds of \(X\) and \(Y\) is \(3: 4\).
II. \(Y\) takes 36 minutes to reach the same destination.
I. If \(Y\) takes \(3 \mathrm{~min}\), then \(X\) takes \(4 \mathrm{~min}\).
II. If \(Y\) takes \(36 \mathrm{~min}\), then \(X\) takes \(\left(\frac{4}{3} \times 36\right) \mathrm{min}=48 \mathrm{~min}\). Thus, I and II together give the answer.
\(\therefore\) Correct answer is \((e)\).
Shweta walked from her home to the bus stop and back again. How long did it take her to make the entire trip?
I. She walked from home to the bus stop at the rate of \(3 \mathrm{~km} / \mathrm{hr}\).
II. She walked back to home @ \(5 \mathrm{~km} / \mathrm{hr}\).
Since the distance between the house and the bus stop is not given, the duration of the trip cannot be calculated. \(\therefore\) Correct answer is \((d)\).
What is the distance between City \(A\) and City \(B\)?
I. Bus starting from A reaches B at \(6: 15\) p.m. at an average speed of \(60 \mathrm{kmph}\).
II. Bus at an average speed of \(40 \mathrm{kmph}\) reaches \(A\) at \(4: 35\) p.m. if it starts from \(B\) exactly at noon.
(Bank P.O., 2009)
I. Only the reaching time is given. So, the duration of the journey and hence the distance between City A and City B cannot be calculated.
II. Required distance \(=\left(40 \times 4 \frac{35}{60}\right) \mathrm{km}=183 \frac{1}{3} \mathrm{~km}\).
\(\therefore\) II alone gives the answer. \(\therefore\) Correct answer is \((b)\).
What is the usual speed of the train? (M.B.A., 2002)
I. The speed of the train is increased by \(25 \mathrm{~km} / \mathrm{hr}\) to reach the destination \(150 \mathrm{~km}\) away in time.
II. The train is late by 30 minutes.
Let the usual speed of the train be \(x \mathrm{kmph}\). Time taken to cover \(150 \mathrm{~km}\) at usual speed \(=\frac{150}{x} \mathrm{hrs}\).
I. Time taken at increased speed \(=\frac{150}{(x+25)}\) hrs.
\(
\begin{aligned}
& \text { II. } \frac{150}{x}-\frac{150}{(x+25)}=\frac{30}{60} \Leftrightarrow \frac{1}{x}-\frac{1}{(x+25)}=\frac{1}{300} \\
& \Leftrightarrow[(x+25)-x] \times 300=x(x+25) \\
& \Leftrightarrow x^2+25 x-7500=0 \Leftrightarrow(x+100)(x-75)=0 \\
& \Leftrightarrow x=75 \text {. } \\
&
\end{aligned}
\)
Thus, I and II together give the answer.
\(\therefore\) Correct answer is \((e)\).
The two towns are connected by railway. Can you find the distance between them?
I. The speed of the mail train is \(12 \mathrm{~km} / \mathrm{hr}\) more than that of an express train.
II. A mail train takes 40 minutes less than an express train to cover the distance.
Let the distance between the two stations be \(x \mathrm{~km}\).
I. Let the speed of the express train be \(y \mathrm{~km} / \mathrm{hr}\). Then, speed of the mail train \(=(y+12) \mathrm{km} / \mathrm{hr}\).
II. \(\frac{x}{y}-\frac{x}{(y+12)}=\frac{40}{60}\).
Thus, even I and II together do not give \(x\).
\(\therefore\) Correct answer is \((d)\).
Sachin jogs at a constant rate for 80 minutes along the same route every day. How long is the route? (M.A.T., 2006)
I. Yesterday, Sachin began jogging at 5 : 00 p.m.
II. Yesterday, Sachin had jogged 5 miles by \(5: 40\) p.m. and 8 miles by \(6: 04\) p.m.
From II, we have :
Distance covered by Sachin from \(5: 40\) p.m. to \(6: 04\) p.m i.e., in \(24 \mathrm{~min}=3\) miles.
\(\therefore\) Length of the route \(=\) Distance covered in \(80 \mathrm{~min}\) \(=\left(\frac{3}{24} \times 80\right)\) miles \(=10\) miles.
So, II alone gives the answer while I alone does not.
\(\therefore\) Correct answer is \((b)\).
The towns \(A, B\) and \(C\) are on a straight line. Town \(C\) is between \(A\) and \(B\). The distance from \(A\) to \(B\) is \(100 \mathrm{~km}\). How far is \(A\) from \(C\)?
(M.B.A., 2003)
I. The distance from \(A\) to \(B\) is \(25 \%\) more than the distance from \(C\) to \(B\).
II. The distance from \(A\) to \(C\) is \(\frac{1}{4}\) of the distance from \(C\) to \(B\).
Let \(\mathrm{AC}=x \mathrm{~km}\).
Then, \(C B=(100-x) \mathrm{km}\)
I. \(\mathrm{AB}=125 \%\) of \(\mathrm{CB}\)
\(
\begin{aligned}
\Leftrightarrow 100 & =\frac{125}{100} \times(100-x) \Leftrightarrow 100-x \\
& =\frac{100 \times 100}{125}=80 \Leftrightarrow x=20 \mathrm{~km} . \\
\therefore A C & =20 \mathrm{~km} .
\end{aligned}
\)
Thus, I alone gives the answer.
II. \(A C=\frac{1}{4} C B \Leftrightarrow x=\frac{1}{4}(100-x) \Leftrightarrow 5 x=100 \Leftrightarrow x=20\).
\(
\therefore A C=20 \mathrm{~km} \text {. }
\)
Thus, II alone gives the answer.
\(\therefore\) Correct answer is \((c)\).
What is the average speed of the car over the entire distance?
I. The car covers the whole distance in four equal stretches at speeds of \(10 \mathrm{kmph}, 20 \mathrm{kmph}, 30 \mathrm{kmph}\), and \(60 \mathrm{kmph}\) respectively.
II. The total time taken is 36 minutes.
Let the whole distance be \(4 x \mathrm{~km}\).
I. Total time taken
\(
\begin{aligned}
& =\left(\frac{x}{10}+\frac{x}{20}+\frac{x}{30}+\frac{x}{60}\right)=\frac{(6 x+3 x+2 x+x)}{60}=\frac{12 x}{60}=\frac{x}{5} . \\
& \therefore \text { Speed }=\frac{\text { Distance }}{\text { Time }}=\frac{4 x}{(x / 5)} \mathrm{kmph}=20 \mathrm{~km} / \mathrm{hr} .
\end{aligned}
\)
\(\therefore\) I alone is sufficient to answer the question.
II. alone does not give the answer.
\(\therefore\) Correct answer is \((a)\).
How long will it take for a jeep to travel a distance of \(250 \mathrm{~km}\)? (M.B.A., 2005)
I. The relative speed of the jeep with respect to the car moving in the same direction at \(40 \mathrm{kmph}\) is \(50 \mathrm{kmph}\).
II. The car started at \(3.00 \mathrm{a} . \mathrm{m}\). in the morning.
I. Speed of the jeep \(=(40+50) \mathrm{kmph}=90 \mathrm{kmph}\). \(\therefore\) Required time \(=\left(\frac{250}{90}\right) \mathrm{hrs}=\frac{25}{9} \mathrm{hrs}=2 \frac{7}{9} \mathrm{hrs}\).
So, I alone gives the answer while II alone does not. \(\therefore\) Correct answer is \((a)\).
A car and a bus start from city \(A\) at the same time. How far is the city \(B\) from city \(A\)?
I. The car travelling at an average speed of \(40 \mathrm{~km} /\) \(\mathrm{hr}\) reaches city \(B\) at \(4: 35\) p.m.
II. The bus reaches city \(B\) at \(6: 15\) p.m. at an average speed of \(60 \mathrm{~km} / \mathrm{hr}\).
Let \(A B=x \mathrm{~km}\). From \(\mathrm{I}\) and II, we get :
\(
\begin{aligned}
\frac{x}{40}-\frac{x}{60} & =1 \frac{40}{60}[(6: 15 \text { p.m. })-(4: 35 \text { p.m. }) \\
& =1 \mathrm{hr} 40 \mathrm{~min}]
\end{aligned}
\)
\(\Leftrightarrow \frac{x}{40}-\frac{x}{60}=\frac{100}{60}\). This gives \(x\).
\(\therefore\) Correct answer is (e).
Two cars pass each other in opposite directions. How long would they take to be \(500 \mathrm{~km}\) apart?
I. The sum of their speeds is \(135 \mathrm{~km} / \mathrm{hr}\).
II. The difference of their speeds is \(25 \mathrm{~km} / \mathrm{hr}\).
I. gives, relative speed \(=135 \mathrm{~km} / \mathrm{hr}\).
\(\therefore \quad\) Time taken \(=\frac{500}{135} \mathrm{hrs}\).
II. does not give the relative speed.
\(\therefore \quad\) I alone gives the answer and II is irrelevant.
\(\therefore\) Correct answer is (a).
Jacob’s house is 60 miles from the town. On Sunday, he went to town and returned home. How long did the entire trip take?
I. He travelled at a uniform rate for the round trip of 30 miles per hour.
II. If Jacob travelled 10 miles per hour faster, it would have taken \(\frac{3}{4}\) of the time for the round trip.
I. Time taken for the round trip \(=\left(\frac{120}{30}\right)\) hrs \(=4\) hrs.
II. Let the time taken for the round trip be \(x\) hours.
Then, \(\frac{120}{\left(\frac{3}{4} x\right)}-\frac{120}{x}=10 \Leftrightarrow \frac{160}{x}-\frac{120}{x}=10\)
\(
\Leftrightarrow 10 x=40 \Leftrightarrow x=4
\)
Thus, I alone or II alone gives the answer.
\(\therefore\) Correct answer is (c).
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