## Mental Ability #5 | Study Material :: General Studies | IAS Help

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#### MORE MISCELLANEOUS PROBLEMS

1. How many litres of a solution that is 15% salt must be added to another solution that is 8% salt so that the resulting solution is 10% salt?
1. 2 litres
2. 3 litres
3. 4 litres
4. 1 litre

Let n represent the number of litres of the 15% solution. Then the amount of salt in the 15% solution = 0.15n
Also the amount of the salt in the 8% solution = 0.08 x 5
Also, the amount of salt in the resulting 10% mixture = 0.10 (n+5)
Now, the total amount of salt in both the solutions combined must equal the amount of salt in the mixture
$0.15n + (0.08 x 5) = 0.10 (n+5)$

This gives $n = 2$

2. If positive integers x and y are not both odd, which of the following must be even?
1. xy
2. x + y
3. x – y
4. x + y – 1

Since it is given that x and y are not both odd, either both must be even or one even and one odd. Using these two alternatives, we can test the outcome of each answer choice to determine when both must be even.

1. (even)(even) = even; (even)(odd) = even. The answer must be
even for both alternatives
2. even + even = even; even + odd = odd. The answer need not be even for both alternatives
3. even – even = even; even – odd = odd. Need not be even
4. even + even – 1 = odd; even + odd – 1 = even. Need not be even
3. Equal amounts of water are poured into two empty jars of different capacities, which made one jar ¼ full and the other jar 1/3 full. If the water in the jar with the lesser capacity is poured into the jar with greater capacity, what fraction of the larger jar will be filled with water?
1. 1/7
2. 2/7
3. 1/12
4. 7/12

It is given that the amounts of water in the two jars are equal. Thus, when the smaller jar is poured into the larger jar, the water in the larger jar will be doubled i.e. the larger jar will become 2 x ¼ = ½ full

4. If u > t, r > q, s > t and t > r, which of the following must be true?
1. u > s
2. s > q
3. u > s and s > q
4. s > q and u > r

going from the given information, u > t > r > q. While it is given that s > t it is not known whether s is between t and u or greater than u. Now, looking at each choice given

1. It is known that u > t and s > t. Select t=2, u=3, s=4. It follows that u < s. Hence the given choice is Not true
2. Since s > t, t > r, r > q, it follows that s > q. Must be true
3. Just shown in A that u > s is not true
4. Since u > t, t > r, it follows that u > r. must be true
1. A certain clock marks every hour by striking a number of times equal to the hour, and the time required for a stroke is exactly equal to the time interval between strokes. At 6:00 the time lapse between the beginning of the first stroke and the end of the last stroke is exactly 22 seconds. At 12:00, how many seconds lapse between the beginning of the first stroke and the end of the last stroke?
1. 72 seconds
2. 50
3. 48
4. 46

At 6:00, there are 6 strokes and 5 intervals between strokes. Thus, there are 11 equal time intervals in the 22 seconds between the beginning and end.

Since the total time lapse is given as 22 seconds, each interval must last 22/11 = 2 seconds

At 12:00, there are 12 strokes and 11 intervals between strokes. Thus, there are 23 equal intervals, each of duration 2 seconds.

Thus, the total time lapsed between beginning and end must be 23 x 2 = 46 seconds

2. A company that ships boxes to 12 distribution centres uses colour coding to indentify each centre. If either one colour or a pair of colours is used to represent each centre, and each centre is uniquely identified by that colour code, what is the minimum number of colours needed to uniquely identify all centres? Assume that the order of the colours in a pair does not matter.
1. 4
2. 5
3. 6
4. 12

Since the question asks for the minimum number of colours needed, let us start with the lowest answer choice available. We can calculate each successive option until finding the minimum number of colours that can identify the 12 distribution centres.

For any given number of colours, the number of centres it can identify is given by the number of combinations possible.

If n is the number of colours available, and r is the number used for a given code, the number of possible combinations is $_nC_r = \frac{n!}{r!(n-r)!}$

 No. of colours Number represented by one colour Number represented by two colours Total represented Decision 4 4 $_4C_2=\frac{4!}{2!(4-2)!} = 6$ 4 + 6 = 10 Not sufficient 5 5 $_5C_2 = \frac{5!}{2!(5-2)!} = 10$ 5 + 10 = 15 Sufficient

1. A box contains 100 balls, numbered 1 to 100. If three balls are selected at random and with replacement from the box, what is the probability that the sum of the three numbers will be odd?
1. ¼
2. 3/8
3. ½
4. 5/8

Since there are 50 odd and 50 even balls, the probability of selecting an odd or an even ball is ½.

For the sum of the three numbers to be odd, the numbers must all be odd or one number must be odd and the other two even.

Probability of selecting three odd balls = $\frac{1}{2}\frac{1}{2}\frac{1}{2} = \frac{1}{8}$

Probability of selecting one odd and two even balls = P{odd, even, even} + P{even, odd, even} + P{even, even, odd}

P{odd,even,even} = P{even, odd, even} = P{even, even, odd} = $\frac{1}{2}\frac{1}{2}\frac{1}{2} = \frac{1}{8}$

Thus, the probability of one odd and two even balls = $\frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{3}{8}$

Hence, the probability that the sum of the three balls will be odd is $\frac{1}{8} + \frac{3}{8} = \frac{1}{2}$

2. A box contains 14 apples and 23 oranges. How many oranges must be removed from the box so that 70% of the contents of the box will be apples?
1. 3
2. 6
3. 14
4. 17

From the given information it can be determined that there is a total of 37 fruit pieces in the box.

If x is the number of oranges that need to be removed, $\frac{14}{37-x} = 0.70$

This gives $x = 17$

3. An airline passenger is planning a trip that involves three connecting flights that leave from airports A, B and C. The first flight leaves airport A every hour beginning at 08:00 and arrives at airport B 2.5 hours later. The second flight leaves airport B every 20 minutes beginning at 08:00 and arrives at C 1 1/6 hours later. The third flight leaves airport C every hour beginning at 08:45. What is the least total amount of time the passenger must spend between flights?
1. 25 min
2. 1 hr 5 min
3. 1 hr 15 min
4. 2 hr 20 min

Assume that the passenger takes the first flight from airport A at 08:00. This would take him to airport B at 10:30

The earliest flight he can take from airport B will leave at 10:40. There is a 10 min wait here. This flight would take him to airport C 1 1/6 hours later i.e. 1 hr 10 min later i.e. at 11:50

The earliest flight he can take from airport C will leave at 12:45. There is a 55 min wait here.

Thus the total wait time is 10 + 55 = 1 hr 5 min

1. Each of 25 people is enrolled in history, mathematics or both. If 20 are enrolled in history and 18 in mathematics, how many are enrolled in both?
1. 12
2. 10
3. 13
4. 15

The 25 people can be divided into three sets: those who history only, those in mathematics only, and those in both

If n is the number of people enrolled in both courses, 20-n is the number in enrolled in history only and 18-n is the number in mathematics only.

Since the total number of students is 25, $n + (20-n) + (18-n) = 25$

This gives $n = 13$

2. In a certain production lot, 40% of toys are red and the remaining is green. Half the toys are small and half is large. If 10% of the toys are red and small, and 40% of toys are green and large, how many of the toys are red and large?
1. 60
2. 40
3. 50
4. 70

Let us organize the information in a table:

 Red Green Total Small 10% 50% Large 50% Total 40% 60% 100%

Based on what is given, we can compute the missing information:

 Red Green Total Small 10% 40% 50% Large 30% 20% 50% Total 40% 60% 100%

Let us assume that n is the total number of toys

It is also given that 40 toys are green and large. Then $0.20n = 40$

This gives $n = 200$

Thus, the number of toys that are red and large is $0.30n = 60$

Please note: This is the last article on Mental Ability. From next week (14 Dec 2009), Mondays will feature articles on Indian Economy.

Keywords: India, ias, upsc, civil service, study material, general studies, mental ability

Previous week: Mental Ability #4 - Miscellaneous Problems

## Mental Ability #4 | Study Material :: General Studies | IAS Help

#### MISCELLANEOUS PROBLEMS

1. The price of sugar is being raised by 10%. By how much percent must a man reduce his consumption so as not to increase his expenditure
1. 9%
2. 9.5%
3. 10%
4. 9 1/11%

Let the original price be Rs. 100 and the original quantity of x

Then the quantity consumed is $100$

Let the new price be Rs 110

Then the man must reduce consumption by: $10 * \frac{100}{110}$

2. If a man loses 4% by selling bananas at Rs 12 a rupee, how many per rupee must he sell them so as to gain 44%?
1. 8
2. 7
3. 6
4. 9

The sale price of one banana is $\frac{1}{12}$

Using the formula: $\frac{S_1}{100+x_1} = \frac{S_2}{100+x_2}$

$\frac{\frac{1}{12}}{100-4} = \frac{S_2}{100+44} = \frac{1}{12 x 96} = \frac{S_2}{144}$

Then $S_2 = \frac{144}{12 x 96} = \frac{1}{8}$

3. I have a certain sum of money to be distributed among certain no. of boys. If I give Rs 3 to each, I shall spend Rs 4 less but if Rs 5 to each, I shall need Rs 6 more. How much do I have with me?
1. 19
2. 18
3. 20
4. 23

Let the number of boys be $x$ and the amount I have be $y$
$3x = y - 4$
$5x = y + 6$
This gives $x = 5$ and $y=19$

4. I multiply a number by 36 and divide the result by 12. The quotient is 374181. What is the number?
1. 124772
2. 124727
3. 134727
4. 174232

Let the number be x
According to the problem, $\frac{36x}{12} = 374181$
This gives $x = \frac{374181}{3} = 124727$

5. Three bells toll at intervals of 1.2, 1.8 and 2.7 seconds beginning together. How often will each bell toll before their tolling together again?
1. 9, 4, 6
2. 8, 6, 4
3. 7, 6, 4
4. 10, 2, 3

The bells will toll together again at the LCM of the three intervals.
The LCM of 1.2, 1.8 and 2.7 is 10.8 sec
Between the beginning and 10.8 sec, each bell will toll the following number of times:
$\frac{10.8}{1.2} = 9$ and $\frac{10.8}{1.8} = 6$ and $\frac{10.8}{2.7} = 4$

6. How many revolutions will be made by a wheel which revolves at the rate of 243 revolutions in 3 minutes, while another wheel revolving 374 times in 11 minutes makes 544 revolutions?
1. 1269
2. 1270
3. 1296
4. 1297

For the second wheel, the time taken for one revolution is $\frac{11}{374}$
The total time taken by the second wheel is $544 x \frac{11}{374}$
For the first wheel, the number of revolutions in one minute is $\frac{243}{3}$
In this same time, the first wheel makes the following number of revolutions: $544 x \frac{11}{374} x \frac{243}{3} = 1296$

7. The digits in the units place and lakh’s place in a number are 3 and 8. What will be the digits in the same places in the remainder when 99999 is subtracted from the number
1. (4,6)
2. (4,5)
3. (7,4)
4. (4,7)

Let the number be 80003
When 99999 is subtracted from this number, the digits will be (7,4)

8. A ship 40 miles from shore springs a leak which admits 3 3/4 tons of water in 12 minutes. 60 tons of water are enough to sink the ship. But the ship’s pumps can throw out 12 tons of water in an hour. What should the average speed of the ship be so that it reaches the shore before sinking?
1. 4 mph
2. 4.5 mph
3. 4.7 mph
4. 7 mph

Amount of water entering the ship in one hour: $3\frac{3}{4} x \frac{60}{12} = \frac{75}{4}$ tons
Amount of water thrown out in one hour: 12 tons
Water accumulating in the ship in one hour = $\frac{75}{4} - 12 = \frac{27}{4}$
Time taken to accumulate 60 tons: $\frac{60}{\frac{27}{4}} = \frac{80}{9}$ hours
Speed for safe arrival: $\frac{40}{\frac{80}{9}} = 4.5 mph$

9. An express train owing to a defect in the engine goes at 5/8 th of its usual speed and arrives at 6:49 p.m. instead of 5:55 p.m. At what hour did the train start?
1. 4:25
2. 4:30
3. 4:50
4. 4:55

The new speed = 5/8 of usual speed
If the usual time taken is $x$, the new time is $\frac{8}{5}x$
Then, $\frac{8}{5}x - x = 6:49 - 5:55 = 54 min$
This gives $x = 90 min$
Thus, the starting time is $5:55 - 90 min = 4:25$

10. A train starts with a certain number of passengers. At the first station, it drops 1/3 of those and takes 20 more. At the next it drops 1/2 of the new total and takes on 10 more. On reaching the third station there are 60 left. With how many passengers did the train start?
1. 110
2. 120
3. 125
4. 130

Let the original number of passengers be x
Then, the number of passengers after the first station is: $x_1 = x - \frac{x}{3} + 20$
After the second station, $x_2 = x_1 - \frac{x_1}{2} + 10$
The number of passengers reaching the third station is given to be 60 i.e. $x_2 = 60$
This gives, $x_1 - \frac{x_1}{2} + 10 = 60$
$x_1 = 100$
This gives,
$x - \frac{x}{3} + 20 = 100$
$x = 120$

## Mental Ability #3 | Study Material :: General Studies | IAS Help

#### PROBLEMS ON WORK, PERCENTAGES AND AGE

1. ‘A’ alone completes a job in 12 days while ‘B’ alone completes the job in 24 days. If ‘A’ and ‘B’ work together, how many days would it take to complete the job?
1. 6
2. 8
3. 7
4. 24
2. 10 men can complete a job in 18 days. After 6 days, 5 more men joined. In how many days would the remaining work be completed?
1. 8
2. 6
3. 2
4. 15
3. A man works twice as fast as a woman and a woman works twice as fast as a child. If 16 men can complete a job in 12 days, how many would be required for 32 women and 64 boys together to complete the same job?
1. 6
2. 7
3. 8
4. 10
4. A contractor undertakes to complete a job in 60 days and employs 90 persons for the purpose. After 45 days he takes a review and finds that 90% of the work is over. In order to finish the job in 60 days, how many persons should he now remove from the job?
1. 40
2. 20
3. 30
4. 60
5. 25 men take 20 days to complete a certain job. 15 men left the work after some days. If the remaining work was completed in 37.5 days, after how many days did the 15 men leave the work?
1. 5
2. 35
3. 7
4. 6
6. In a competetive exam 10000 boys and 12000 girls appeared. If 26% of boys and 15% of girls could qualify, what is the overall % of students who could not qualify?
1. 80
2. 40
3. 60
4. 70
7. At an election contested by two candidates, the loser loses by a 20% margin equivalent to 20000 votes. If the polling was only 50% of the total eligible number, how many people were eligible for casting their votes?
1. 100000
2. 200000
3. 300000
4. 40000
8. Of the total amount received by Kiran 10% was spent on purchases and 5% of the remaining on transportation. If he is left with 1400, the initial amount was
1. 1637
2. 1637.5
3. 1637.43
4. 1638
9. In an exam, 60% failed in the GK test and 40% failed in quantitative aptitude test. 15% failed in both tests. What percent passed both tests?
1. 10%
2. 17%
3. 15%
4. 20%
10. A solution of 70 L of wine and water contains 10% water. How much water must be added to make the water 12.5% of the resulting solution?
1. 5
2. 3
3. 4
4. 2
11. The ratio of ages of a mother to her daughter today is 7:3. After 5 years, this ratio would be 2:1. How old was the mother at the time of birth of the daughter?
1. 18
2. 21
3. 22
4. 20
12. The average age of a class of 50 students is 24 years. If the average age of 10 of them is 22, while the average age of another 10 is 26, what is the average age of the remaining 30 students?
1. 22 years
2. 32
3. 24
4. 42
13. The average age of a class of 27 students is 10 years. The average increases by 1 if the teachers age is included. What is the age of the teacher?
1. 35
2. 36
3. 38
4. 37
14. The sum of ages of a father and his son is 56. If they both live until the son is as old as the father is at present, their ages together will then be 96. Find the present age of the father.
1. 48 years
2. 37
3. 28
4. 38
15. My brother is 6 years younger than me. If after 7 years our ages will be in the ratio 8:7, what is my brother’s age now?
1. 35
2. 25
3. 30
4. 40

## UPSC IAS :: Study Material :: General Studies :: Mental Ability #2

#### STREAMS AND BOATS

• Assume speed of a boat is $u$kmph and speed of a stream is $v$kmph
• Then speed of the boat downsteam $r = u + v$. Speed of the boat upstream $r = u - v$

#### PIPES AND CISTERNS

• If one pipe can fill a tank in $t_1$ minutes and another can fill it in $t_2$ minutes, then amount of water in one minute when both pipes are opened

$Q = \frac{1}{t_1} + \frac{1}{t_2}$

• Time taken by both pipes to fill the tank $t = \frac{t_1t_2}{t_1+t_2}$
• One pipe fills a cistern in $t_1$ minutes and another empties it in $t_2$ minutes. When both pipes are opened, amount of water in the cistern in one minute

$Q = \frac{1}{t_1} - \frac{1}{t_2}$

• Time taken to fill the cistern $t = \frac{t_1t_2}{t_2-t_1}$

#### PROFIT AND LOSS

• $Profit prct = \frac{profit}{cost price} 100$
• $Loss prct = \frac{loss}{cost price} 100$
• $Sale price = \frac{100 + profit}{100} CP$
• $Cost price = \frac{100}{100 + profit}SP$
• If a person sells $n_1$ part of his goods at profit $p_1$% and $n_2$ part of his goods at profit $p_2$% and the rest at a loss of $l$%, the net profit or loss is

$Q = \frac{n_1p_1 + n_2p_2 - (rest)l}{n_1+n_2+rest}$

#### REVIEW QUESTIONS

1. A boat travels downstream through a distance of 20 km in 1 hr, but takes 2 hrs to travel the same distance upstream. What is the ratio of the speed of the boat to the speed of the current?
1. 3:1
2. 1:3
3. 2:3
4. 3:2
2. A man can row 3 kmph in still water. If the river is running at 0.5 kmph, it takes him 1 hr to row to a place and back. How far is the place?
1. 1.4 km
2. 1.47 km
3. 1.46 km
4. 1.458 km
3. A man can row 15 kmph in still water. It takes him twice as long to row up the river as it takes him to row down the river. Find the speed of the stream.
1. 15 kmph
2. 10 kmph
3. 8 kmph
4. 5 kmph
4. A stream has a current 2 kmph. A boat goes 10 km upstream and back again to the starting point in 55 min. What is the speed of the boat in still water?
1. 22
2. 23
3. 25
4. 24
5. A boat takes 6 hrs to go upstream but return in just two hours. If the speed of the current is 2 kmph, what is the speed of the boat?
1. 4 kmph
2. 3 kmph
3. 5 kmph
4. 6 kmph
6. A pipe can fill a tank in 12 hrs. Due to a leak in the bottom it is filled in 15 hrs. If the tank is full, how much time will it take for the leak to empty it?
1. 30 hrs
2. 60 hrs
3. 1/30 hrs
4. 1/60 hrs
7. Two pipes can fill a cistern in 4 hrs and 6 hrs respectively. When the pipes are opened simulatenously it is found that due to leakage in the bottom, 36 extra min are taken for the cistern to be filled up. When the cistern is full, in how much time will the leak empty it?
1. 10
2. 1/10
3. 1/12
4. 12
8. Three pipes A, B, C can independently fill a tank in 12, 15 and 20 hrs respectively. They were operated simulateneously for 4 hrs and then pipe B is closed. In what extra time would the tank fill up now?
1. 1.6 hrs
2. 1.7 hrs
3. 1.5 hrs
4. 2 hrs
9. Two pipes A and B can fill a tank in 20 mins, 25 mins respectively. If both pipes are opened simultaneously, after how much time should B be closed so that the tank fills in 15 min?
1. 1 1/4
2. 3 1/4
3. 5 1/4
4. 6 1/4
10. If two pipes function simultaneously, a reservoir will be filled in 12 hrs. One pipe fills the reservoir 10 hrs faster than the other. How many hours does it take the second to fill the reservoir?
1. 30 hrs
2. 12
3. 40
4. 3
11. A man sold a table at a profit of 15%. Had he bought it at 10% less and sold for Rs 21 less, he would have gained 25%. What is the cost price of the table?
1. Rs 800
2. Rs 840
3. Rs 420
4. Rs 1000
12. A girl sold 4/5th of her articles at a gain of 50% and the rest at par. Find her net loss or gain percent.
1. 10
2. 15
3. 17
4. 24
13. A man buys 5kg of apples for Rs 30 and sells at 3 kg for Rs 45. To gain Rs 180, how many kg must he sell?
1. 15 kg
2. 20 kg
3. 25 kg
4. 18 kg
14. A man has a gain of 25% when he sells a table at a certain price. If he had sold it at double the price, what would the gain percent have been?
1. 120
2. 140
3. 130
4. 150
15. A merchant sells a cloth at a 10% loss. He uses a duplicate scale and gains 15%. Find the false length of the scale. (Assume original measure 100 cm)
1. 78.25 cm
2. 78 cm
3. 77.5 cm
4. 91.5 cm

## Notes

• Speed

$Speed = \frac{distance}{time}$

• Two objects A and B moving at speeds $v_{A}$ and $v_{B}$….

When A and B are moving in the same direction: $v_{R} = v_{A} - v_{B}$

When A and B are moving in opposite directions: $v_{R} = v_{A} + v_{B}$

• Two objects of length $l_{A}$ and $l_{B}$….time taken for the objects to cross each other

$t = \frac {l_{A} + l_{B}} {relative speed}$

• Two objects A and B moving in opposite directions with speeds $v_{A} and v_{B}$, after crossing each other take time $t_{A}$ and $t_{B}$ to reach their respective destinations. Then,

$\frac{v_{A}}{v_{B}} = \sqrt{\frac{t_{B}}{t_{A}}}$

• If a train covers a certain distance at average speed ‘u’, without stopping and covers the same distance at average speed ‘v’ with stopping, then the time spent at the stop ‘t’ is given by

$t = \frac{v-u}{max(u, v)}$

## Questions

1. Find the time taken by a train 150 m long running at 54 kmph in crossing an electric pole (assume width of the pole is negligible)
1. 10 sec
2. 15 sec
3. 20 sec
4. 17 sec
2. A man standing on a 150 m long bridge finds that a train crosses the bridge in 19 sec and crosses himself in 9 sec. What is the speed of the train?
1. 50 kmph
2. 45 kmph
3. 54 kmph
4. 8 kmph
3. Two trains travel in the same direction at speeds of 36 kmph and 54 kmph. The faster train overtakes the slower in 60 sec. What is the length of either train, if both trains are equally long?
1. 100 m
2. 160 m
3. 300 m
4. 150 m
4. A train travels a certain distance making three stops of 20 min each. The overall speed of the train comes to 40 kmph. Without stopping, the speed is 45 kmph. How much distance did the train travel?
1. 180 km
2. 360 km
3. 90 km
4. 300 km
5. Two stations A and B are 125 km apart. A train starts from A at 07:00 hrs and travels towards B at 10 kmph. Another train starts from B at 08:00 hrs and travels towards A at 15 kmph. At what time will they cross each other?
1. 10:00 hrs
2. 12 3/5 hrs
3. 11:00 hrs
4. 09:00 hrs