1.3 Marks

Three vessels having volumes in the ratio of 1 : 3 : 5 are full of a mixture ofwater and milk. In the first vessel, ratio of water and milk is 7 : 13, in second 9 :11 and in third 11 : 14. If the liquid in all the three vessels were mixed in abigger container, what is the resulting ratio of water and milk?

1) 3 : 4

2) 11 : 17

3) 39 : 51

4) 29 : 41

5) 7 : 9

Solution:

Let the three vessels contain 1 litre, 3 litres and 5 litres of water and milkmixture.

In the 1st vessel,

In the 2nd vessel,

In the 3rd vessel,

Ratio of water and milk in the final mixture

Hence, option 3.

2.3 Marks

Hari, Gopi and Sufi enter into a partnership. Gopi contributes one-fourth of thewhole capital while Hari contributes twice as much as Gopi and Sufi togethercontribute. If the profit at the end of the year is Rs. 4800, how much would eachreceive?

1) Rs. 3600, Rs. 1000 and Rs. 200

2) Rs. 3600, Rs. 800 and Rs. 400

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Section I

2) Rs. 3600, Rs. 800 and Rs. 400

3) Rs. 3200, Rs. 1200 and Rs. 400

4) Rs. 3000, Rs. 1000 and Rs. 800

5) Rs. 3200, Rs. 1000 and Rs. 600

Solution:

Let the total capital be 12x.

Then Gopi contributes 3x.

Hari contributes twice as much as Gopi and Sufi together contribute.

∴ If Sufi contributes y, then Hari contributes 2(3x + y)

∴ 3x + y + 2(3x + y) = 12x

∴ y = x

∴ Sufi contributes x and Hari contributes 8x.

∴ Gopi’s, Hari’s and Sufi’s contributions are in the ratio 3 : 8 : 1

∴ Their profits also would be in the same proportion and would be Rs. 1200,Rs. 3200 and Rs. 400.

Hence, option 3.

3.3 Marks

Sonia, Rita and Sushma enter into partnership. Sonia contributes one-third ofthe capital for half of the time, Rita contributes one-third of the capital for one-third of the time and Sushma contributes the remaining capital for the wholetime. How should they divide a profit of Rs. 1650?

1) Rs. 450, Rs. 300 and Rs. 900

2) Rs. 615, Rs. 410 and Rs. 615

3) Rs. 360, Rs. 600 and Rs. 680

4) Rs. 615, Rs. 650 and Rs. 375

5) Rs. 400, Rs. 650 and Rs. 690

Solution:

Sonia’s share : Rita’s share : Sushma’s share

= 3 : 2 : 6

Hence, option 1.

4.3 Marks

Mukesh and Anil enter into a partnership. Mukesh puts in Rs. 2400 and Anilputs in Rs. 3000. At the end of 6 months, Mukesh withdraws one-third of hiscapital and at the end of 8 months, Anil withdraws two-third of his capital. Sunilthen enters into the partnership with a capital of Rs. 6000. In what ratio will theprofit be divided, at the end of 12 months?

1) 2 : 3 : 2

2) 6 : 7 : 6

3) 3 : 4 : 6

4) 4 : 6 : 5

5) 6 : 5 : 6

Solution:

Mukesh’s share : Anil’s share : Sunil’s share

= 24000 : 28000 : 24000

= 6 : 7 : 6

Hence, option 2.

5.3 Marks

A father divided an amount of Rs. 82000 between his two sons aged 12 yearsand 14 years respectively and deposited their shares in a bond. If the interestrate is 25% compounded annually and if each received the same amount asthe other when he attained the age of 18 years, their shares are:

1) Rs. 34000 and Rs. 48000

2) Rs. 24000 and Rs. 58000

3) Rs. 32000 and Rs. 50000

4) Rs. 41000 each

5) None of these

Solution:

Let the principle amount for younger son = Rs. x and for elder son = Rs. y

It is given that they receive the same amount when they attain the age of 18years.

∴ The share of younger son is Rs. 32000 and of elder son is Rs. 50000.

Hence, option 3.

6.3 Marks

Mr. Daulatram decided to gift gold watches to three teams who participated ina competition in such a way that for every 15 watches team A gets, team Bshould get 9 watches and team C should get 6 watches. Now if there were1300 gold watches in total, how much did team A get?

1) 520

2) 580

3) 650

4) 710

5) 780

Solution:Ratio of magic watches given to team A, team B and team C = 15 : 9 : 6 = 5 :3 : 2

Hence, option 3.

7.3 Marks

Zebisko wants to give 550 roses to princess Alice, princess Cinderella andprincess Fiona such that if princess Cinderella gets 15 roses, princess Aliceshould get 24 roses and princess Fiona should get 11 roses. How many rosesdid each of the princess receive?

1) 250, 180 and 120

2) 264, 165 and 121

3) 280, 140 and 130

4) 224, 185 and 141

5) None of these

Solution:Ratio of roses received by Alice, Cinderella and Fiona = 24 : 15 : 11

Hence, option 2.

8.3 Marks

In the famous Palm Island, there are seven men for every nine women and twochildren for a man. How many women are there in the island if it has 504children?

1) 240

2) 275

3) 284

4) 206

5) 324

Solution:Ratio of men to women is 7 : 9

Ratio of children to men is 2 : 1 = 14 : 7

∴ The ratio of men : women : children = 7 : 9 : 14

∴ For 14 children there are 9 women.

Hence, option 5.

9.3 Marks

The monthly income of Sujay is equal to 4/5 times that of Rupesh whereas themonthly income of Dhanesh is equal to 6/11 times that of Rupesh. If themonthly income of Dhanesh is Rs. 6600, find the monthly income of Sujay.

1) Rs. 9680

2) Rs. 9240

3) Rs. 5460

4) Rs. 6800

5) Rs. 6230

Solution:

Let the monthly income of Rupesh, Sujay and Dhanesh are Rs. r, Rs. s and Rs.d respectively.

From the given information, we can write,

∴ s : r = 4 : 5 and d : r = 6 : 11

∴ s : r : d = 44 : 55 : 30

Let the total income of all the three be Rs. x

∴ x = Rs. 28380

Hence, option 1.10.

3 MarksA can is full of paint. Out of this, 5 litres are removed and substituted by athinning liquid. The process is repeated one more time. Now the ratio of paintto thinner is 49 : 15. What is the full capacity of the can?

1) 20 litres

2) 60 litres

3) 40 litres

4) 50 litres

Solution:

If there is P volume of pure liquid initially and in each operation, Q volume is

taken out and replaced by Q volume of another liquid, then at the end of n suchoperations, then

So, based on the information given in the problem, we get

Hence, option 3.

11.3 Marks

The cost of a certain diamond varies directly as the square of its weight. Once,this diamond broke into four pieces with weights in the ratio 1 : 2 : 3 : 4. Whenthe pieces were sold, the merchant got Rs. 70,000 less. Find the original priceof the diamond.

1) Rs. 1.4 lakhs

2) Rs. 2.0 lakhs

3) Rs. 1.0 lakh

4) Rs. 2.1 lakhs

Solution:Let the total weight be 10 grams, then the given diamond would be broken into1, 2, 3 and 4 grams.

Also, let the cost and weight be denoted by c and w respectively, then C α w2

⇒ c = kw2 … (k is the proportionality constant.)

Total cost = C = k (102) = 100k

Cost of broken pieces is as follows:

(i) C1 = k(12) = k

(i) C1 = k(12) = k

(ii) C2 = k(22) = 4k

(iii) C3 = k(32) = 9k

(iv) C4 = k(42) = 16k

Thus, total cost of the broken pieces = C1 + C2 + C3 + C4 = 30k

Loss in value = 100k – 30k = 70k = Rs. 70000.

Hence, option 3.12.

3 MarksA student gets an aggregate of 60% marks in the five subjects in the ratio 10 :9 : 8 : 7 : 6. If the passing marks are 50% of the maximum marks and eachsubject has the same maximum marks, in how many subjects did he pass theexam?

1) 2

2) 3

3) 4

4) 5

Solution:Let the maximum marks in each of the 5 subjects = 100.

Also, let the marks in each subject be given by: 10x, 9x, 8x, 7x and 6xrespectively.

Aggregate = 60% of 500 = 300 = (10 + 9 + 8 + 7 + 6)x = 40x

Thus, marks in each subject are as listed below:

Subject 1 = 10 × 7.5 = 75 > 50

Subject 2 = 9 × 7.5 = 67.5 > 50

Subject 3 = 8 × 7.5 = 60 > 50

Subject 4 = 7 × 7.5 = 52.5 > 50

Subject 5 = 6 × 7.5 = 45 ≯ 50

Thus, it can be seen that in 4 subjects, he gets more than 50% marks.

Hence, option 3.

13.3 Marks

I have a number of one rupee coins, fifty paise coins and twenty-five paisecoins. The number of coins are in the ratio 2.5 : 3 : 4. If the total amount withme is Rs. 210, then find the number of one rupee coins.

1) 90

2) 85

3) 100

4) 105

Solution:

Let the number of 1 Re. coins, 50 p coins and 25 p coins be 5x, 6x and 8xrespectively.

The total value = (5x × 1) + (6x × 0.5) + (8x × 0.25) = 210

i.e. 5x + 3x + 2x = 210

∴ 10x = 210

∴ x = 21

i.e. 5x = 105

Hence, the number of 1 Re. coins = 105.

Hence, option 4.

Group Question

Answer the following questions based on the information given below.

Krishna distributed 10 acres of land to Gopal and Ram who paid him the totalamount in the ratio 2 : 3. Gopal invested a further Rs. 2 lakhs in the land andplanted coconut and lemon trees in the ratio 5 : 1 on equal area of land. There was,a total of 100 lemon trees. The cost of one coconut was Rs. 5. The crop took 7years to mature and when the crop was reaped in 1997, the total revenuegenerated was 25% of the total amount put in by Gopal and Ram together. Therevenue generated from the coconut and lemon trees was in the ratio 3 : 2 and itwas shared equally by Gopal and Ram as the initial amount spent by them wereequal.

14.3 Marks

What was the total output of coconuts?

1) 24,000

2) 36,000

3) 18,000

3) 18,000

4) 48,000

Solution:

Let the initial investment of Gopal and Ram towards the land be 2x and

3x lakhs.

The total amount invested by Gopal and Ram is equal.

So, 2x + 2 = 3x ⇒ x = 2

Gopal’s investment towards the land = Rs. 4 lakhs.

Ram’s investment towards the land = Rs. 6 lakhs.

Gopal’s investment towards the crop = Rs. 2 lakhs.

∴ Total investment by Ram and Gopal = Rs. 12 lakhs.

Total revenue generated in 1997 = 25% of 12 lakhs = Rs. 3 lakhs.

Gopal and Ram had equal share in the revenue.

∴ Each received Rs. 1.5 lakhs.

It is given that the value of a coconut is Rs. 5

Hence, option 2.

15.3 Marks

What was the value of output per acre of lemon trees planted? (inlakh/acre)

1) 0.24

2) 2.4

3) 24

4) Indeterminate

Solution:

Hence, option 1.

16.3 Marks

What is the amount received by Gopal in 1997?

1) Rs.1.5 lakh

2) Rs.3.0 lakh

3) Rs.6 lakh

4) None of these

Solution:The amount received by Gopal = Rs. 1.5 lakhs.

Hence, option 1.

17.3 Marks

What was the value of output per tree for coconuts?

1) Rs. 36

2) Rs. 360

3) Rs. 3600

4) Rs. 240

Solution:

The number of coconut trees is 5 times that of the number of lemontrees.

∴ The number of coconut trees = 500

Hence, option 2.

18.3 Marks

What was the ratio of yields per acre of land for coconuts and lemons (interms of the number of lemons and coconuts)?

1) 3 : 2

2) 2 : 3

3) 1 : 1

4) Indeterminate

Solution:Since we do not know the value of each lemon, we cannot determine the

Since we do not know the value of each lemon, we cannot determine thenumber of lemons.

Hence, option 4.

19.3 Marks

I used 6 litres of oil paint to paint a map of India 6 meters high. How manylitres of paint would I need to paint a proportionally scaled map 18 metreshigh?

1) 54

2) 18

3) 30

4) Indeterminate

Solution:

Let the width of the map whose height is 6 m be 1 m.

If the proportionally scaled up map has a height of 18 m, then its width will be 3m.

The amount of oil paint spent for an area of 6 m2 is 6 litres.

∴ The amount of oil paint spent for an area of 3 × 18 i.e. 54 m2 is 54 litres.

Hence, option 1.

20.3 Marks The value of each of a set of coins varies as the square of its diameter, if itsthickness remains constant, and it varies as the thickness, if the diameterremains constant. If the diameter of two coins are in the ratio of 4 : 3, whatshould the ratio of their thickness’ be if the value of the first is 4 times that ofthe second?

1) 16 : 9

2) 9 : 4

3) 9 : 16

4) 4 : 9

Solution:

Let the value, diameter and thickness be denoted by v, d and t respectively,then

Hence, option 2.

21.3 Marks

One year’s payment to the servant is Rs. 90 plus one turban. The servantleaves after 9 months and receives Rs. 65 and a turban. Find the price of theturban.

1) Rs. 10

2) Rs. 15

3) Rs. 7.5

4) Indeterminate

Solution:

Let the salary per month and the price of the turban be denoted by m and trespectively.

For 1 year: 12m = t + 90 … (i)

For 9 months: 9m = t + 65 … (ii)

Subtracting (ii) from (i),

3m = 90 – 65 = 25

Using this value in (ii), we get,

⇒ t = 75 – 65 = Rs. 10

Hence, option 1.

22.3 Marks

The speed of a railway engine is 42 km per hour when no compartment isattached, and the reduction in speed is directly proportional to the square rootof the number of compartments attached. If the speed of the train carried bythis engine is 24 km per hour when 9 compartments are attached, themaximum number of compartments that can be carried by the engine is :

1) 49

2) 48

2) 48

3) 46

4) 47

Solution:

Let the number of compartments and the reduction in the speed of the train bedenoted by c and r respectively.

It is known that:

When c = 9, the reduction in speed = c = 42 – 24 = 18 km/hr.

Maximum speed reduction possible = r = 42 km/hr.

Hence, the maximum number of compartments that can be added such thatthe speed does not become zero = 49 – 1 = 48.

Hence, option 2.

23.3 Marks

Total expenses of a boarding house are partly fixed and partly varying linearlywith the number of boarders. The average expense per boarder is Rs. 700when there are 25 boarders and Rs. 600 when there are 50 boarders. What isthe average expense per boarder when there are 100 boarders?

1) 550

2) 560

3) 540

4) 500

Solution:

Let the total expenses, fixed charge, number of boarders and the variablecharge per boarder be denoted by E, F, V and n respectively.

∴ E = F + n × V

When n = 25: E = 700 × 25 = 17500 = F + 25V … (i)

When n = 25: E = 700 × 25 = 17500 = F + 25V … (i)

When n = 50: E = 600 × 50 = 30000 = F + 50V … (ii)

Solving equations (i) and (ii),

25V = 12500

Substituting in (i), we get

17500 = F + 25 × 500

∴ F = 17500 – 12500 = 5000

When n = 100: E = F + 100V = 5000 + 100 × 500 = 55,000

Hence, option 1.

24.3 Marks

A man buys spirit at Rs. 60 per litre, adds water to it and then sells it at Rs. 75per litre. What is the ratio of spirit to water if his profit in the deal is 37.5%?

1) 9 : 1

2) 10 : 1

3) 11 : 1

4) None of these

Solution:

Cost price of spirit = Rs. 60 per litre

Selling price of spirit mixture = Rs. 75 per litre

Profit percentage = 37.5%

Hence, option 2.

25.3 Marks

Two liquids A and B are in the ratio 5 : 1 in container 1 and in the ratio 1 : 3 incontainer 2. In what ratio should the contents of the two containers be mixedso as to obtain a mixture of A and B in the ratio 1 : 1?

1) 2 : 3

2) 4 : 3

3) 3 : 2

4) 3 : 4

Solution:

By rule of alligation:

Hence, option 4.26.

3 MarksThere are two containers: the first contains 500 ml of alcohol, while the secondcontains 500 ml of water. Three cups of alcohol from the first container areremoved and mixed well in the second container. Then three cups of thismixture are removed and mixed in the first container. Let A denote theproportion of water in the first container and B denote the proportion of alcoholin the second container. Then _____.

1) A > B

2) A < B

3) A = B

4) Indeterminate

Solution:

Initially, container 1 has 500 ml of alcohol and container 2 has 500 ml of waterin it. Let the cup size be 100 ml.

Stage 1: Initial Condition

Stage 2: After pouring 3 cups from container 1 to 2

Stage 3: After pouring 3 cups from container 2 to 1

After stage 2, a total of 300 + 500 = 800 ml mixture of alcohol and water ispresent

In 300 ml of this mixture which is transferred to container 1,

From above table, it can be seen that the proportion of water in container 1 isexactly equal to the proportion of alcohol in container 2

i.e. A = B

Hence, option 3.

Group Question

Answer the following questions based on the information given below.

The following table presents the sweetness of different items relative to sucrose,whose sweetness is taken to be 1.00

27.3 Marks

What is the maximum amount of sucrose (to the nearest gram) that can beadded to one-gram of saccharin to make a mixture that will be at least100 times as sweet as glucose?

1) 72) 8

3) 9

4) 100

Solution:

x g of sucrose gives x × 1.0 = x units,

1 g of saccharin gives 1 × 675.0 = 675 units

In all, (x + 1) g of the mixture gives (x + 675) units

Thus, the maximum amount that can be added = 8 g of sucrose

Hence, option 2.

28.3 Marks

Approximately how many times sweeter than sucrose is a mixture consistingof glucose, sucrose and fructose in the ratio of 1 : 2 : 3 ?

1) 1.3

2) 1

3) 0.6

4) 2.3

Solution:

Let the amount of glucose, sucrose and fructose be equal to 1, 2 and 3 grespectively

Hence, option 1.

29.3 Marks

A, B and C individually can finish a work in 6, 8 and 15 hours respectively.They started the work together and completing the work got Rs. 94.60 in all.When they divide the money among themselves, A, B and C will respectivelyget (in Rs.) …after

1) 44, 33, 17.60

2) 43, 27, 24.40

3) 45, 30, 19.60

4) 42, 28, 24.60

Solution:

The wages will be divided among A, B and C in the ratio or the rate of work ofA, B and C

So, let us assume that the total work is 120 units

So, Rs. 94.60 should be divided among A, B and C in the ratio 20 : 15 : 8

Hence, option 1.

30.3 Marks

Three machines, A, B and C can be used to produce a product. Machine Awill take 60 hours to produce a million units. Machine B is twice as fast asMachine A. Machine C will take the same amount of time to produce a millionunits as A and B running together. How much time will be required to producea million units if all the three machines are used simultaneously?

1) 12 hours

2) 10 hours

2) 10 hours

3) 8 hours

4) 6 hours

Solution:Machine A alone takes 60 hrs to produce a million units

Machine B alone takes 30 hrs to produce a million units

∴ Machine C alone takes 20 hrs to produce a million units

Hence, option 2.

31.3 Marks

If 200 soldiers eat 10 tonnes of food in 200 days, how much will 20 soldiers eat in20 days?

1) 1 ton

2) 10 kg

3) 100 kg

4) 50 kg

Solution:The amount of food consumed by 200 soldiers in 200 days = 10 tonnes

= 10 × 1000 = 10000 kg

Hence, option 3.

32.3 Marks

A supply of water lasts for 150 days, if 7.5 gallons leak out every day, but onlyfor 100 days if 15 gallons leak out daily. What is the total quantity of water inthe supply?

1) 2250 gallons

2) 1125 gallons

3) 3350 gallons

4) 1250 gallons

Solution:

Let the every day supply of water be x gallons.

(x +7.5) × 150 = (x + 15) × 100

∴ (x + 7.5) × 3 = (x + 15) × 2

∴ 3x + 22.5 = 2x + 30

∴ x = 7.5

So, the total quantity of water supply = (7.5 + 7.5) × 150 = 2250 gallons.

Hence, option 1.

33.3 Marks

Anand and Bharat can cut 5 kg of wood in 20 minutes. Bharat and Chandracan cut 5 kg of wood in 40 minutes. Chandra and Anand can cut 5 kg of woodin 30 minutes. How much time will Chandra alone take to cut 5 kg of wood?

1) 120 minutes

2) 48 minutes

3) 240 minutes

4)

Solution:Based on the given information, we have

Anand and Bharat can cut 30 kg of wood in 2 hrs.

∴ a + b = 30 … (i)

Bharat and Chandra can cut 15 kg of wood in 2 hrs.

∴ b + c = 15 … (ii)

Chandra and Anand can cut 20 kg of wood in 2 hrs.

∴ c + a = 20 … (iii)

Adding (ii) and (iii),

a + b + 2c = 35 … (iv)

Subtracting (i) from (iv),

c = 2.5

i.e. Chandra can cut 2.5 kg of wood in 2 hrs.

So, Chandra can cut 5 kg of wood in 4 hrs, i.e. 240 minutes.

Hence, option 3.

34.3 Marks

A group of workers was put on a job. From the second day onwards, oneworker was withdrawn each day. The job was finished when the last workerwas withdrawn. Had no worker been withdrawn at any stage, the group wouldhave finished the job in two thirds the time. How many workers were there inthe group?

1) 2

2) 3

3) 5

4) 10

Solution:

Let the number of workers be n

The number of workers decreases by 1 everyday

∴ On the second day, the number of workers = (n − 1)

On the third day, the number of workers = (n − 2)

Similarly, on the nth day, the number of workers = [n − (n − 1)]= 1 worker

So, the job is completed in n days

Let the amount of work done by each worker each day be 1 unit

∴ Work done on the first day = n units

Work done on the second day = (n − 1) units

Work done on the nth day = 1 unit

If all the n workers worked together,

From (i) and (ii),

∴ n = 3

Hence, option 2.

35.3 Marks

One man can do as much work in one day as a woman can do in 2 days. Achild does one-third the work done by a woman in a day. If an estate-ownerhires 39 pairs of hands - men, women and children in the ratio 6 : 5 : 2 andpays them a total of Rs. 1,113 at the end of the day’s work. What must thedaily wages of a child be, if the wages are proportional to the amount of workdone?

1) Rs. 14

2) Rs. 5

3) Rs. 20

4) Rs. 7

Solution:

Every day, let the amount of work done by a man, a woman and a child be 6units, 3 units and 1 unit respectively

In all, 39 people are hired with the men, women and children ratio being 6: 5:2,

∴ In one day, the amount of work done = 18 × 6 + 15 × 3 + 6 × 1 = 159 units

i.e. The wages paid for 159 units of work = Rs 1113

Hence, option 4.

36.3 Marks

A water tank has three taps A, B and C. A fills 4 buckets in 24 minutes, B fills8 buckets in 1 hour and C fills 2 buckets in 20 minutes. If all the taps areopened together, a full tank is emptied in 2 hours. If a bucket can hold 5 litres

opened together, a full tank is emptied in 2 hours. If a bucket can hold 5 litresof water, what is the capacity of the tank?

1) 120 litres

2) 240 litres

3) 180 litres

4) 60 litres

Solution:

Tap A can fill 4 buckets in 24 minutes

Tap B can fill 8 buckets in 60 minutes

Tap C can fill 2 buckets in 20 minutes

Thus, in 2 hrs, taps A, B and C can fill 20 + 16 + 12 = 48 buckets = 48 × 5 =240 litres

Hence, option 2.

37.3 Marks

There is a leak in the bottom of the tank. This leak can empty a full tank in 8hours. When the tank is full, a tap is opened into the tank which admits 6 litresper hour and the tank is now emptied in 12 hours. What is the capacity of thetank?

1) 28.8 litres

2) 36 litres

3) 144 litres

4) Indeterminate

Solution:

Let the capacity of the tank be 24 units

The rate at which the leak, leaks the water = 3 units/hr

When both the tap and the leak were open, the tank is emptied in 12 hours

In 12 hours, the leak would have emptied 36 units of water

In 12 hours, the tap would have filled in 12 × 6 = 72 litres of water

So, 24 units + 72 litres = 36 units

∴ 12 units of water = 72 litres

The capacity of the tank is 24 units, i.e. 144 litres

Hence, option 3.

38.3 Marks

A company has a job to prepare a certain number of cans and there are threemachines A, B and C for this job. A can complete the job in 3 days, B cancomplete the job in 4 days and C can complete the job in 6 days. How manydays will the company take to complete the job if all the machines are usedsimultaneously?

1) 4 days

2)

3) 3 days

4) 12 days

Solution:

Let the total number of cans that needs to be prepared be 12c.

Thus, in 1 day, A, B and C can together prepare 4c + 3c + 2c = 9c cans.

Hence, option 2.

39.3 Marks

Two friends A and B can finish a certain piece of work in 12 and 18 daysrespectively, while working alone. But, when they work together, they starttalking and their efficiency comes down by 20%. If one of them works alone fora few days, then the second works alone for a few days and then the both worktogether, the work gets over in 13 days. If the number of days for which A andB work alone is a prime number then which of the following can bedetermined?

1) The number of days for which A worked alone

2) The number of days for which B worked alone

3) The number of days for which A and B worked together

4) All of these

5) None of these

Solution:

Let A and B work on the project alone for x and y days respectively.

∴ They work together for (13 – x – y) days.

A can finish the work alone in 12 days.

B can finish the work alone in 18 days.

When they work together, their efficiency is reduced by 20%, so they work at80% of their efficiency.

∴ 52 – x – 2y = 36

∴ x + 2y = 16

x and y can take values (2, 7), (4, 6), (6, 5), (8, 4), (10, 3), (12, 2) and (14, 1).

The maximum value that x + y can take is 13. Also, (13 − x − y) > 0.

∴ x and y can take values (2, 7), (4, 6), (6, 5) and (8, 4).

By the condition given in the question, x + y should be a prime number,

By the condition given in the question, x + y should be a prime number,

therefore, x = 6 and y = 5.

∴ A works alone for 6 days, B works alone for 5 days and A and B worktogether for 2 days.

Hence, option 4.

40.3 Marks

Two painters, A and B are under contract to paint a certain wall every day.Their rate of painting is constant and never varies, even from day to day. Onday 1, they notice that they can paint the wall together in 'x' minutes. The nextday, A does not turn up and B works alone. B notices that he takes 5 minutesmore than they had taken the previous day to paint the wall. On day 3, B doesnot turn up. A calculates the time he took to paint the wall alone and tells B thathe had taken 40 minutes more than B had taken on day 2. What is the value ofx?

1) 15

2) 25

3) 35

4) 45

5) 55

Solution:

B alone takes (x + 5) minutes to paint the wall.

A alone takes (x + 5 + 40) = (x + 45) minutes to paint the wall.

However, we also know that A and B together paint the wall in x minutes.

Hence, option 1.