Explanation / Important formulas:
Direct Proportion:
 Two quantities are said to be directly proportional, if on the increase (or decrease) of the one, the other increases (or decreases) to the same extent.
Example: Cost is directly proportional to the number of articles. (More Articles, More Cost)
Indirect Proportion:
Two quantities are said to be indirectly proportional, if on the increase of the one, the other decreases to the same extent and viceversa.
Example: The time taken by a car is covering a certain distance is inversely proportional to the speed of the car. (More speed, less is the time taken to cover a distance).
Note: In solving problems by chain rule, we compare every item with the term to be found out
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Test  Chain Rule
Chain rule  Question and Answers
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Question 1

A

20 hours

B

60 hours

C

80 hours

D

70 hours

Less man, more hours (indirect proportion)
15:36:: 25: x
15 * x = 36 * 25
X= (36 *25) / 15 = 60 hours
Hence 15 men’s can do it in 60 hours
Question 2

A

Rs. 2520

B

Rs. 3065

C

Rs. 7658

D

Rs 2765

More men, more wages( Direct proportion)
Less days, Less wages ( Direct proportion)
Men = 6:9 ::2100 😡
Days = 15:12 :: 2100 : x
(6*15*x) = (9* 12* 2100)
X = [(9*12*2100) / (6* 15)]
X = 420 * 6
X= 2520
Hence, the required wages are Rs. 2520
Question 3

A

16 days

B

12 days

C

22 days

D

26 days

(8 children + 12 men) = 16 men
Less men, more days( Indirect proportion )
12 : 16 :: 9: x
12x= (16*9)
12x = 144
X= 144/12
X= 12 days
Question 4

A

1500

B

1600

C

1800

D

1964

More machines, more bottles(direct proportion)
Machines = 6: 10 ::270 : x
Time(in minutes) = 1:4 :: 270 : x
6 * 1* x= 10* 4* 270
X= (10*4*270) / (6*1)
= 20 * 90
X = 1800
Question 5

A

16

B

17
Hint:

C

19

D

64

More mat weavers, more mats(Direct proportion)
More days, more mats ( Direct proportion)
Mat weavers = 4:8 :: 4 : x
Days = 4:8 :: 4:x
4*4*x=8*8*4
X= (8*8*4) / (4*4)
X= 2*2*4
X= 16
Question 6

A

87

B

56

C

48

D

65

Less weight, less cost (Direct proportion)
250 : 200 :: 60 : x
250 * x = (200 * 60)
X= (200 * 60) / 250
X= 8*6
X= 48
Question 7

A

100 patients

B

120 patients

C

95 patients

D

140 patients

= More liters, more patients (Direct proportion)
=Less days, more patients (Indirect proportion)
Litres = 1350: 1710: : 70 : x
Days = 20: 8::70: x
X= (1710*30*70) / (1350 * 28)
= 19 * 5
95 patients
Question 8

A

13 days

B

27 days

C

17 days

D

18 days

Less persons, More days(Indirect proportion)
More working hours per day, Less days(Indirect proportion)
Person = 30:39 ::12 : x
Working hours/days = 6: 5 ::12 : x
(30*6*x)= (39*5*12)
X= (39*5*12) / (30*6)
X= 13 days
Question 9

A

25

B

16

C

20

D

12

Or = [20*20/25] = 16
Required number of days = 16
Question 10

A

46

B

32

C

35

D

30

14*12* x = 16* 7 * 48 or x= 32
Question 11

A

6 days

B

12 days

C

3 days

D

4 days

2x *1 * y = x* ½ *12 or y = 3
Question 12

A

2000

B

1200

C

600

D

12000

Number of wheel moves in 1 hour = 10 * 60 = 600
Distance moved = 600 * 200 = 12000
Question 13

A

5 days

B

4 days

C

10 days

D

20 days

Or 8x = 10 * 8
Or x= 10
Required number of days 10
Question 14

A

18

B

24

C

21

D

28

140 * 30 * x = 100 * 18 * 42
Or x= 18
Question 15

A

300

B

400

C

500

D

250

Or x= 96 * 1384.08 / 438 = 303.36
Approx cost = Rs. 300
Question 16

A

1260

B

920

C

1320

D

1380

12*6 * 24*x = 18 * 8* 36* 460
Or x= 1380
Question 17

A

Rs. 3000

B

Rs. 4000

C

Rs. 3500

D

Rs. 2500

357: 49 * 12:: 1517.25 : x
Or 375* x= 49 * 12 * 1517.25
49* 12 * 157.25 / 357 = 2499
Approximate cost = Rs. 2500
Question 18

A

486

B

546

C

626

D

686

Then, more toys, more cost(Direct proportion)
15: 35:: 234 : x
(15 x x) = (35 x 234)
X= (35 X 234) / 15 = 546
Hence, the cost of 35 toys is Rs. 546
Question 19

A

40 hrs

B

50 hrs

C

60 hrs

D

70 hrs

Then, less men, more hours(Indirect proportion)
15:36 :: 25 : x
(15 X x) = (36 X 25)
(36 X 25)/15 = 60
Hence, 15 men can do it in 60 hours
Question 20

A

2200

B

2520

C

2720

D

3000

More men, more wages (Direct proportion)
Less days, less wages (Direct proportion)
Men 6:9:: 2100: x
Days 15:12
Therefore (16 x 15 x X) = (9 x 12 x 2100)
X= (9 x 12 x 2100) / (6 x 15) = 2520
Hence the required wages are Rs. 2520
Question 21

A

44 m

B

49 m

C

54 m

D

59 m

More men, more length built (Direct proportion)
Less days, less length built (Direct proportion)
Men 20:35
Days 6:3:: 56: x
Therefore (20 x 6 x X) = (35 x 3x 56)
X= (35 x 3x 56) / 120 = 49
Hence, the required length is 49 m
Question 22

A

19

B

12

C

15

D

18

More men, less days (Indirect proportion)
Less hours per day, more days (Indirect proportion)
Men 18:15
Hours per day 8:9 :: 16 : x
(18 x 8 x X) = (15 x 9 x 16)
X = (44 x 15)144 = 15
Hence, the required number of days = 15
Question 23

A

1

B

3

C

7

D

14

Let the required number days be x
Less spider, More days (Indirect proportion)
Less webs, less days (Direct Proportion)
1 x7 x X = 7 x 1 x 7
X = 7
Question 24

A

22 metric tonnes

B

24 metric tonnes

C

26 metric tonnes

D

28 metric tonnes

Then, 4 engines of latter type consume 1 unit in 1 hour
Therefore 1 engine of former type consumes (1/3) unit in 1 hour
1 engine of latter type consumes (1/4) unit in 1 hour
Let the required consumption of coal be x units
Less engines, less coal consumed (Direct proportion)
More working hours, more coal consumed (Direct proportion)
Less rate of consumption, less coal consumed(Direct proportion)
Number of engines 9:8
Working hours 8: 13:: 24 : x
Rate of consumption (1/3) : (1/4)
[9x8x(1/3)x X] = (8 x 13 x (1/4) x 24)
24x = 624
X= 26
Hence, the required consumption of coal= 26metric tonnes
Question 25

A

1500

B

1700

C

1900

D

2100

3300 men taking 850 gms per head have provisions for(32  7) or 25 days
How many men taking 825 gms each have provisions for 17 days?
Less ration per head, more men(Indirect proportion)
Less days, more men(Indirect proportion)
Ratio 825: 850
Days 17 :25 ::3300 : x
(825 x 17 x X) = 850 x 25 x 3300
Or x= (850 x 25 x 3300) / (825 x 17) = 5000
Strength of reinforcement = (5500  3300) = 1700
Question 26

A

12

B

18

C

22

D

24

Less man, more days (Indirect proportion)
27 :36 ::18 : x 27 X x = 36 x
X =36 x 18/27
X= 24
Question 27

A

48 paise

B

54 paise

C

56 paise

D

72 paise

Less weight, less cost (Direct proportion) 250:200::60: x 250 x
X= (200 x 60)
X= (200 x 60) /250
X= 48
Question 28

A

Rs. (xy/d)

B

Rs. (xd)

C

Rs.(yd)

D

Rs. (yd/x)

Cost of 1 meter = Rs.(d/x)
Cost of y meters = Rs.[(d/x).y] = Rs.(yd/x)
Question 29

A

10

B

13

C

14

D

15

Less persons, more days (Indirect proportion)
More working hours per day, less days (Indirect proportion)
30 x 6 x X = 39 x 5 x12
X= 13
Question 30

A

50

B

40

C

30

D

20

Now , 200 children = 120 men
50 children = 30 men
Question 31

A

3

B

5

C

6

D

9

This is 5 days behind schedule
Question 32

A

Rs.(b*x)

B

Rs. {(b*x)/a}

C

Rs.{(a*b)/x}

D

none of these

Therefore, each meter of cloth costs Rs. (x/a).
The cost of ‘b’ meters of cloth = Rs. b*(x/a) = Rs. {(b*x)/a}
Question 33

A

20

B

36

C

25

D

42

Now 50 more personnel are added in the camp. hence, the food reserve for 300 personnel will last for:
250:300:: x:30 ...(It is an indirect proportion as less men means more days)
X= (250*30)/300
X= 25 days
Question 34

A

42

B

54

C

90

D

70

Now, 70 adults = 90 children
7 adult = 9 children
Therefore, 42 adults = 54 children
Hence, the meal can be catered to 54 children
Question 35

A

10

B

12

C

6

D

16

(3/8):6:: (5/8):x
X= 6*(5/8) / (3/8)
X= 10
Question 36

A

12

B

18

C

6

D

1

Hence it’s a direct proportion.
(100*25*30): (75*20*50):: 12 😡
75000:75000:: 12:x
X = 12
Question 37

A

20

B

30

C

24

D

16

Hence, x workers can complete the work in 25 days and (x9) workers can complete the work in 40 days
Now, the number of men and the amount of work are in direct proportion and number of men and number of days in indirect proportion
X*25 = (x 9)*40
25x=40x – 360
15x=360
X=24
Question 38

A

12

B

24

C

6

D

18

10*24*W1/2 = 20* D2*W1...... (W2=W1 / 2)
D2 = 6 days
Question 39

A

4.5

B

6

C

12

D

4

Hence, 4 men = 12 women
4 men and 4 women = 12 + 4 = 16 women
M1 D1 W2= M2 D2 W1
M1 D1 = M2 D2 ....(W1 = W2)
6*12 = 16*D2
D2 = 4.5 day
Question 40

A

36

B

90

C

40

D

50

Now, let x workers be employed to work 8 hours a day for the rest 18 days to complete 4/9th of the work
M1 D1 T1 W2 = M2 D2 T2 W1
54*30*10*4/9 = x*18* 8*5/9
X= 90
Hence, 36 more workers should be employed
Question 41

A

Rs. (xd/y)

B

Rs. xd

C

Rs.(yd/x)

D

Rs. yd

Cost of 1 meter of wire= Rs.(d/x)
Cost of y meter of wire = Rs. (y*dx) = Rs. (yd/x)
Question 42

A

50

B

30

C

40

D

10

Meal for 1 child = Meal for 120/200 men
Meal for 150 children = Meal for 120 x 150/200 men = Meal for 90 men
Total meal available = meal for 120 men
Remaining meal = Meal for 120 men – Meal for 90 men = Meal for 30 men
Question 43

A

65 paise

B

70 paise

C

52 paise

D

48 paise

Cost of ¼ kg potato = 60 paise
Cost of 250gm potato = 60 paise
More quantity, more prise
Hence we can write as
Quality(200:250)::x:60
200 x 60 = 250 x X
4 x 60 = 5 x X
4 x 12 = X
X= 48
Question 44

A

15

B

12

C

21

D

9

More cogs, less revolutions (indirect proportion)
Hence we can write as
Cogs(6:14)::x:21
6 x 21 = 14 x X
6 x 3 = 2 x X
3 x 3 = X
X= 9
Question 45

A

205

B

200

C

195

D

190

More cloth, more time,(direct proportion)
Hence we can write as
Cloth 0.128:25}:: 1: x
0.128x= 25
X= 25/128 = 25000/128
= 3125/16 = 195
Question 46

A

49

B

36

C

50

D

41

1 cow = 21/15 goats
35 cows
= 21 x 35/15 goats = 21 x 7/3 goats = 7 x 7 goats
= 49 goats
Question 47

A

12.5 m

B

10.5 m

C

14

D

12

More shadow length, more height
Hence we can write as
Shadow length {40.25:28.75}:: 17.5 : x
40.25x X = 28.75 x 17.5
X= 28.75 x17.5/40.25
2875 x 7/1610
2875/230
575/46
12.5
Question 48

A

44

B

42

C

36

D

52

Hence, after 10 days, the remaining food is sufficient for 150 men for 35 days
Remaining men after 10 days =150 – 25 = 125
Assume that after 10 days, the remaining food is sufficient for 125 men for x days
More men, less days
Men {150:125}:: x:35
150 x 35 = 125x=> 6x35= 5x=>x = 6x7 = 42
The remaining food is sufficient for 125 men for 42 days
Question 49

A

7

B

21

C

14

D

28

As more pumps are used the time to fill the tank will decrease
4:6:: 14: x
Or 4x = 6*14
Or x= 6*14/4 = 21
Extra pumps needed = (2114) = 7
Question 50

A

1

B

40

C

26

D

20

Less cows, more days
Less bags, less days
Hence, we can write as
[Cow 1:40 bags 40:1]
1 x 40 * x = 40*1 * 40
X= 40
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