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41. If 12 carpenters, working 6 hours a day, can make 460 chairs in 24 days, how many chairs will 18 carpenters make in 36 days, each working 8 hours a day ?

A. 1260

B. 1320

C. 920

D. 1380

Discuss

Solution:

Let the required number of chairs be x
Then, More carpenters, More chairs (Direct proportion)
More hours per day, More chairs (Direct proportion)
More days, More chairs (Direct proportion)

\begin{matrix} Carpenters 12 : 18 \\ Hours per day 6 : 8 \\ Days 24 : 36 \end{matrix}} :: 460 : x

\begin{aligned} ∴ 12 \times 6 \times 24 \times x = 18 \times 8 \times 36 \times 460 \\ \Leftrightarrow x = \frac{(18 \times 8 \times 36 \times 460)}{(12 \times 6 \times 24)} \\ \Leftrightarrow x = 1380 \end{aligned}

∴ Required number of chairs = 1380

42. If 5 spiders can catch five files in five minutes, how many files can hundred spiders catch in 100 minutes ?

A. 100

B. 500

C. 1000

D. 2000

Discuss

Solution:

Let the required number of chairs be x.
Then, More spiders, More flies (Direct proportion)
More time, More flies (Direct proportion)

\begin{matrix} Spiders 5 : 100 \\ Minutes 5 : 100 \end{matrix}} :: 5 : x

\begin{aligned} ∴ 5 \times 5 \times x = 100 \times 100 \times 5 \\ \Leftrightarrow x = \frac{(100 \times 100 \times 5)}{(5 \times 5)} \\ \Leftrightarrow x = 2000 \end{aligned}

43. 2 persons working 2 hours a day assemble 2 machine in 2 days. The number of machines assemble by 6 persons working 6 hours a day in 6 day is ?

A. 6

B. 18

C. 27

D. 54

Discuss

Solution:

Let the required number of machine be x
More persons, More machines (Direct proportion)
More working working hours, More machines (Direct proportion)
More days, More machines (Direct proportion)

\begin{matrix} Persons 2 : 6 \\ Working hours 2 : 6 \\ Days 2 : 6 \end{matrix}} :: 2 : x

\begin{aligned} ∴ 2 \times 2 \times 2 \times x = 6 \times 6 \times 6 \times 2 \\ \Leftrightarrow x = \frac{(6 \times 6 \times 6 \times 2)}{(2 \times 2 \times 2)} \\ \Leftrightarrow x = 54 \end{aligned}

44. A wall of 100 meters can be built by 7 men or 10 women in 10 days. How many days will 14 men and 20 women take to build a wall of 600 metres ?

A. 15

B. 20

C. 25

D. 30

Discuss

Solution:

Let the required number of days be x
7 men = 10 women
(14 men and 20 women) = (20 + 20) women = 40 women
More length, More days (Direct proportion)
More women, Less days (Indirect proportion)

\begin{matrix} Length 100 : 600 \\ Women 40 : 10 \end{matrix}} :: 10 : x

\begin{aligned} ∴ 100 \times 40 \times x = 600 \times 10 \times 10 \\ \Leftrightarrow x = \frac{(600 \times 10 \times 10)}{(100 \times 40)} \\ \Leftrightarrow x = 15 \end{aligned}

45. 21 binders can bind 1400 books in 15 days. How many binders will required to bind 800 books in 20 days ?

A. 7

B. 9

C. 12

D. 14

Discuss

Solution:

Let the required number of binders be x
Less books, Less binders (Direct proportion)
More days, Less binders (Indirect proportion)

\begin{matrix} Books 1400 : 800 \\ Days 20 : 15 \end{matrix}} :: 21 : x

\begin{aligned} ∴ 1400 \times 20 \times x = 800 \times 15 \times 21 \\ \Leftrightarrow x = \frac{(800 \times 15 \times 21)}{(1400 \times 20)} \\ \Leftrightarrow x = 9 \end{aligned}

46. If 17 labourers can dig a ditch 26 m long in 18 days, working 8 hours a day; how many more labourers should be engaged to dig a similar ditch 39 m long in 6 days, each labourer working 9 hours a day ?

A. 34

B. 51

C. 68

D. 85

Discuss

Solution:

Let the total number of men to be engaged be

x

More length, More labourers (Direct proportion)
Less days, More labourers (Indirect proportion)

\begin{matrix} Lentgh 26 : 39 \\ Days 6 : 18 \\ Hours per day 9 : 8 \end{matrix}} :: 17 : x

\begin{aligned} ∴ 26 \times 6 \times 9 \times x = 39 \times 18 \times 8 \times 17 \\ \Leftrightarrow x = \frac{(39 \times 18 \times 8 \times 17)}{(26 \times 6 \times 9)} \\ \Leftrightarrow x = 68 \end{aligned}

∴ Number of more labourers = (68 -17) = 51

47. 20 men complete one-third of a piece of work in 20 days. How many more men should be employed to finish the rest of the work in 25 more days ?

A. 10

B. 12

C. 15

D. 20

Discuss

Solution:

Let the total number of men be x
Work done =

\frac{1}{3}

Remaining work
=

(1 - \frac{1}{3}) = \frac{2}{3}

More work, More men (Direct proportion)
More days, Less men (Indirect proportion)

\begin{matrix} Work \frac{1}{3} : \frac{2}{3} \\ Days 25 : 20 \end{matrix}} :: 20 : x

\begin{aligned} ∴ (\frac{1}{3} \times 25 \times x) = (\frac{2}{3} \times 20 \times 20) \\ \Leftrightarrow x = \frac{800}{25} \\ \Leftrightarrow x = 32 \end{aligned}

∴ More men to be employed = (32 - 20) = 12

48. A rope makes 70 rounds of the circumference of a cylinder whose radius of the base is 14 cm. How many times can it go round a cylinder with radius 20 cm ?

A. 40

B. 49

C. 100

D. None of these

Discuss

Solution:

Let the required number of rounds be x
More radius, Less rounds (Indirect proportion)

\begin{aligned} ∴ 20 : 14 :: 70 : x \\ \Leftrightarrow (20 \times x) = (14 \times 70) \\ \Leftrightarrow x = \frac{(14 \times 70)}{20} \\ \Leftrightarrow x = 49 \end{aligned}

49. A contract is to be complete in 46 days and 117 men were set to work, each working 8 hours a day. After 33 days, $\frac{4}{7}$ of the work is completed. How many additional men may be employed so that the work may be completed in time, each man now working 9 hours a day ?

A. 80

B. 81

C. 82

D. 83

Discuss

Solution:

Remaining work
=

(1 - \frac{4}{7}) = \frac{3}{7}

Remaining period = (46 - 33) = 13 days
Let the total men working at it be x
Less work, Less men (Direct proportion)
Less days, More men (Indirect proportion)
More hour/day, Less men (Indirect proportion)

\begin{matrix} Work \frac{4}{7} : \frac{3}{7} \\ Men 13 : 33 \\ Hour/day 9 : 8 \end{matrix}} :: 117 : x

\begin{aligned} ∴ \frac{4}{7} \times 13 \times 9 \times x = \frac{3}{7} \times 33 \times 8 \times 117 \\ \Leftrightarrow x = \frac{(3 \times 33 \times 8 \times 117)}{(4 \times 13 \times 9)} \\ \Leftrightarrow x = 198 \end{aligned}

∴ Addition men to be employed = (198 - 117) = 81

50. 12 persons can do a piece of work in 4 days. How many persons are required to complete 8 times the work in half the time ?

A. 144

B. 180

C. 190

D. 192

Discuss

Solution:

Let the required number of persons be x
Less days , More persons (Indirect proportion)
More work , More persons (Direct proportion)

\begin{matrix} Days 2 : 4 \\ Work 1 : 8 \end{matrix}} :: 12 : x

\begin{aligned} ∴ 2 \times 1 \times x = 4 \times 8 \times 12 \\ \Leftrightarrow x = \frac{(4 \times 8 \times 12)}{(2)} \\ \Leftrightarrow x = 192 \end{aligned}

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