Practice MCQ Questions and Answer on Chain Rule

11.

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

Show Answer

 Let the total number of men to be engaged be $$x$$ More length, More labourers (Direct proportion) Less days, More labourers (Indirect proportion) \[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,{\text{Lentgh 26}}:39 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 6}}:18 \hfill \\ {\text{Hours per day }}9:8 \hfill \\ \end{gathered} \right\}::17:x\] $$\eqalign{ & \therefore {\text{ }}26 \times 6 \times 9 \times x = 39 \times 18 \times 8 \times 17 \cr & \Leftrightarrow x = \frac{{\left( {39 \times 18 \times 8 \times 17} \right)}}{{\left( {26 \times 6 \times 9} \right)}} \cr & \Leftrightarrow x = 68 \cr} $$ ∴ Number of more labourers = (68 -17) = 51

12.

A fort has provisions for 50 days. If after 10 days they are strengthened by 500 men and the food lasts for 35 days longer, the number of men originally in the fort were ?

• (A) 2500
• (B) 3000
• (C) 3500
• (D) 4000

Show Answer

 Let there be x men originally So, x men had provisions 40 days whereas (x + 500) men consumed it in 35 days More men, Less days (Indirect proportion) $$\eqalign{ & \therefore \,\left( {x + 500} \right):x::40:35 \cr & \Leftrightarrow 35 \times \left( {x + 500} \right) = 40x \cr & \Leftrightarrow 5x = 35 \times 500 \cr & \Leftrightarrow x = \left( {\frac{{35 \times 500}}{5}} \right) \cr & \Leftrightarrow x = 3500 \cr} $$

13.

If 5 men or 7 women can earn Rs. 5250 per day, how much would 7 men and 13 women earn per day ?

• (A) Rs. 11600
• (B) Rs. 11700
• (C) Rs. 16100
• (D) Rs. 17100

Show Answer

 Let the required earning be Rs. x 5 men = 7 women 7 men = $$ \left( \frac{7}{5} \times 7 \right) $$   women = $$\frac{49}{5}$$ women ∴ (7 men and 13 women) $$\eqalign{ & = \left( {\frac{{49}}{5} + 13} \right){\text{women}} \cr & = \frac{{114}}{5}{\text{ women}} \cr} $$ Now, More women, More earning (Direct proportion) $$\eqalign{ & \therefore {\text{ }}7 : \frac{{114}}{5}::{\text{ 5}}250 : x \cr & \Leftrightarrow 7x = \left( {\frac{{114}}{5} \times 5250} \right) \cr & \Leftrightarrow 7x = 119700 \cr & \Leftrightarrow x = 17100 \cr} $$

14.

If 7 spiders make 7 webs in 7 days, then 1 spider will make 1 web in how many days?

• (A) 1
• (B) $$\frac{7}{2}$$
• (C) 7
• (D) 49

Show Answer

 Let the required number days be x. Less spiders, More days (Indirect Proportion) Less webs, Less days (Direct Proportion) \[\left. \begin{gathered} {\text{Spiders}}\,\,\,\,\,1:7 \hfill \\ \,\,\,{\text{Webs}}\,\,\,\,\,7:1 \hfill \\ \end{gathered} \right\}::7:x\] $$\eqalign{ & \therefore 1 \times 7 \times x = 7 \times 1 \times 7 \cr & \Rightarrow x = 7 \cr} $$

15.

Running at the same constant rate, 6 identical machines can produce a total of 180 bottles per hour. How many bottles could 15 such machines produce on 30 minutes ?

• (A) 225
• (B) 250
• (C) 300
• (D) 350

Show Answer

 Let the required number of bottles be x More machines, More bottles produced (Direct proportion) Less time, Less bottles produce (Direct proportion) \[\left. \begin{gathered} {\text{Machines 6}}:15 \hfill \\ \,\,\,\,\,\,{\text{Times 60}}:30 \hfill \\ \end{gathered} \right\}::180:x\] $$\eqalign{ & \therefore {\text{ }}6 \times 60 \times x = 15 \times 30 \times 180 \cr & \Leftrightarrow x = \frac{{\left( {15 \times 30 \times 180} \right)}}{{\left( {6 \times 60} \right)}} \cr & \Leftrightarrow x = 225 \cr} $$

16.

30 labourers working 7 hours a day can finish a piece of work in 18 days. If the labourers work 6 hours a day, then the number of labourers to finish the same piece of work in 30 days, will be ?

• (A) 15
• (B) 21
• (C) 22
• (D) 25

Show Answer

 Let the required number of labourers be x Less working hour/day, More labours (Indirect proportion) More days, Less labourers (Indirect proportion) \[\left. \begin{gathered} {\text{Working hours/day 6}}:7 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 30}}:18 \hfill \\ \end{gathered} \right\}::30:x\] $$\eqalign{ & \therefore 6 \times 30 \times x = 7 \times 18 \times 30 \cr & \Leftrightarrow 6x = 126 \cr & \Leftrightarrow x = 21 \cr} $$

17.

12 men and 18 boys, working $$7\frac{1}{2}$$ hours a day, can do a piece of work in 60 days. If a man works equal to 2 boys, then how many boys will be required to help 21 men to do twice the work in 50 days, working 9 hours a day ?

• (A) 30
• (B) 42
• (C) 48
• (D) 90

Show Answer

 1 man ≡ 2 boys ⇔ (12 men + 18 boys)           ≡ (12 × 2 ×18) boys = 42 boys Let required number of boys = x ⇒ (21 men + x boys) ≡ (21 × 2 × x) boys = (42 + x) boys Less days, More boys (Indirect proportion) More hours per day, Less boys (Indirect proportion) More work, More boys (Direct proportion) \[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 50}}:60 \hfill \\ {\text{Hours per day 9}}:\frac{{15}}{2} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Work }}1:2 \hfill \\ \end{gathered} \right\}::42:\left( {42 + x} \right)\] $$\therefore \left[ {50 \times 9 \times 1 \times \left( {42 + x} \right)} \right] = $$     $$\left( {60 \times \frac{{15}}{2} \times 2 \times 42} \right)$$ $$\eqalign{ & \Leftrightarrow \left( {42 + x} \right) = \frac{{37800}}{{450}} \cr & \Leftrightarrow 42 + x = 84 \cr & \Leftrightarrow x = 42 \cr} $$

18.

A wheel that has 6 cogs is meshed with a larger wheel of 14 cogs. When the smaller wheel has made 21 revolutions, then the number of revolutions mad by the larger wheel is:

• (A) 4
• (B) 9
• (C) 12
• (D) 49

Show Answer

 Let the required number of revolutions made by larger wheel be x. Then, More cogs, Less revolutions (Indirect Proportion) $$\eqalign{ & \therefore 14:6::21:x \cr & \Rightarrow 14 \times x = 6 \times 21 \cr & \Rightarrow x = \frac{{6 \times 21}}{{14}} \cr & \Rightarrow x = 9 \cr} $$

19.

Four gardeners with four grass mowers mow 400 sq. m of ground in 4 hours. How long would it take for eight gardeners with eight grass mowers to mow 800 sq. m of ground ?

• (A) 4 hours
• (B) 6 hours
• (C) 8 hours
• (D) 12 hours

Show Answer

 Since each gardener would work with one grass mower, So, we shall consider 1 gardener and 1 grass mower as one unit. Let the required time be x hours More gardeners and grass mowers, Less time (Indirect proportion) More area, More time (Direct proportion) \[\left. \begin{gathered} {\text{Gardeners & grass mowers 8}}:4 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Area 400}}:800 \hfill \\ \end{gathered} \right\}::4:x\] $$\eqalign{ & \therefore {\text{ }}8 \times 400 \times x = 4 \times 800 \times 4 \cr & \Leftrightarrow x = \frac{{\left( {4 \times 800 \times 4} \right)}}{{\left( {8 \times 400} \right)}} \cr & \Leftrightarrow x = 4 \cr} $$

20.

56 men can complete a piece of work in 24 days. In how many days can 42 men complete the same piece of work ?

• (A) 18
• (B) 32
• (C) 48
• (D) 98

Show Answer

 Let the required number of days be x Less men, More days (Indirect proportion) $$\eqalign{ & \therefore 42:56::24:x \cr & \Leftrightarrow 42 \times x = 56 \times 24 \cr & \Leftrightarrow x = \left( {\frac{{56 \times 24}}{{42}}} \right) \cr & \Leftrightarrow x = 32 \cr} $$