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21. On a certain map of India the actual distance of 1450 km between two cities Delhi and Kolkata is shown as 5 cm. What scale is used to draw the map ?

A. $1 : 15 \times 10^{6}$

B. $1 : 20 \times 10^{6}$

C. $1 : 25 \times 10^{6}$

D. $1 : 29 \times 10^{6}$

Discuss

Solution:

5 cm on the map represents 1450 km
∴ 1 cm on the map represents

(\frac{1450}{5})

km = 290 km
Hence, the scale is 1 cm : 290 km i.e.,

1 : 29 \times 10^{6}

[∵ 290 km = (29 \times 10^{6}) cm]

22. A flagstaff 17.5 m high caste a shadow of length 40.25 m. The height of the building, which casts a shadow of length 28.75 m under similar conditions will be ?

A. 10 m

B. 12.5 m

C. 17.5 m

D. 21.25 m

Discuss

Solution:

Let the height of the building be x metres
Lees lengthy shadow, Less is the height (Direct proportion)

\begin{aligned} ∴ 40.25 : 28.75 :: 17.5 : x \\ \Leftrightarrow 40.25 \times x = 28.75 \times 17.5 \\ \Leftrightarrow x = \frac{(28.75 \times 17.5)}{40.25} \\ \Leftrightarrow x = 12.5 \end{aligned}

23. A man completes $\frac{5}{8}$ of a job in 10 days. At this rate, how many more days will it take him to finish the job ?

A. 5

B. 6

C. 7

D. $7 \frac{1}{2}$

Discuss

Solution:

Work done =

\frac{5}{8}

Balance work

= (1 - \frac{5}{8})

= \frac{3}{8}

Less work, Less days ( Direct proportion)
Let the required number of days be x
Then,

\begin{aligned} \Leftrightarrow \frac{5}{8} : \frac{3}{8} :: 10 : x \\ \Leftrightarrow \frac{5}{8} \times x = \frac{3}{8} \times 10 \\ \Leftrightarrow x = (\frac{3}{8} \times 10 \times \frac{8}{5}) \\ \Leftrightarrow x = 6 \end{aligned}

24. 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

Discuss

Solution:

Let the required number of days be x
Less men, More days (Indirect proportion)

\begin{aligned} ∴ 42 : 56 :: 24 : x \\ \Leftrightarrow 42 \times x = 56 \times 24 \\ \Leftrightarrow x = (\frac{56 \times 24}{42}) \\ \Leftrightarrow x = 32 \end{aligned}

25. 30 men can do a piece of work in 16 days. How many men would be required to do the same work in 20 days ?

A. 12

B. 24

C. 36

D. 48

Discuss

Solution:

Let the required number of men be x
More days, Less men (Indirect proportion)

\begin{aligned} ∴ 20 : 16 :: 30 : x \\ \Leftrightarrow 20 x = 16 \times 30 \\ \Leftrightarrow x = (\frac{16 \times 30}{20}) \\ \Leftrightarrow x = 24 \end{aligned}

26. 4 mat-weavers can weave 4 mats in 4 days. At the same rate, how many mats would be woven by 8 mat-weavers in 8 days ?

A. 4

B. 8

C. 12

D. 16

Discuss

Solution:

Let the required number of mats be x
More weavers , More mates ( Direct proportion )
More days, More mats ( Direct proportion )

\begin{matrix} Weavers 4 : 8 \\ Days 4 : 8 \end{matrix}} :: 4 : x

\begin{aligned} ∴ 4 \times 4 \times x = 8 \times 8 \times 4 \\ \Leftrightarrow x = \frac{(8 \times 8 \times 4)}{(4 \times 4)} \\ \Leftrightarrow x = 16 \end{aligned}

27. If 7 maids with 7 mops cleaned 7 floors in 7 hours, how long would it take 3 maids to mop 3 floors with 3 mops ?

A. $\frac{7}{3} hours$

B. $3 hours$

C. $\frac{49}{3} hours$

D. $7 hours$

Discuss

Solution:

Since each maid would work with one mop,
So,we shall consider 1 maid and 1 mop as 1 unit
Let the required time be x hours
Less maids and mops, More time (Indirect proportion)
Less floor, Less time (Direct proportion)

\begin{matrix} Maids &  Mops 3 : 7 \\ Floors 7 : 3 \end{matrix}} :: 7 : x

\begin{aligned} ∴ 3 \times 7 \times x = 7 \times 3 \times 7 \\ \Leftrightarrow x = \frac{(7 \times 3 \times 7)}{(3 \times 7)} \\ \Leftrightarrow x = 7 \end{aligned}

28. 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

Discuss

Solution:

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)

\begin{matrix} Gardeners &  grass mowers 8 : 4 \\ Area 400 : 800 \end{matrix}} :: 4 : x

\begin{aligned} ∴ 8 \times 400 \times x = 4 \times 800 \times 4 \\ \Leftrightarrow x = \frac{(4 \times 800 \times 4)}{(8 \times 400)} \\ \Leftrightarrow x = 4 \end{aligned}

29. 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

Discuss

Solution:

Let the required number of bottles be x
More machines, More bottles produced (Direct proportion)
Less time, Less bottles produce (Direct proportion)

\begin{matrix} Machines 6 : 15 \\ Times 60 : 30 \end{matrix}} :: 180 : x

\begin{aligned} ∴ 6 \times 60 \times x = 15 \times 30 \times 180 \\ \Leftrightarrow x = \frac{(15 \times 30 \times 180)}{(6 \times 60)} \\ \Leftrightarrow x = 225 \end{aligned}

30. If 6 persons working 8 hours a day earn Rs. 8400 per week, then 9 persons working 6 hours a day will earn per week ?

A. Rs. 8400

B. Rs. 9450

C. Rs. 16200

D. Rs. 16800

Discuss

Solution:

Let the weekly earning be Rs. x
More persons , More earning (Direct proportion)
Less working hours, Less earning (Direct proportion)

\begin{matrix} Persons 6 : 9 \\ Working hours 8 : 6 \end{matrix}} :: 8400 : x

\begin{aligned} ∴ 6 \times 8 \times x = 9 \times 6 \times 8400 \\ \Leftrightarrow x = \frac{(9 \times 6 \times 8400)}{(6 \times 8)} \\ \Leftrightarrow x = 9450 \end{aligned}

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