Objective Questions-

Marks the correct answer in each of the following:

Q1: The ratio 92 : 115 in its simplest form is

(a) 23 : 25 (b) 18 : 23 (c) 3 : 5 (d) 4 : 5

Sol:

(d) 4 : 5

92 :115 = \(\frac{92 \div 23}{115 \div 23} = \frac{4}{5}\)

Q2: If 57 : x :: 51 : 85, then the value of x is

(a) 95 (b) 76 (c) 114 (d) none of these

Sol:

(a) 95

57 : x :: 51 : 85

\(\frac{57}{x} = \frac{51}{85}\)

\(\Rightarrow x = \frac{57 \times 85}{51}\)

\(\Rightarrow x = \frac{4845}{51} = 95\)

Q3: If 25 : 35 :: 45 : x, then the value of x is

(a) 63 (b) 72 (c) 54 (d) none of these

Sol:

(a) 63

25 : .35 :: 45 : x

\(\frac{25}{35} = \frac{45}{x}\)

\(\Rightarrow x = \frac{45 \times 35}{25} = \frac{1575}{25} = 63\)

Q4: If 4 : 5 :: x : 35, then the value of x is

(a) 42 (b) 32 (c) 28 (d) none of these

Sol:

(c) 28

4 : 5 :: x : 35

\(\Rightarrow \frac{4}{5} = \frac{x}{35}\)

\(\Rightarrow x = \frac{4 \times 35}{5} = 4 \times 7 = 28\)

Q5: If a, b, c, d are in proportion, then

(a) ac = bd (b) ad = bc (c) ab = cd (d) none of these

Sol:

(b) ad = bc

Given :

a, b, c, d are in proportion.

a : b :: c : d

\(\Rightarrow \frac{a}{b} = \frac{c}{d}\)

\(\Rightarrow ad = bc\)

Q6: If a, b, c are in proportion, then

(a) \(a^{2} = bc\)

Sol:

(b) \(b^{2} = ac\)

Given:

a, b, c are in proportion.

a : b :: b : c

Product of means = Product of extremes

\(\Rightarrow b^{2} = ac\)

Q7: Choose the correct statement:

(a) (5 : 8) > ( 3 : 4) (b) (5 : 8) < ( 3 : 4) (c) Two ratios cannot be compared

Sol:

(b) (5 : 8) < ( 3 : 4)

We can write

\(5 : 8 = \frac{5}{8} \;\; and \;\; 3 : 4 = \frac{3}{4}\)

Making the denominator equal:

\(\frac{5}{8} \;\; and \;\; \frac{3 \times 2}{4 \times 2} = \frac{6}{8}\)

As 6 > 5, \(\frac{5}{8} < \frac{3}{4}\)

Q8: If Rs.760 is divided between A and B in the ratio 8 : 11 then B’s share is

(a) Rs. 440 (b) Rs. 320 (c) Rs. 430 (d) Rs. 330

Sol:

(a) Rs. 440

A : B = 8 : 11

Sum of ratio terms = 8 + 11 = 19

B’s share = \(\frac{11}{19} \times 760 = \frac {8360} {19}\)

Q9: Two numbers are in the ratio 5 : 7 and the sum of these numbers is 252. The larger of these numbers is

(a) 85 (b) 119 (c) 105 (d) 147

Sol:

(d) 174

Ratio = 5 : 7

Let x be any number such that we have:

5x + 7x = 252

\(\Rightarrow\)

\(\Rightarrow\)

Now, 5x = 5 \(\times\)

7x = 7 \(\times\)

The largest number is 147.

Q10: The side of a triangle are in the ratio 1 : 3 : 5 and its perimeter is 90 cm. The length of its largest side is

(a) 40 cm (b) 50 cm (c) 36 cm (d) 54 cm

Sol:

(b) 50 cm

The sided of the triangle are in the ratio 1 : 3 : 5.

Let x be any number such that the sides are 1x cm, 3x cm and 5x cm.

x + 3x + 5x = 90

\(\Rightarrow\)

\(\Rightarrow x = 10\)

First side = x = 10 cm

Second side = 3 \(\times\)

Third side = 5x = 5 \(\times\)

The length of the largest side is 50 cm.

Q11: The ratio of boys and girls in a school is 12 : 5. If the number of girls is 840, the total strength of the school is

(a) 1190 (b) 2380 (c) 2856 (d) 2142

Sol:

(c) 2856

Ratio of boys and girls = 12 : 5

Let x be any number such that the number of boys and girls are 12x and 5x respectively.

Number of girls = 840

5x = 840

\(\Rightarrow x = \frac{840}{5} = 168\)

Number of boys = 12x = 12 \(\times\)

Number of girls = 840

Total strength of the school = 2016 + 840 = 2856

Q12: If the cost of 12 pens is Rs. 138, then the cost of 14 such pens is

(a) Rs. 164 (b) Rs. 161 (c) Rs. 118.30 (d) Rs. 123.50

Sol:

(b) Rs. 161

Cost of 12 pens = Rs. 138

Cost of 1 pen = Rs. \(\frac{138}{12}\)

Cost of 14 pens = Rs. \(14 \times \frac{138}{12} = Rs. \;\frac{1932}{12} = Rs. \; 161\)

Q13: If 24 workers can build a wall in 15 days, how many days will 8 workers take to build a similar wall?

(a) 42 days (b) 45 days (c) 48 days (d) none of these

Sol:

(b) 45 days

Time taken by 24 workers to build a wall = 15 days

Time taken by 1 worker to build a wall = \(24 \times 15 = 360\)

Time taken by 8 worker to build a wall = \(\frac{360}{8} = 45\)

Q14: If 40 men can finish a piece of work in 26 days, how many men will be required to finish it in 20 days?

(a) 52 (b) 31 (c) 13 (d) 65

Sol:

(a) 52

Number of men required to finish the work in 26 days = 40

Number of men required to finish it in 1 day = \(40 \times 26 = 1040\)

Number of men required to finish it in 20 days = \(\frac{1040}{20} = 52\)

Q15: In covering 111 km , a car consumes 6 L of petrol. How many kilometers will it go in 10 L of petrol?

(a) 172 km (b) 185 km (c) 205 km (d) 266.5 km

Sol:

(b) 185 km

Distance covered in 6 L of petrol = 111 km

Distance covered in 1 L of petrol = \(\frac{111}{6}\)

Distance covered in 10 L of petrol = \(\frac{111}{6} \times 10 = \frac{1110}{6} = 185\)

Q16: In a fort, 550 men had provision for 28 days. How many days will it last for 700 men?

(a) 22 days (b) \(35 \frac{7}{11}\)

Sol:

(a) 22 days

Number of days for which 550 men had provision = 28 days

Number of days for which 1 man had provision = \(28 \times 550 = 15400\)

Number of days for which 700 men had provision = \(\frac{15400}{700} = 22\)

Q17: The angles of a triangle are in the ratio 3 : 1 : 2. The measure of the largest angle is

(a) \(30^{\circ}\)

Sol:

(c) \(90^{\circ}\)

Ratio of the angles of a triangle is 3 : 1 : 2.

Let x be any number such that the three angles are \((3x)^{\circ} , (1x)^{\circ} \;\; and \;\; (2x)^{\circ}\)

We know that the sum of the angles of a triangle is \(180^{\circ}\)

3x + x + 2x = \(180^{\circ}\)

\(\Rightarrow 6x = 180\)

\(\Rightarrow x = 30\)

Therefore, \((3x)^{\circ} = (3 \times 30)^{\circ} = 90^{\circ}\)

\( (x)^{\circ} = (1 \times 30)^{\circ} = 30^{\circ}\)

\((2x)^{\circ} = (2 \times 30)^{\circ} = 60^{\circ}\)

The measure of the largest angle is \(90^{\circ}\)

Q18: Length and breadth of a rectangle field are in the ratio 5 : 4. If the width of the field is 36 m , what is its length?

(a) 40 m (b) 45 m (c) 54 m (d) 50 m

Sol:

(b) 45 m

Length : Breadth = 5 : 4

Let x be any real number such that the length and the breadth are 5x and 4x respectively.

Now, 4x = 36

x = 0

Length = 5x = 45 m

Q19: If a bus covers 195 km in 3 hours and a train covers 300 km in 4 hours, then the ratio of their speed is

(a) 13 : 15 (b) 15 : 13 (c) 13 : 12 (d) 12 : 13

Sol:

(a) 13 : 15

Speed = \(\frac{Distance}{Time}\)

Speed of the bus = \(\frac{195}{3}\)

Speed of the train = \(\frac{300}{4} \)

Ratio = \(\frac{65}{75} = \frac{65 \div 5}{75 \div 5} = \frac{13}{15} = 13 : 15\)

Q20: If the cost of 5 bars of soap is Rs. 82.50, then the cost of one dozen such bars is

(a) Rs. 208 (b) Rs. 192 (c) Rs. 198 (d) Rs. 204

Sol:

(c) Rs. 198

Cost of 5 bars of soap = Rs. 82.50

Cost of 1 bar of soap = \(\frac{82.50}{5} = Rs. \; 16.5\)

Cost of 12 bars of soap = \(16.5 \times 12 = Rs. \; 198\)

Q21: If the cost of 30 packets of 8 pencils each is Rs. 600, what is the cost of 25 packets of 12 pencils each?

(a) Rs. 725 (b) Rs. 750 (c) Rs. 480 (d) Rs. 720

Sol:

(b) Rs. 750

Cost of 30 packets of 8 pencils each = Rs. 600

Cost of 1 packet of 8 pencils = \(\frac{600}{30} = Rs. \;\; 20\)

Cost of 1 pencil = Rs. \(\frac{20}{8}\)

Cost of 1 packet of 12 pencils = \(12 \times \frac{20}{8} = \frac{240}{8} = Rs. \;\; 30\)

Cost of 25 packets of 12 pencils each = 25 \(\times\)

Q22: A rail journey of 75 km costs Rs. 215. How much will a journey of 120 km cost?

(a) Rs. 344 (b) Rs. 324 (c) Rs. 268.75 (d) None of these

Sol:

(a) Rs. 344

Cost of rail journey of 75 km = Rs. 215

Cost of rail journey of 1 km = Rs. \(\frac{215}{75}\)

Cost of rail journey of 120 km = \(120 \times \frac{215}{75} = \frac{25800}{75} = Rs. \;\; 344\)

Q23: The 1st, 2nd and 4th items of a proportion are 12, 21 and 14 respectively. Its third term is

(a) 16 (b) 18 (c) 21 (d) 8

Sol:

(d) 8

Let the third term be x.,

Then, we have :

12 : 21 :: x : 14

We know:

Product of means = Product of extremes

21x = 12 \(\times\)

\(\Rightarrow 21x = 168\)

\(\Rightarrow x = \frac{168}{21} = 8\)

The third term is 8.

Q24: 10 boys can dig a pitch in 12 hours. How long will 8 boys take to do it?

(a) 9 h 36 min (b) 15 h (c) 6 h 40 min (d) 13 h 20 min

Sol:

(b) 15 h

Time taken by 10 boys to dig a pitch = 12 hours

Time taken by 1 boy to dig a pitch = 12 \(\times\)

Time taken by 8 boys to dig a pitch = \(\frac{120}{8} = 15\)