2 Mark Question

Question 12 Marks

Find the mean proportional between:
0.4 and 0.9

Answer

$0.4 : \text{x} :: \text{x} : 0.9$
$\Rightarrow \text{x}^2 = 0.4 \times 0.9$
$\text{x} = \sqrt{0.36} = 0.6$
Mean proportional = 0.6

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Question 22 Marks

Find the third proportional to:
4.5 and 6

Answer

Let x be the third proportional, then
$4.5 : 6 : : 6 : \text{x}$
$\Rightarrow4.5:6::6:\text{x}$
$\Rightarrow4.5\times\text{x}=6\times6$
$\Rightarrow\text{x}=\frac{6\times6}{4.5}=\frac{6\times6\times10}{45}=\frac{360}{45}=8$
$\therefore$ Third proportional = 8

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Question 32 Marks

Find the fourth proportional to the number:
8, 36, 6

Answer

Let x be the fourth proportional, then:
$8:36::6:\text{x}$
$\therefore8\times\text{x}=36\times6$
$\Rightarrow\text{x}=\frac{36\times6}{8}$
$\text{x}=9\times3=27$
$\therefore$ Fourth proportional = 27

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Question 42 Marks

Express the following ratios in the simplest form:
8 months : 1 year

Answer

8 months : 1 year = 8 months : 12 months
(Converting into the same kind)
HCF of 8 and 12 = 4
Dividing by 4
8 ÷ 4 : 12 ÷ 4
= 2 : 3

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Question 52 Marks

Two numbers are in the ratio 5 : 7. If the sum of these numbers is 84, find the numbers.

Answer

Suppose that the numbers are 5x and 7x
Then, 5x + 7x = 84
12x = 84
x = 7
Hence, the numbers are (5 × 7) = 35 and (7 × 7) = 49.

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Question 62 Marks

Find the fourth proportional to the number:
5, 7, 30

Answer

Let x be the fourth proportional, then:
$5:7::30:\text{x}$
$\therefore5\times\text{x}=7\times30$
$\Rightarrow\text{x}=\frac{7\times30}{5}=7\times6$
$\text{x}=42$
$\therefore$ Fourth proportional = 42

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Question 72 Marks

If a car covers 54km in an hour, how much distance will it cover in 40 minutes ?

Answer

Distance covered in 60 min = 54km
Distance covered in 1 min $=\frac{54}{60}\text{km}$
Distance covered in 40 min $=-\frac{54}{60}\times40=36\text{km}$

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Question 82 Marks

Find the fourth proportional to the number:
2, 8, 14, 3.5

Answer

Let x be the fourth proportional, then:
$2.8 : 14 ::3.5 : \text{x}$
$\therefore2.8\times\text{x}=14\times3.5$
$\Rightarrow\text{x}=\frac{14\times3.5}{2.8}=\frac{14\times5}{4}=\frac{7\times5}{2}$
$\text{x}=\frac{35}{2}=17.5$
$\therefore$ Fourth proportional = 17.5

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Question 92 Marks

Find the mean proportional between:
3 and 27

Answer

$3 : \text{x} : : \text{x} : 27$
$\Rightarrow\text{x}^2 = 3 \times 27 = 81$
$\text{x} = \sqrt{81} = 9$
Mean proportional = 9

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Question 102 Marks

Find the mean proportional between:
6 and 24

Answer

Let x be the mean proportional, then
$6:\text{x}::\text{x}:24$
$\Rightarrow\text{x}^2=6\times24=144$
$\text{x}=\sqrt{144}=12$
Mean proportional = 12

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Question 112 Marks

Express the following ratios in the simplest form:
(2kg 250g) : (3kg)

Answer

2kg 250g : 3kg = 2250g : 3000g (converting into the same kind)
HCF of 2250 and 3000 = 750
Dividing by 750,
2250 ÷ 750 : 3000 ÷ 750 = 3 : 4

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Question 122 Marks

If 8 : x :: 16 : 35, find the value of x.

Answer

Product of the extremes = 8 × 35 = 280
Product of the means = 16 × x = 16x
Since 8 : x :: 16 : 35
We have:
Product of the extremes = product of the means
⇒ 280 = 16x
⇒ x = 17.5

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Question 132 Marks

If (4x + 5) : (3x + 11) = 13 : 17, find the value of x.

Answer

(4x + 5) : (3x + 11) = 13 : 17
$\frac{4\text{x}+5}{3\text{x}+11}=\frac{13}{17}$
By cross multiplication,
68x + 85 = 39x + 143
⇒ 68x - 39x = 143 - 85
⇒ 29x = 58
x = 2
Hence x = 2

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Question 142 Marks

Show that 30, 40, 45, 60 are in proportion.

Answer

We know that a, b, c, d are in proportion if ad = bc
Now 30, 40, 45, 60 are in proportion
if 30 × 60 = 40 × 45
if 1800 = 1800
which is true
30, 40, 45, 60 are in proportion.

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Question 152 Marks

Express the following ratios in the simplest form:
1m 5cm : 63cm

Answer

1m 5cm : 63cm = 105cm : 63cm
(Converting into same kind)
HCF of 105 and 63 = 21
105 ÷ 21 : 63 ÷ 21 (Dividing by 21)
= 5 : 3

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Question 162 Marks

Find the third proportional to:
8 and 12

Answer

Let x be the third proportional, then
$8 : 12 : : 12 : \text{x}$
$\Rightarrow8\times\text{x}=12\times12$
$\Rightarrow\text{x}=\frac{12\times12}{8}=18$
$\therefore$ Third proportional = 18

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Question 172 Marks

The ratio of monthly income to the savings of a family is 11 : 2. If the savings be Rs. 2500, find the income and expenditure.

Answer

Ratio in income and savings = 11 : 2
Let income = 11x
Then savings = 2x
But savings = Rs. 2500
2x = 2500
⇒ x = 1250
Then income = 1250 × 11 = Rs. 13750
and expenditure = Total income – savings = 13750 - 2500 = Rs. 11250

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Question 182 Marks

Express the following ratios in the simplest form:
2.5 : 6.5 : 8

Answer

2.5 : 6.5 : 8
= 2.5 : 6.5 : 8.0
= 25 : 65 : 80
HCF of 25, 65, 80 = 5
Dividing by 5,
5 : 13 : 16

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Question 192 Marks

If x, 18, 108 are in continued proportion, find the value of x.

Answer

x : 18 :: 18 : 108
⇒ x × 108 = 18 × 18
(Product of extremes = Product of means)
⇒ 108x = 324
⇒ x = 3
Hence, the value of x is 3.

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Question 202 Marks

If 40 men can finish a piece of work in 60 days, in how many days will 75 men finish the same work?

Answer

40 men can finish the work in 60 days
1 man can finish the work in 60 x 40 days [Less men, more days]
75 men will finish the work in $=\frac{60\times40}{75}=32$ days
Hence, 75 men will finish the same work in 32 days.

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Question 212 Marks

If the third proportional to 7 and $x$ is 28 , find the value of $x$.
Hint. $7: x:: x: 28$ Find $x$.

Answer

Third proportional $=28$ then
$7: x:: x: 28$
$\Rightarrow 7 \times 28=x \times x$
$\Rightarrow x^2=28 \times 7=196$
$\Rightarrow x=\sqrt{ } 196=14$
$x = 14$

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Question 222 Marks

Express the following ratios in the simplest form:
75 paise : 3 rupees

Answer

75 paise : 3 rupees = 75 paise : 300 paise
(Converting into same kind)
HCF of 75, 300 = 75
75 : 300 = 75 ÷ 75 : 300 ÷ 75 (Dividing by 75) = 1 : 4

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Question 232 Marks

Show that 36, 49, 6, 7 are not in proportion.

Answer

We know that if a, b, c, d are in proportion if ad = bc
Now 36, 49, 6, 7 are in proportion
if 36 × 7 = 49 × 6
if 252 = 294
But 252 ≠ 294
36, 49, 6, 7 are not in proportion.

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Question 242 Marks

Express the following ratios in the simplest form:
1km : 750m

Answer

1km : 750m = 1000m : 750m
(Converting into metre)
= 4 : 3 (dividing by 250)
= 4 : 3

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Question 252 Marks

The ratio of boys and girls in a school is 8 : 3. If the total number of girls be 375, find the number of boys in the school.

Answer

Ratio in boys and girls = 8 : 3
And total number of girls = 375
Let number of boys = 8x
Then number of girls = 3x
3x = 375
⇒ x = 125
Number of boys = 8x = 8 × 125 = 1000

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Question 262 Marks

Find the third proportional to 8 and 12.

Answer

Suppose that the third proportional to 8 and 12 is x
Then, 8 : 12 :: 12 : x
⇒ 8x = 144 (Product of extremes = Product of means)
x = 18
Hence, the third proportional is 18.

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Question 272 Marks

If x : 35 :: 48 : 60 find the value of x.

Answer

Product of the extremes = x × 60 = 60x
Product of the means = 35 × 48 = 1680
Since x : 35 :: 48 : 60
We have:
Product of the extremes = product of the means
⇒ 60x = 1680
⇒ x = 28

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Question 282 Marks

Find the third proportional to:
12 and 18

Answer

Let x be the third proportional, then
$12 : 18 : : 18 : \text{x}$
$\Rightarrow12\times\text{x}=18\times18$
$\Rightarrow\text{x}=\frac{18\times18}{12}=\frac{3\times18}{2}=3\times9=27$
$\therefore$ Third proportional = 27

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Question 292 Marks

Express the following ratios in the simplest form:
1 hour 5 minutes : 45 minutes

Answer

1 hour 5 minutes : 45 minutes = 65 minutes : 45 minutes
(Converting into minutes)
13 : 9 (dividing by 5)
= 13 : 9

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Question 302 Marks

Show that the numbers 25, 36, 5, 6 are not in proportion.

Answer

Product of the extremes = 25 × 6 = 150
Product of the means = 36 × 5 = 180
The product of the extremes is not equal to that of the means.
Hence, 25, 36, 5 and 6 are not in proportion.

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