Ratio & Proportion - Exercise 7(C)

The ratio between 6 cm and 20 mm is

1. 3 : 1
2. 3 : 10
3. 6 : 10
4. 3 : 2

Answer

Given:

First quantity = 6 cm

Second quantity = 20 mm

First, convert to the same unit (1 cm = 10 mm).

6 cm = 6 x 10 = 60 mm.

Ratio = 60 : 20 = $\dfrac{60}{20} = \dfrac{3}{1}$ = 3 : 1

Hence, option 1 is the correct option.

The ratio $\dfrac{1}{8} : \dfrac{1}{12}$ converted to the simplest form is

1. 12 : 8
2. 3 : 2
3. 2 : 5
4. 8 : 20

Answer

Given:

Ratio: $\dfrac{1}{8} : \dfrac{1}{12}$

Let us find the L.C.M. of denominators 8 and 12:

$\begin{array}{l|rr} 2 & 8, & 12 \\ \hline 2 & 4, & 6 \\ \hline 2 & 2, & 3 \\ \hline 3 & 1, & 3 \\ \hline & 1, & 1 \end{array}$

L.C.M. = 2 x 2 x 2 x 3 = 24

Multiply both terms by 24:

$\Big(\dfrac{1}{8} \times 24\Big) : \Big(\dfrac{1}{12} \times 24\Big) = 3 : 2$

Hence, option 2 is the correct option.

If A : B = 2 : 5 and B : C = 4 : 5, then C : A = ?

1. 5 : 2
2. 2 : 1
3. 15 : 8
4. 25 : 8

Answer

Given:

A : B = 2 : 5

B : C = 4 : 5

To find C : A, first find A : C by multiplying the ratios:

$\dfrac{A}{C} = \dfrac{A}{B} \times \dfrac{B}{C} \\[1em] = \dfrac{2}{5} \times \dfrac{4}{5} \\[1em] = \dfrac{8}{25}$

So, A : C = 8 : 25.

For C : A, we reverse the ratio = 25 : 8.

Hence, option 4 is the correct option.

By increasing 91 in the ratio 7 : 13, we get :

1. 182
2. 169
3. 121
4. 116

Answer

Given:

Original number = 91

Increased ratio = 7 : 13

The original part is 7.

7 parts = 91

1 part = 91 ÷ 7 = 13

The increased value is 13 parts:

New value = 13 x 13 = 169

Hence, option 2 is the correct option.

If 12 : x : : 15 : 25, then the value of x is

1. 10
2. 15
3. 18
4. 20

Answer

Given:

Proportion: 12 : x : : 15 : 25

Product of Means = Product of Extremes

x × 15 = 12 × 25

15x = 300

⇒ x = $\dfrac{300}{15}$

⇒ x = 20

Hence, option 4 is the correct option.

The third proportional to 9 and 18 is

1. 21
2. 24
3. 27
4. 36

Answer

Given:

Numbers = 9 and 18

Let the third proportional be x.

9 : 18 : : 18 : x

Product of Means = Product of Extremes

9 × x = 18 × 18

⇒ x = $\dfrac{18 \times 18}{9}$

⇒ x = 2 x 18

⇒ x = 36

Hence, option 4 is the correct option.

The mean proportional between 5 and 45 is

1. 10
2. 12
3. 15
4. 25

Answer

Given:

Numbers = 5 and 45

Mean Proportion = $\sqrt{a \times b}$

= $\sqrt{5 \times 45} = \sqrt{225}$

= 15

Hence, option 3 is the correct option.

Which of the following are in continued proportion?

1. 4, 8, 12
2. 5, 15, 25
3. 6, 36, 216
4. 9, 12, 18

Answer

Let us consider the given numbers as: a, b, c

Check each option using b² = a x c:

8² = 64, 4 × 12 = 48 (No)

15² = 225, 5 × 25 = 125 (No)

36² = 1296, 6 × 216 = 1296 (Yes)

12² = 144, 9 × 18 = 162 (No)

Hence, option 3 is the correct option.

Fill in the blanks :

(i) Ratio has ............... unit.

(ii) To convert a ratio a : b in its simplest form, we divide a and b by ............... of a and b.

(iii) If a : b : : b : c, then a, b, c are said to be in ............... proportion.

(iv) If a, b, c are in continued proportion, then c is called the ............... proportional to a and b.

(v) In a proportion, the first and fourth terms are called the ............... .

Answer

(i) Ratio has no unit.

(ii) To convert a ratio a : b in its simplest form, we divide a and b by H.C.F. of a and b.

(iii) If a : b : : b : c, then a, b, c are said to be in continued proportion.

(iv) If a, b, c are in continued proportion, then c is called the third proportional to a and b.

(v) In a proportion, the first and fourth terms are called the extremes.

Write true (T) or false (F) :

(i) If a, b, c, d are in proportion, then ac = bd.

(ii) If a : b : : c : d, then a, b, c, d are said to be in absolute proportion.

(iii) If a, b, c, are in continued proportion, then the mean proportion b = $\dfrac{a + c}{2}$.

(iv) If x is the third proportional to a, b, then a : b : : b : x.

(v) 1, 2, 3, 4, are in proportion.

Answer

(i) False
Reason — For a, b, c, d to be in proportion (a : b :: c : d), the rule is Product of Extremes = Product of Means. This means a x d = b x c, or ad = bc. The statement says ac = bd, which is incorrect.

(ii) False
Reason — When four terms are in the form a : b :: c : d, they are simply said to be in proportion. There is no standard mathematical term called "absolute proportion" used in this context.

(iii) False
Reason — If a, b, c are in continued proportion, then a : b :: b : c. This means b² = ac or $b = \sqrt{ac}$. The formula $\dfrac{a+c}{2}$ is for the arithmetic mean, not the mean proportional.

(iv) True
Reason — By definition, if x is the third proportional to a and b, then a, b, and x are in continued proportion. In this sequence, b is the mean (repeated) term.

(v) False
Reason — To check if 1, 2, 3, 4 are in proportion, we test if 1 x 4 = 2 x 3:

Product of Extremes (1 x 4) = 4

Product of Means (2 x 3) = 6

Since 4 ≠ 6, they are not in proportion.

Ram Nath sold one of his properties worth ₹ 38,00,000. He wished to divide this money between his two daughters Priya and Seema in the ratio 7 : 12. He sold another property for ₹ 60,00,000. He divided this money between Priya and Seema in the ratio $\dfrac{1}{5} : \dfrac{1}{7}$

(1) What amount did Priya receive from the sale of second property ?

1. ₹ 14,00,000
2. ₹ 24,00,000
3. ₹ 25,00,000
4. ₹ 35,00,000

(2) What amount did Seema receive from the sale of first property ?

1. ₹ 14,00,000
2. ₹ 24,00,000
3. ₹ 25,00,000
4. ₹ 49,00,000

(3) The difference between the total amounts received by Priya and Seema is :

1. ₹ 0
2. ₹ 1,00,000
3. ₹ 2,00,000
4. ₹ 5,00,000

(4) The ratio between the amounts received by Seema from the sale of the first and the second properties is :

1. 1 : 1
2. 12 : 7
3. 24 : 25
4. 14 : 35

Answer

(1) Given:

Value of second property = ₹ 60,00,000

Ratio (Priya : Seema) = $\dfrac{1}{5} : \dfrac{1}{7}$

Let us find L.C.M. of 5 and 7:

$\begin{array}{l|rr} 5 & 5, & 7 \\ \hline 7 & 1, & 7 \\ \hline & 1, & 1 \end{array}$

L.C.M. = 5 x 7 = 35

Priya : Seema = $(\dfrac{1}{5} \times 35) : (\dfrac{1}{7} \times 35)$

Priya : Seema = 7 : 5

Total parts = 7 + 5 = 12

Value of 1 part = ₹ 60,00,000 ÷ 12 = ₹ 5,00,000

Priya's amount = 7 parts x ₹ 5,00,000 = ₹ 35,00,000

Hence, option 4 is the correct option.

(2) Given:

Value of first property = ₹ 38,00,000

Ratio (Priya : Seema) = 7 : 12

Total parts = 7 + 12 = 19

Value of 1 part = ₹ 38,00,000 ÷ 19 = ₹ 2,00,000

Seema's amount = 12 parts x ₹ 2,00,000 = ₹ 24,00,000

Hence, option 2 is the correct option.

(3)

Calculate total for Priya.

From 1st property:

Ratio (Priya : Seema) = 7 : 12 $\quad$Given

Value of 1 part = ₹ 2,00,000 $\quad$[From previous step]

∴ 7 x 2,00,000 = ₹ 14,00,000

From 2nd property:

Priya's amount = ₹ 35,00,000 $\quad$[From step 1]

Total = ₹ 14,00,000 + ₹ 35,00,000 = ₹ 49,00,000

Calculate total for Seema.

From 1st property:

Seema's amount = ₹ 24,00,000 $\quad$[From step 2]

From 2nd property:

Ratio (Priya : Seema) = $\dfrac{1}{5} : \dfrac{1}{7}$

L.C.M. of 5 and 7 is 35.

Priya : Seema = $(\dfrac{1}{5} \times 35) : (\dfrac{1}{7} \times 35)$

Priya : Seema = 7 : 5

Value of 1 part = ₹ 5,00,000

∴ 5 x ₹ 5,00,000 = ₹ 25,00,000

Total = ₹ 24,00,000 + ₹ 25,00,000 = ₹ 49,00,000

Difference = Priya - Seema

= ₹ 49,00,000 - ₹ 49,00,000 = ₹ 0

Hence, option 1 is the correct option.

(4)

Seema's 1st amount = ₹ 24,00,000 $\quad$[From step 2]

Seema's 2nd amount = ₹ 25,00,000 $\quad$[From previous step]

Ratio = 24,00,000 : 25,00,000

Ratio = 24 : 25

Hence, option 3 is the correct option.

Ranjan Singh makes statues of brass. Brass is an alloy of copper and zinc. Ranjan uses two varieties of brass for different kinds of statues. Variety 1 contains copper and zinc mixed in the ratio 7 : 4 and variety 2 contains these metals in the ratio 5 : 3. Ranjan makes an elephant statue from variety 1 and a horse statue from variety 2. The elephant statue weighs 176 g and it is known that the brass used in the horse statue contains 135 g zinc.

(1) Find the quantity of copper present in the brass used to make the elephant statue.

1. 98 g
2. 112 g
3. 121 g
4. 132 g

(2) How much copper is contained in the brass used to make the horse statue ?

1. 165 g
2. 175 g
3. 205 g
4. 225 g

(3) How much zinc is contained in the brass used to make the two statues ?

1. 169 g
2. 179 g
3. 189 g
4. 199 g

(4) The ratio of the quantities of copper and zinc used to make the two statues is :

1. 113 : 98
2. 337 : 148
3. 221 : 199
4. 337 : 199

Answer

(1) Given:

Elephant statue is made from variety 1.

Variety 1 ratio (Copper : Zinc) = 7 : 4

Total weight of statue = 176 g

Total parts = 7 + 4 = 11

Value of 1 part = 176 g ÷ 11 = 16 g

Copper = 7 parts x 16 g = 112 g

Hence, option 2 is the correct option.

(2) Given:

Horse statue is made from variety 2.

Variety 2 ratio (Copper : Zinc) = 5 : 3

Quantity of Zinc = 135 g

3 parts of Zinc = 135 g

Value of 1 part = 135 g ÷ 3 = 45 g

Copper = 5 parts x 45 g

Copper = 225 g

Hence, option 4 is the correct option.

(3) Given:

Zinc in Horse = 135 g

Zinc in Elephant:

Variety 1 ratio (Copper : Zinc) = 7 : 4

Value of 1 part = 16 g $\quad$[From step 1]

∴ 4 parts x 16 g = 64 g

Zinc in Elephant = 64 g

Total Zinc = 135 g + 64 g = 199 g

Hence, option 4 is the correct option.

(4)

Calculate Copper in Elephant:

Variety 1 ratio (Copper : Zinc) = 7 : 4

Value of 1 part = 16 g $\quad$[From step 1]

Copper in Elephant = 7 x 16 g = 112 g

Copper in Horse = 225 g $\quad$[From step 2]

Total Copper = Copper in Elephant + Copper in Horse

Total Copper = 112 g + 225 g = 337 g $\quad$[Substituting the values]

Total Copper = 337 g

Total Zinc = 199 g $\quad$[From previous step]

The ratio of the quantities of copper and zinc = 337 : 199

Hence, option 4 is the correct option.

Assertion: If we divide ₹ 1250 between Dinesh and Anmol in the ratio 3 : 7, then the difference between their shares is ₹ 500.

Reason: Ratio is a fraction. It has no units.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

Explanation

Given:

₹1250 is divided between Dinesh and Anmol in the ratio 3 : 7.

Total parts: 3 + 7 = 10

Value of one part: 1250 ÷ 10 = 125

Shares:

Dinesh = 3 × 125 = 375

Anmol = 7 × 125 = 875

Difference: 875 − 375 = 500

So the Assertion is true.

A ratio is indeed a comparison of two quantities of the same kind, so it is a fraction and has no units. The Reason is True.

This statement is true, but it does not explain why the difference is ₹500.

Hence, option 2 is the correct option.

Assertion: The numbers 4, 8, 16 are in continued proportion.

Reason: Three numbers a, b, c are in continued proportion, if a : b = b : c.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation

Given:

Numbers 4, 8, 16 are in continued proportion.

Numbers to be in continued proportion, it should satisfy 4 : 8 = 8 : 16

Ratio 1 (4 : 8) = $\dfrac{4}{8} = \dfrac{1}{2}$

Ratio 2 (8 : 16) = $\dfrac{8}{16} = \dfrac{1}{2}$

Since the ratios are equal, the Assertion is True.

The reason states that a, b, c are in continued proportion if a : b = b : c. This is the mathematical definition of continued proportion.

This statement is correct and explains the assertion.

Hence, option 1 is the correct option.

If a bus travels 126 km in 3 hours and a train travels 315 km in 5 hours, then the ratio of their speeds is:

1. 2 : 5
2. 2 : 3
3. 5 : 2
4. 25 : 6

Answer

Given:

Distance covered by bus = 126 km

Time taken by bus = 3 hours

Distance covered by train = 315 km

Time taken by train = 5 hours

Step 1: Find the speed of the bus and the train

Speed = $\dfrac{\text{Distance}}{\text{Time}}$

Speed of bus = $\dfrac{126}{3}$ = 42 km/h

Speed of train = $\dfrac{315}{5}$ = 63 km/h

Step 2: Find the ratio of their speeds

Ratio of speeds = Speed of bus : Speed of train

= 42 : 63

= $\dfrac{42}{63}$ $\hspace{2cm}$[Writing the ratio as a fraction]

= $\dfrac{42 \div 21}{63 \div 21}$

[Dividing numerator and denominator by H.C.F. of 42 and 63, which is 21]

= $\dfrac{2}{3}$

= 2 : 3

∴ The ratio of speeds of the bus and the train is 2 : 3.

Hence, option 2 is the correct option.

If 4, a, a, 36 are in proportion, then a =

1. 24
2. 12
3. 3
4. 14

Answer

Given:

4, a, a, 36 are in proportion.

In a proportion, product of extremes = product of means.

Here, extremes are 4 and 36, and means are a and a.

According to the question, the equation can be written as:

4 × 36 = a × a

⇒ a² = 144

⇒ a = $\sqrt{144}$

⇒ a = 12 $\hspace{2cm}$ [∵ 12 × 12 = 144]

∴ The value of a is 12.

Hence, option 2 is the correct option.

u : v = 4 : 7 and v : w = 9 : 7. If u = 72, then the value of w is:

1. 98
2. 77
3. 63
4. 49

Answer

Given:

u : v = 4 : 7

v : w = 9 : 7

u = 72

Step 1: Find the value of v

u : v = 4 : 7

$\dfrac{u}{v}$ = $\dfrac{4}{7}$

$\dfrac{72}{v}$ = $\dfrac{4}{7}$ $\hspace{2cm}$[Substituting the value of u]

⇒ v = $\dfrac{72 \times 7}{4}$ $\hspace{2cm}$[Cross multiplying]

⇒ v = 18 × 7

⇒ v = 126

Step 2: Find the value of w

v : w = 9 : 7

$\dfrac{v}{w}$ = $\dfrac{9}{7}$

$\dfrac{126}{w}$ = $\dfrac{9}{7}$ $\hspace{2cm}$[Substituting the value of v]

⇒ w = $\dfrac{126 \times 7}{9}$ $\hspace{2cm}$[Cross multiplying]

⇒ w = 14 × 7

⇒ w = 98

∴ The value of w is 98.

Hence, option 1 is the correct option.

If x : y = 2 : 3 and y : z = 4 : 5, then z : x is equal to:

1. 8 : 15
2. 15 : 8
3. 6 : 5
4. 8 : 5

Answer

Given:

x : y = 2 : 3

y : z = 4 : 5

To find z : x, we make y the same in both the ratios.

L.C.M. of 3 and 4 = 12.

So, we make y equal to 12 in each case.

Step 1: Make y = 12 in both ratios

x : y = 2 : 3 = $\dfrac{2 \times 4}{3 \times 4}$ = $\dfrac{8}{12}$ = 8 : 12

y : z = 4 : 5 = $\dfrac{4 \times 3}{5 \times 3}$ = $\dfrac{12}{15}$ = 12 : 15

Step 2: Combine to get x : y : z

x : y : z = 8 : 12 : 15

Step 3: Find z : x

z : x = 15 : 8

∴ The ratio z : x is 15 : 8.

Hence, option 2 is the correct option.

A bag contains ₹2, ₹5 and ₹10 coins in the ratio 5 : 7 : 8, whose total value is ₹1250. The number of ₹5 coins in the bag is:

1. 70
2. 91
3. 84
4. 78

Answer

Given:

Ratio of number of ₹2, ₹5 and ₹10 coins = 5 : 7 : 8

Total value of coins = ₹1250

Let the number of ₹2 coins = 5k, the number of ₹5 coins = 7k and the number of ₹10 coins = 8k.

Step 1: Find the total value in terms of k

Value of ₹2 coins = 2 × 5k = ₹10k

Value of ₹5 coins = 5 × 7k = ₹35k

Value of ₹10 coins = 10 × 8k = ₹80k

Total value = 10k + 35k + 80k = ₹125k

Step 2: Apply the total value condition

According to the question, the equation can be written as:

125k = 1250

⇒ k = $\dfrac{1250}{125}$

⇒ k = 10

Step 3: Find the number of ₹5 coins

Number of ₹5 coins = 7k = 7 × 10 = 70

∴ The number of ₹5 coins in the bag is 70.

Hence, option 1 is the correct option.

If A : B = 7 : 8 and B : C = 7 : 9, then A : B : C is:

1. 56 : 49 : 72
2. 49 : 56 : 72
3. 56 : 72 : 49
4. 72 : 56 : 49

Answer

Given:

A : B = 7 : 8

B : C = 7 : 9

To find A : B : C, we make B the same in both the ratios.

L.C.M. of 8 and 7 = 56.

So, we make B equal to 56 in each case.

Step 1: Make B = 56 in both ratios

A : B = 7 : 8 = $\dfrac{7 \times 7}{8 \times 7}$ = $\dfrac{49}{56}$ = 49 : 56

B : C = 7 : 9 = $\dfrac{7 \times 8}{9 \times 8}$ = $\dfrac{56}{72}$ = 56 : 72

Step 2: Combine to get A : B : C

A : B : C = 49 : 56 : 72

∴ The ratio A : B : C is 49 : 56 : 72.

Hence, option 2 is the correct option.

A man has 25 paise, 50 paise and 1 rupee coins. There are 220 coins in all and the total amount is ₹160. If there are thrice as many 1 rupee coins as there are 25 paise coins, then the number of 50 paise coins is:

1. 60
2. 120
3. 40
4. 80

Answer

Given:

Total number of coins = 220

Total amount = ₹160 = 16000 paise $\hspace{2cm}$[∵ ₹1 = 100 paise]

Number of 1 rupee coins = 3 × Number of 25 paise coins

Step 1: Express the number of each type of coin

Let the number of 25 paise coins = x.

Then, the number of 1 rupee coins = 3x.

Number of 50 paise coins = 220 − x − 3x

= 220 − 4x

Step 2: Set up the equation using the total amount

Value of 25 paise coins = 25x paise

Value of 50 paise coins = 50 × (220 − 4x) = (11000 − 200x) paise

Value of 1 rupee coins = 100 × 3x = 300x paise

According to the question, the equation can be written as:

25x + (11000 − 200x) + 300x = 16000

25x + 11000 − 200x + 300x = 16000

125x + 11000 = 16000

125x = 16000 − 11000

125x = 5000

⇒ x = $\dfrac{5000}{125}$

⇒ x = 40

Step 3: Find the number of 50 paise coins

Number of 50 paise coins = 220 − 4x

= 220 − 4 × 40

= 220 − 160

= 60

∴ The number of 50 paise coins is 60.

Hence, option 1 is the correct option.

Deepa and Mahima both start reading the same book on the same day. Deepa reads 6 pages a day and Mahima reads 9. What page will Mahima be on when Deepa is on page 72?

1. 81
2. 99
3. 102
4. 108

Answer

Given:

Pages read by Deepa per day = 6

Pages read by Mahima per day = 9

Page number Deepa is on = 72

Since both started on the same day, they have been reading for the same number of days.

Step 1: Find the number of days Deepa has been reading

Number of days = $\dfrac{\text{Total pages read}}{\text{Pages per day}}$

Number of days = $\dfrac{72}{6}$ = 12 days

Step 2: Find the page Mahima is on

Pages read by Mahima in 12 days = 9 × 12 = 108

∴ Mahima will be on page 108 when Deepa is on page 72.

Hence, option 4 is the correct option.

5 mangoes and 4 oranges cost as much as 3 mangoes and 7 oranges. The ratio of the cost of 1 mango to that of 1 orange is:

1. 4 : 3
2. 1 : 3
3. 3 : 2
4. 5 : 2

Answer

Given:

Cost of 5 mangoes and 4 oranges = Cost of 3 mangoes and 7 oranges

Let the cost of 1 mango = ₹m and the cost of 1 orange = ₹o.

Step 1: Set up the equation

Cost of 5 mangoes and 4 oranges = ₹(5m + 4o)

Cost of 3 mangoes and 7 oranges = ₹(3m + 7o)

According to the question, the equation can be written as:

5m + 4o = 3m + 7o

Step 2: Solve for the ratio m : o

5m − 3m = 7o − 4o $\hspace{2cm}$[Rearranging the terms]

2m = 3o

$\dfrac{m}{o}$ = $\dfrac{3}{2}$ $\hspace{2cm}$[Dividing both sides by 2o]

m : o = 3 : 2

∴ The ratio of the cost of 1 mango to that of 1 orange is 3 : 2.

Hence, option 3 is the correct option.

In class 6, the ratio of number of girls to the number of boys is 3 : 5. In 7th class, the ratio is 9 : 13. In which class there are more girls?

1. 6th class
2. 7th class
3. Both have equal number of girls
4. Can't say

Answer

Given:

In class 6, ratio of girls to boys = 3 : 5

In class 7, ratio of girls to boys = 9 : 13

A ratio only compares two quantities; it does not tell us their actual values.

The ratio 3 : 5 in class 6 means that for every 3 girls there are 5 boys, but the total number of students in class 6 is not given.

Similarly, the ratio 9 : 13 in class 7 means that for every 9 girls there are 13 boys, but the total number of students in class 7 is not given.

Without knowing the total number of students in each class, the actual number of girls in each class cannot be determined.

∴ We can't say in which class there are more girls.

Hence, option 4 is the correct option.