Ratio and Proportion

Q1.| There are three containers of equal capacity. The ratio of Sulphuric acid to water in the first container is 3 : 2, that in the second container is 7 : 3 and in the third container it is 11 : 4. If all the liquids are mixed together, then the ratio of Sulphuric acid to water in the mixture will be?

(a) 61 : 29 (b) 61 : 28

Ans- (a)

Let 30L be the capacity of Container

Sulphuric Acid. Water.

1. 18. 12. (3:2)

2. 21. 9. (7:3)

3. 22. 8. (11:4)

Total = 61. 29.

Q2.| 200 litres of a mixture contains milk and water in the ratio 17 : 3. After the addition of some more milk to it, the ratio of milk to water in the resulting mixture becomes 7 : 1. The quantity of milk added to it was?

(a) 20 litres (b) 40 litres

Ans- (b)

Milk. Water

200L. 17. 3 (Water remains

200 + x L. 3. 1. Constant)

200 L → 17 : 3

200 + x L → 21 : 3

Difference is Ratio of milk = 21 – 17 = 4

20 unit → 200 L

1 unit → 10 L

4 unit → 40 L

Q3.| A Can contains a mixture of two liquids A and B in the ratio 7 : 5. When 9 liters of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7 : 9. Liters of liquid A contained by the can initially was?

(a) 10 (b) 20

Ans- (c)

A. B

X-9. 7. 5

X-9+9. 7. 9

Difference between ratio of B = 9 – 5 = 4

4 unit → 9 liters

1 unit → 9/4 liters

12 unit → 9/4× 12

= 27 liters

Liquid Initially in the can = 27 + 9

= 36 liters

Liters of liquid A in can = 36 × 7/12

= 21 liters

Q4.| Two alloys are both made up of copper and tin. The ratio of copper and tin in the first alloy is 1 : 3 and in the second alloy is 2 : 5. In what ratio should the two alloys be mixed to obtain a new alloy in which the ratio of tin and copper be 8 : 3?

(a) 3 : 5 (b) 4 : 7

Ans- (b)

Part of Copper in Ist alloy = 1/4

Part of Copper in IInd alloy = 2/7

Part of Copper in Resultant mixture = 3/11

Using Allegation

I. II.

1. 2

4. 7

22 - 21 : 12 - 11

77. : 44

1:77. ;. 1/44

4:7

Q5.| In two alloys A and B, the ratio of zinc to tin is 5 : 2 and 3 : 4 respectively. Seven kg of the alloy A and 21 kg of the alloy B are mixed together to form a new alloy. What will be the ratio of zinc and tin in the new alloy?

(a) 2 : 1 (b) 1 : 2

Ans- (b)

Quantity of zinc in alloy A = 7 × 5/7 = 5

Quantity of tin in Alloy A = 7 × 2/7 = 2

Quantity of zinc in Alloy B = 21 × 3/7 = 9

Quantity of tin in Alloy B = 21 × 4/7 = 12

zinc Tin

Ratio in New Alloy → 5+9. 12+2

14. 14

1 : 1

Q6.| A mixture contains wine and water in the ratio 3 : 2 and another mixture contains them in the ratio 4 : 5. How many liters of the later must be mixed with 3 liters of the former so that the resulting mixture may contain equal quantities of wine and water?

(a) 5×2/5 laters. (b) 5×2/3 laters

Ans- (a)

Wine : Water

I 3 : 2. 3 liters

II 4 : 5. x liters

In 3 liters , wine = 3 × 3/5 = 9/5 ,

Water = 2 × 3/5 = 6/5

In x liters , wine = x × 4/9 = 4x/9 ,

Water = x × 5/9 = 5x/9

In mixture → wine = 9/4 + 4x/9

Water = 6/5 + 5x/9

Wine : Water = 1 : 1

9/4 + 4x/9 : 6/5 + 5x/9

X= 27/5 = 5×2/5 liters

Q7.| A vessel contains 20 liters of acid. 4 liters of acid is taken out of the vessel and replaced by the same quantity of water. Next 4 liters of the mixture are withdrawn, and again the vessel is filled with the same quantity of water, ratio of acid left in the vessel with the quantity of acid initially in the vessel is?

(a) 4 : 5 (b) 4 : 25

Ans- (c)

Final quantity = Initial quantity

{ 1 - Quantity take out } power n

{. Initial quantity. }

n → no. of times repeated

Final = Initial ( 1-4/20 ) Power 2

Final = 16

Initial = 25

Q8.| 60 kg of an alloy A is mixed with 100 kg of alloy B. If alloy A has lead and tin in the ratio 3 : 2 and alloy B has tin and lead in the ratio 1 : 4, the amount of tin in the new alloy is?

(a) 53 kg (b) 44 kg

Ans- (b)

Lead in 60 kg of Alloy A = 3/5 × 60 = 36

Tin in 60 kg of Alloy A = 2/5 × 60 = 24

Lead in 100 kg of Alloy B = 4/5 × 100 = 80

Tin in 100 kg of Alloy B = 1/5 × 100 = 20

Tin in the mixture = 24 + 20 = 44 kg

Q9.| Annual incomes of Amit and Veeri are in the ratio 3 : 2, while the ratio of their expenditure is 5 : 3. If at the end of the year each saves Rs. 1,000, the annual income of Amit is?

(a) Rs. 9,000 (b) Rs. 8,000

Ans- (d)

Amit. Veeri.

Income - 3. 2

Expenditure -. 5. 3

Amit’s saving = 3x – 5y

Veeri’s saving = 2x – 3y

3x – 5y = 1000 …(i)

2x – 3y = 1000 …(ii)

Solving (i) & (ii)

x = 2000

Income of Amit = 3x

= 3 × 2000

= 6000 Rs.

Q10.| A and B have their monthly incomes in the ratio 8 : 5, while their monthly expenditures are in the ratio 5 : 3. If they have saved Rs. 12,000 and Rs. 10,000 monthly respectively, then the difference in their monthly incomes is?

(a) Rs. 52,000 (b) Rs. 42,000

Ans- (b)

A. B

Income - 8. 5

Expenditure -. 5. 3

Income of A = 8x

Expenditure of A = 5y

8x – 5y = 12000 …(i)

Income of B = 5x

Expenditure of B = 3y

5x – 3y = 10000 …(ii)

From (i) & (ii)

x = Rs. 14000

Difference between salaries of A & B

= 8x – 5x

= 3x = 3 × 14000 = Rs. 42000

Q11.| A man divides his property so that his son’s share to his wife’s and wife’s share to his daughter’s are both as in the ratio 3 : 1. If the daughter gets Rs. 10,000 less than son, the value (in rupees) of the whole property is?

(a) Rs. 16,250 (b) Rs. 16,000

Ans- (a)

S : W = 3 : 1

W : D = 3 : 1

S : W : D = 9 : 3 : 1 ⇔ 9 + 3 + 1 = 13

Difference in Son & Daughter = 9-1=8

8 unit → 10,000

1 unit → 10000/8

= 1250

13 unit → 13 × 1250

= 16250 Rs.

Q12.| A bag contains 1rs , 50 paise & 25 paise coins and the ratio of their value is 30 : 11 : 7 and the total no. of coins are 480. Find the number of 50p coins?

(a) 142 (b) 132

Ans- (b)

1 Rs. 50 paise 25 paise

Value 30. : 11. : 7

No. of coins 30 : 22. :. 28

30 + 22 + 28 → 480

80 → 480

1 → 6

22 → 132

Q13.| The ratio of ages of Ram &Shyam 5 years ago is 2 : 3 and the Ratio of their ages after 5 years should be 3 : 4. Find the sum of their present ages?

(a) 70 (b) 80

Ans- (c)

Ram Shyam

-5 year. 2. : 3

+5 year. 3 : 4

10 years 1 1

1 unit → 10 years

Ram’s age 5 years ago = 20,

Shyam’s age 5 years ago = 30 years

Present age of Ram + Shyam = 20 + 30 + 10 = 60 years

Q14.| The ratio of the numbers of boys and girls in a school was 5 : 3. Some new boys and girls were admitted to the school, in the ratio 5 : 7. At this, the total no. of students is the school become 1200 and the ratio of boys to girls changed to 7 : 5. The number of students in the school before new admissions was?

(a) 700 (b) 720

Ans- (d)

Boy. Girls

5. :. 3

5x. 3x. = 8x

Boys. Girls

New comers. 5. : 7

5x. 7x. = 12y

8x + 12y = 1200

2x + 3y = 300 …(i)

5x+5y = 7

3x+7y. = 5

25x + 25y = 21x + 49y

x = 6y …(ii)

From (i) & (ii)

12y + 3y = 300

y = 20 & x = 120

No. of students initially = 8x

= 8 × 120

= 960

Q15.| 555 Rs. was to be divided among A, B and C in the ratio of 1/4; 1/5; 1/6. But by mistake it was divided in the ratio 4 : 5 : 6. The amount in excess received by C was?

(a) Rs. 72 (b) Rs. 75

Ans- (a)

A + B + C = 555

A : B : C = 1/4 : 1/5 : 1/6

= 30 : 24 : 20

= 15 : 12 : 10

15x + 12x + 10x = 555

37x = 555

x = 15

C = 15 × 10 = 150

A : B : C = 4 : 5 : 6

4x + 5x + 6x = 555

15x = 555

x = 37

C gets = 37 × 6 = 222

Difference = 222 – 150

= 72 Rs.

Q16.| Four years ago, the ratio of A’s age to B’s age was 11: 14 and four years later their ages will be in the ratio 13 : 16. The present age of A is?

(a) 48 years (b) 26 years

Ans- (a)

A. : B.

Four years ago - 11. :. 14.} (2r= Four years later -. 13. :. 16} difference)

This 2r = 8 years

1r = 4 years

So, present age of A = (13 × 4) – 4

= 48 years

Q17.| In an alloy, zinc and copper are in the ratio 1: 2. In the second alloy, the same elements are in the ratio 2 : 3. If these two alloys be mixed to form a new alloy in which two elements are in the ratio 5 : 8, the ratio of these two alloys in the new alloys is?

(a) 3 : 10 (b) 3 : 7

Ans- (a)

I alloy → Zn: Cu = 1: 2] × 65 ⇒ Zinc = 65

II alloy → Zn: Cu = 2: 3] × 39 × 2⇒ Zinc = 78

New alloy → Zn: Cu = 5: 8] × 15 ⇒ Zinc = 75

LCM of (3, 5, 13) = 195

By the rule of Alligation,

65. 78

\ /

/. \

3. 10

Required ratio = 3 : 10

Q18.| The salaries of A, B and C are in the ratio 1: 3 : 4. If the salaries are increased by 5%, 10% and 15% respectively, then the increased salaries will be in the ratio?

(a) 20 : 66 : 95 (b) 21 : 66 : 95

Ans- (c)

Let the salaries be 100, 300 and 400.

New salary after increament of 5%, 10% & 15% respectively.

105, 330 and 460

New ratio,

21 : 66 : 92

Q19.| In a mixture of 75 litres, the ratio of milk to water is 2 : 1. The amount of water to be further added to the mixture so as to make to the ratio of the milk to water 1 : 2 will be?

(a) 45 litres (b) 60 litres

Ans- (c)

1. -. M : W = 2 : 1

New ratio -. M : W =( 1 : 2 ) × 2

2 : 1. (3r)

2 : 4

According to the question,

(2+1)r = 3r = 75 l

Amount of water to be added = 3r

= 75 litres

Q20.| The ratio in which two sugar solutions of the concentrations 15% and 40% are to be mixed to get a solution of concentration 30% is?

(a) 2 : 3 (b) 3 : 2

Ans- (a)

S : W

Sugar type I 15:85 3:17

Sugar type II 40:60 2:3

Sugar mixture III 30:70 3:7

L.C.M (20,5,10) = 20

3. 8

\. /

/. \

2. : 3

Q21.| A boy has a few coins of denominations 50 paise, 25 paise and 10 paise in the ratio 1 : 2 : 3. If the total amount of the coins is Rs. 6.50, the number of 10 paise coins is?

(a) 5 (b) 10

Ans- (c)

50p :. 25p. :. 10p

Ñ. -. 1. :. 2. :. 3

Value. -. 50p : 50p : 30p

Total value= Rs. 1.3 → Rs. 6.50

Re 1 → 5

The no. of 10 paise coins = 15

Q22.| The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The measure of the largest angle of the quadrilateral is?

(a) 144° (b) 108°

Ans- (a)

According to the question,

(1 + 2 + 3 + 4)r → 360°

10r → 360°

1r → 36°

Largest angle = 4r = 4 36

= 144°

Q23.| The speeds of A and B are in the ratio 3: 4. A takes 20 minutes more than B to each a

destination. In what time does A reach the destination?

(a) 1×1/3 hours (b) 2 hours

Ans- (a)

A. :. B

Speed. -. 3. :. 4

Time. -. 4. :. 3

According, to the question,

(4 – 3) r = 20 m

1r → 20 m

Time taken by A → 4r = 80 min.

= 1 ×1/3 hr.

Q24.| The expenses on rice, fish and oil of a family are in the ratio 12: 17: 3. The prices of these articles are increased by 20%, 30% and 50% respectively. The total expenses of family on these articles are increased by?

(a) 14 × 1/8% (b) 7 × 1/8%

Ans- (d)

Rice fish. Oil

Experience. - 12. :. 17. :. 3

Let the expenses be → 120, 170 and 30 respectively.

Total → 320

Increased price → 144, 221 and 45 respectively.

Total → 410

% increment = 90/320 × 100

= 28 × 1/8 %

Q25.| In a school having roll strength 286, the ratio of boys and girls is 8 : 5. If 22 more girls get admitted into the school, the ratio of boys and girls becomes?

(a) 12 : 7 (b) 10 : 7

Ans- (d)

Strength of school = 286

B: G → 8: 5

13r → 286

1r → 22

New Girls admitted = 22

New ratio = 8: 6

= 4: 3

Q26.| The bus fare and train fare of a place from Kolkata were Rs. 20 and Rs. 30 respectively. Train fare has been increased by 20% and the bus fare has been increased by 10%. The ratio of new train fare to new bus fare is?

(a) 11 : 18 (b) 18 : 11

Ans - (b)

Bus fare = Rs. 20

Increment = 10%

New Bus fare = Rs. 22

Train fare = Rs. 30

Increment = 20%

New train fare = Rs. 36

Train. Bus

New ratio. -. 36. :. 22

18 :. 11

Q27.| With a given rate of simple interest, the ratio of principal and amount for a certain period of time is 4 : 5. After 3 years, with the same rate of interest, the ratio of the principal and amount becomes 5 : 7. The rate of interest is?

(a) 4% (b) 6%

Ans- (c)

Principal: Amount = 4: 5

After three years → P: A = 5: 7

Principal will remain same

So, equating principal,

4: 5] ×5

5: 7] ×4

P :. A

-. 20 : 25

-. 20 :. 28

Now,

This difference of amount will give difference of interest.

In 3 years interest = 3r

So, for 1-year interest = 1r

I = P × r × t

100

1r = 20r × Rate × 1

100

Rate = 5 %

Q28.| A, B and C completed a work costing Rs. 1,800. A worked for 6 days, B for 4 days and C for 9 days. If their daily wages are in the ratio of 5 : 6 : 4, how much amount will be received by A?

(a) Rs. 800 (b) Rs. 600

Ans- (b)

A → 6, B → 4, C → 9

Daily wages = 5 : 6 : 4

Days working = 6 : 4 : 9

Amount = 30 : 24 : 36

According to the question,

(30 + 24 + 36) r → Rs. 1800

90r → Rs. 1800

1r → Rs. 20

Amount received by A = Rs. 20 × 30

= 600

Q29.| The volumes of a sphere and a right circular cylinder having the same radius are equal, The ratio of the diameter of the sphere to the height of the cylinder is?

(a) 3 : 2 (b) 2 : 3

Ans- (a)

Volume (cylinder) = Volume of sphere

πr power 2 ×h = 4/3 πr power 3

h = 4r/3

Diameter of sphere =. 2r.

h. (4r/3)

3 : 2

Q30.| An employer reduces the number of his employees in the ratio 9: 7 and increases their wages in the ratio 14: 15. State whether his bill of total wages increases of decreases, and in what ratio?

(a) 11: 15 (b) 20: 21

Ans- (c)

We know the bill = wage per person x no. of total employees

Ratio of change in bill = 9 × 14: 7 × 15

= 6: 5

Then, there is decrease in bill.