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RATIO AND PROPORTION

Ratio and Proportion topic majorly based on fractions. When a fraction is represented in the form of x : y, then it is a ratio whereas a proportion states that two ratios are equal. Here, x and y are any two integers. Here, ‘x’is called the first term or antecedent, and ‘y’ is called the second term or consequent. In proportion, the two ratios are x : y & u : v. The two terms ‘y’ and ‘u’ are called ‘means or mean term.’ whereas the terms ‘x’ and ‘v’ are known as ‘extremes or extreme terms.’

        If x : y = u : v, then v is called the fourth proportional to x, y, u. u is called the third proportion to x and y. Mean proportional between x and y is √(xy).

RATIO: Ratio means Comparison. The number of times one quantity contains another quantity of the same kind.
            Thus the ratio between 5 litres of oil and 15 litres of oil can be possible, but not between 10 apples and 25 kg of rice. 
* The ratio between one quantity to another is measured by  a : b or a/b
Ex: 8 : 9   or   5 : 7 etc.
* The two quantities in the ratio are called its terms. The first is called the antecedent and the second term is called consequent.
* The terms of the ratio can be multiplied or divided by the same number.

Types of Ratios:
1. Duplicate ratio: The ratio of the squares of the two numbers.
     Ex: 9 : 16 is the duplicate ratio of 3 : 4.
2. Triplicate Ratio: The ratio of the cubes of the two numbers.
     Ex: 27 : 64 is the triplicate ratio of 3 : 4.
3. Sub-duplicate Ratio: The ratio between the square roots of the two numbers.
      Ex: 4 : 5 is the sub-duplicate ratio of 16 : 25.
4. Sub-triplicate Ratio: The ratio between the cube roots of the two numbers.
      Ex: 4 : 5 is the sub-triplicate ratio of 64 : 125.
5.Inverse ratio: If the two terms in the ratio interchange their places, then the new ratio is inverse ratio of the first.
      Ex: 9 :5 is the inverse ratio of 5 : 9.
6. Compound ratio: The ratio of the product of the first terms to that of the second terms of two or more ratios.
      Ex: The compound ratio of  
PROPORTION: If two ratios are equal, then they make a proportion.
            Thus  
Each term of the ratios 4/5 and 8/10  is called proportional.
The middle terms 5 and 8 are called means and the end terms 4 and 10 are called extrems.

Product of Means = Product of Extremes

Continued Proportion: In the proportion 8/12=12/8,  8, 12, 18 are in the continued proportion.
Fourth proportion: If a : b = c : x, then x is called fourth proportion of a,b and  c.
There fore fourth proportion of  a, b, c  = bc/a
Third proportion: If a : b = b : x, then x is called third proportion of a and b.
Therefore third proportion of a, b =  bb/a
Second or mean proportion: If a : x = x : b , then x is called second or mean proportion of a and b.
Therefore mean proportion of a and b =  root(ab)

 

Model Questions

1. Three friends A, B and C divided Rs.5200 among themselves in the ratio 1/2 : 1/3 : 1/4 the share of A is? (in Rs.)
1) 1600                 2) 1200                 3) 1800            4) 2400               5) 1500
Explanation: Share of A = 6 : 4 : 3 
= 6/13 x  5200 = 2400 
Ans: 4

 

2. 20% of a one number is equal to 3/4th  of another number then find the ratio of the two numbers?
1) 15 : 4                2) 4 : 1                3) 7 : 4           4) 7 : 5                5) 4 : 3 
Explanation: 20% of x = 3/4 y

Ans: 1

 

3. A sum of money is divided between two persons in the ratio of 12 : 23. If the share of one person is Rs.220 less than that of the other, what is the sum?
1) Rs.500              2) Rs.600              3) Rs.700            4) Rs.800          5) Rs.900
Explanation: 11p 220
                        35p
Sum = 35p/11p × 220 = 700 
Ans: 3

 

4. Two numbers A and B are such that the sum of 5% of A and 4% of B is two-third of the sum of 6% of A and 8% of B. Find the ratio of A : B?
1) 1 : 2                2) 3 : 1                3) 3 : 4             4) 4 : 3               5) 1 : 6 

 

5. 120 apples is to be divided among three friends A, B and C. C gets half of what A and B together gets and B gets 5/3  times as much as A. How many apples does A have?
1) 10       2) 20       3) 30       4) 60       5) 50
Explanation: A = 3x, B = 5x, C = 4x 
Apples does A have = 3x/12x × 120 = 30 
Ans: 3

 

6. The ratio between a two - digit number and the sum of the digits of that number is 4 : 1. If the digit at the unit’s place is 3 more than the digit at the ten’s place, then the number is.....
1) 47       2) 69       3) 36       4) 25       5) 32
Explanation: The digit at the unit place is 3 more than the digit at the ten’s place, then the numbers can be 14, 25, 36, 47, 58, 69. Only 36 satifies the given condition (4 : 1)
Ans: 3

 

7. 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......
1) 20 : 66 : 95          2) 21 : 66 : 95          3) 21 : 66 : 92        4) 19 : 66 : 92         5) 11 : 36 : 92

 

8. Rs.8500 is divided among 4 men, 5 women and 6 boys. So that the share of a man, a woman and a boy are in the ratio 9 : 8 : 4. What is the share of a boy?
1) Rs.300         2) Rs.250         3) Rs.340           4) Rs.400       5) Rs.150
                                       men    women    boy
Explanation: Share = 9x × 4 : 8x × 5 : 4x × 6
                   = 36x : 40x : 24x
                   = 9x : 10x : 6x
                    25x 8500
                   6x
8500/25 × 6  =  2040
6 boys = 2040 
1boy = 2040/6 = 340 
Ans: 3

 

9. In a mixture of 90 liters, the ratio of milk and water is 2 : 1. What amount of water must be added to make the ratio of milk and water is 3 : 4?
1) 50       2) 72       3) 80       4) 30       5) 25 
Explanation: 60/30 + x = 3/4   
240 = 90 + 3x ⇒ x  = 50
Ans: 1

 

10. In what ratio must a person mix three kinds of tea costing Rs.60/kg, Rs.75/kg and Rs.100/kg so that the resultant mixture when sold at Rs.96/kg yields a profit of 20%?
1) 1 : 2 : 4          2) 3 : 7 : 6          3) 1 : 4 : 2           4) 1 : 3 : 5          5) 1 : 3 : 4
Explanation: Sp (120% 96 Rs/kg)
Cp (100% 80 Rs/kg)
Through options
1. 1(60) + 2(75) + 4(100) = 610
             ≠ (1 + 2 + 4)80 = 560
2. 3(60) + 7(75) + 6(100) = 1305    
            ≠ (3 + 7 + 6)80 = 1280
3. 1(60) + 4(75) + 2(100) = 560
          = (1 + 4 + 2)80 = 560
4. 1(60) + 3(75) + 5(100) = 785
            ≠ (1 + 3 + 5)80 = 720
5. 1(60) + 3(75) + 4(100) = 685
            ≠ (1 + 3 + 4) 80 = 640
Option 3 is correct
Ans: 3 

 

11. A bag contains one rupee, 50 paisa and 25 paisa coins and ratio of their value is 30 : 11 : 7 and the total number of coins are 480. Find the number of 50 paisa coins?
1) 142              2) 132            3) 122          4) 112         5) 100
Explanation: Value ratio = 30 : 11 : 7
No. of coins ratio = 30 : 22 : 28
(30p + 22p + 28p)
80p 480
22p ?
480/80 × 22 = 132 
 Ans: 2

 

12. Some parrots and pigeons are living in a room of Nehru Zoological Park. Each parrot can eat 7 carrots a day and each pigeon can eat 9 carrots a day. One day 35 birds can eat 295 carrots, then find how many parrots in the room?
1) 25           2) 10         3) 15         4) 20         5) 35
Explanation: Let No. of parrots = x
No. of pigeons = (35 - x)
7(x) + 9(35 - x) = 295
2x = 20 ⇒ x = 10
Ans: 2

 

13. A rabbit takes 7 steps for every 6 steps of a cat, but 4 steps of a cat are equal to 6 steps of a rabbit. What is the ratio of speed of rabbit to that of cat?
1) 7 : 3                2) 7 : 9                3) 2 : 3              4) 7 : 6               5) 4 : 5
Explanation: 4 steps of cat = 6 steps of rabbit Ratio of speed of rabbit and cat = 7 × 4/ 6 × 6 = 7/9
Ans: 2

 

14. Certain sum of money was to be distributed among P, Q and R in the ratio of 2 : 3 : 4. Instead, it was distributed among P, Q, R and S in the ratio of 2 : 3 : 4 : 5. If the loss to P is Rs.40 then what is the amount received by S?
1) 100           2) 60          3) 120          4) 150          5) 180
Explanation: Amount received by P is 
⇒ 2P/9 - 2P/14 = 40 
P/9 - P/14 = 20
(14 − 9)P/126  = 20 ⇒  5P/126 = 20 ⇒ P = 504
Amount received by S = 5/14 (504) = 180 
Ans: 5

 

15. 20 Boys and 25 girls form a group of social workers. During their membership drive, the same number of boys and girls joined the group (e.g. if 7 boys joined, 7 girls joined). How many members does the group have now, if the ratio of boys to girls is 7 : 8?
1) 75            2) 80           3) 15          4) 85          5) 70
Explanation: 20 + x/25 + x = 7/8 
160 + 8x = 175 + 7x ⇒ x = 15
Total members are = 20 + 25 + 15 + 15 = 75
Ans: 1

 

16. When 15 male employees went on leave, then for every male there are 2 female employees in an office. After that 45 females went on leave. Now, for every females, there are 5 males. Find out the number of male employees initially?
1) 40             2) 50           3) 30          4) 45           5) 25
Explanation: Let total number of male employees is x
2(x - 15) = 5(x + 15 - 45)
2x - 30 =  5x −150 ⇒ 3x = 120 ⇒ x = 40
Ans: 1

 

17. In a cricket match the total number of runs scored by Sachin, Vinod and Sourav is 285. The ratio of the number of runs scored by Sachin and Sourav is 3 : 2 and that of the runs scored by Sourav and Vinod is also 3 : 2. The number of runs scored by Sachin in that match is.......
1) 135            2) 90             3) 60         4) 140          5) 120
Explanation: Sachin & Saurav = 3 : 2    × 3 
Sachin & vinod =  3 : 2 × 2/9 : 6 : 4 
NO. of runs scored by sachin = 9/19 × 285
= 135
Ans: 1

 

Ratio, Proportion and Partnership

1. If 28 : x : : x : 7, then find the value of x.
a) 7            b) 14            c) 28            d) 196 
Sol: 28/x = x/7  ⇒ x2 = 28 × 7 ⇒ x = 14
Ans: b

2. If Rs.195 is divided in the ratio 1/2 :1/3 : 1/4 then what is the largest part?
a) Rs.60     b) Rs.45     c) Rs.90     d) Rs.150 
Sol: Given ratio =  1/2, : 1/3 : 1/4 = 6/12 : 4/12 : 3/12
i.e. 6 : 4 : 3

6. If Rs.1,030 is divided among A, B, C and D in such a way that A : B = 4 : 5, B : C = 3 : 5 and C : D = 3 : 2. Who will get the maximum amount?
a) A             b) B             c) C             d) D
Sol: A : B = 4 : 5 and B : C = 3 : 5 ⇒ A : B : C = 4 × 3 : 5 × 3 : 5 × 5 = 12 : 15 : 25 A : B : C = 12 : 15 : 25 and C : D = 3 : 2
⇒ A : B : C : D = 12 × 3 : 15 × 3 : 25 × 3 : 25 × 2 = 36 : 45 : 75 : 50
Hence, C gets the maximum amount.
Ans: c

7. The age of a man and his son is in the ratio 4 : 1. Five years hence the respective ratio of their ages will be 3 : 1. How old is the father at present?
a) 40 years       b) 60 years       c) 75 years        d) 100 years
Sol: Let father’s age be 4x and son’s age be x. 

9. How many one rupee, 50 paisa and 25 paisa coins respectively are there, if these are in the ratio 4 : 5 : 6 respectively and together amount to Rs.32?
a) 16, 24, 20           b) 24, 16, 20           c) 24, 20, 16             d) 16, 20, 24
Sol: Let 4x, 5x and 6x be the number of coins of one rupee, 50 paisa and 25 paisa respectively. 

 Ans: d

10. Father's age was five times his son's age 5 years ago and will be three times the son's age after 2 years. What is the ratio of the present age of father to that his son?
a) 10 : 3       b) 7 : 2       c) 4 : 1       d) 9 : 2
Sol: Let ‘f’ be the age of father and ‘s’ be the age of son.
f - 5 = 5(s − 5) ⇒ f - 5s = −20 ...(i)
and f + 2 = 3(s + 2) ⇒ f − 3s = 4 ...(ii)
Subtracting (i) from (ii), we get 2s = 24 ⇒ s = 12 years
∴ f = 40 years
∴ Required ratio =  40/12 = 10/3 i. e.10 : 3 
Ans: a

11. Earnings of A and B are in the ratio 7 : 9 respectively and their expenditures are in the ratio 9 : 7 respectively. What is the ratio of their savings?
a) 1 : 7       b) 7 : 5        c) 1 : 1            d) Can't be determined
Sol: Either incomes or expenditure must be given.
Hence, it cannot be determined.
Ans: d

13. A bag contains 25 paisa, 10 paisa and 5 paisa coins in the ratio of 1 : 2 : 3 respectively. If their total value is Rs.45, then number of 10 paisa coins is.....
a) 75          b) 150          c) 225          d) 175
Sol: Let x, 2x and 3x be the number of coins of 25 paisa, 10 paisa and 5 paisa respectively. 
Then,  25/100  × x +  10/100 × 2x +  5/100  × 3x = 45 
⇒ 25x + 20x + 15x = 4500 ⇒ 60x = 4500 ⇒ x = 75 ∴ Number of 10 paisa coins = 2 × 75 = 150
Ans: b 

 


15. The salaries of A, B and C are in the ratio 1 : 2 : 5. If the increments of 20%, 15% and 10% respectively are allowed in their salaries, then what will be the new respective ratio of their salaries?
a) 12 : 23 : 55       b) 11 : 23 : 55      c) 23 : 11 : 55        d) 12 : 55 : 23
Sol: Let A = k, B = 2k, C = 5k

18. Two numbers are in the ratio of 7 : 4. If the largest number is 81 more than the smaller number, then find the numbers?
a) 179 and 98       b) 189 and 108       c) 211 and 292         d) 290 and 371
Sol: Let 7x and 4x be the numbers. Then,
7x − 4x = 81 ⇒ 3x = 81 ⇒ x = 27
∴ Numbers are 189 and 108.            
Ans: b

 

20. Find the fourth proportional to 7, 9 and 14?
a) 10           b) 12           c) 16           d) 18
Sol: Let fourth proportional be x. Then,
7 : 9 = 14 : x
∴ x = 14 × 9/ 7 = 18 
Ans: d

21. The ratio of the third proportion of 3 and 4 with the fourth proportion of 6, 8 and 10 respectively is....
a) 2 : 5        b) 5 : 2        c) 3 : 5        d) 5 : 3 

22. 465 coins consists of 1 rupee, 50 paisa and 25 paisa coins. Their values are in the ratio 5 : 3 : 1. The number of each type of coins respectively is....
a) 155, 186, 124       b) 154, 187, 124       c) 154, 185, 126        d) 150, 140, 175
Sol: Ratio of respective number of coins = 5 : 6 : 4 
|
Ans: a

23. A starts business with Rs.7,000 and after 5 months, B joined as a partner. After a year the profit is divided in the ratio 2 : 3. The capital of B is?
a) Rs.10,000        b) Rs.6,500           c) Rs.18,000           d) Rs.9,000
Sol: Let amount invested by B be Rs.x, then

Ans: c

 

Posted Date : 19-07-2021

 

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