### RATIO AND PROPORTION

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)

EXAMPLES

1. a : b = 3: 4; b : c = 6 : 7. Find a : b : c.
Sol:    a     b      c
3     4
6      7
a : b : c = 3 × 6 : 6 × 4 : 4 × 7 = 9 : 12 : 14

2. A sum of Rs.4960 has been divided among A, B and C in the ratio of 5:4:7. Find the share of B.
Sol: B's share =  4/5+4+7*4960
= Rs.1240

3. 36% of first number is 28% of the second number. What is the respective ratio of the first number to the second number?
Sol:  Let the numbers be x and y.
36% of x = 28% of y
x/y=28/36=7/9
∴   x : y = 7 : 9

4. Two numbers are in 4:7 ratio. The difference between them is 27. What is the bigger number?
Sol:  Let the numbers be 4x and 7x.
7x − 4x = 27
⇒ 3x = 27  ⇒  x = 9
∴ Bigger number is 7x = 7 × 9 = 63
Short cut: The difference of the terms of the ratio = 7 − 4 = 3.
But the actual difference between the numbers is 27
∴  3 parts is equal to 27
7 parts (Bigger number) = 7/3 × 27 = 63

5. The ratio of the ages of a man and his son is 7: 3. The average of their ages is 30 years. What will be the ratio of their ages after 4 years?
Sol: Average age = 30 years
Total age = 2 × 30 = 60 years.
Let their present ages be 7x and 3x years
∴ 7x + 3x = 60 ⇒ x = 60/10 = 6
∴ Their present ages are
7 × 6 and 3 × 6 = 42 and 18.
∴ Their ages after 4 years
= 42 + 4 and 18 + 4 = 46 and 22 years
∴ ratio = 46 : 22 = 23 : 11

6. Two numbers are in the ratio of 3:4. If 4 is subtracted from each, the remainders are in the ratio of 5:7. What are the numbers?
Sol: Let the numbers be 3x and 4x.
If 4 is subtracted from each, the numbers will be (3x −4) and (4x −4).
∴ (3x −4) : (4x −4) = 5: 7
Product of means = Product of extremes
(3x –4) 7 = (4x – 4) 5
⇒ 21x – 28 = 20x – 20
⇒ x = 8
∴ The numbers are 3 × 8 and 4 × 8
= 24 and 32

7. In a bowl there is 30 litre mixture of milk and water. The ratio of milk and water is 7:3. How much water must be added to it so that the ratio of milk to the water be 3:7?
Sol : Milk quantity in the mixture
= 7/10 × 30 = 21 litres
∴ Water = 30 - 21 = 9 litres
New ratio = 3 : 7
∴ 3 parts of milk is 21 litres (There is no difference in the milk quantity of new mixture)
∴ Water quantity in the mixture
=7/3 × 21 = 49 litres
∴ 49 - 9 = 40 litres  water is to be added in the new mixture

8. A bag contains of one rupee, 50 paise and 25 paise coins. if these coins are in the ratio of 5 : 6 : 8, and the total amount of coins is Rs. 210, find the number of 50 paise coins in the bag.
Sol : Let the number of one rupee, 50 paise, 25 paise coins be 5, 6 and 8 respectively
The value of one rupee coins
= Rs. 1 × 5 = Rs. 5
The value of fifty  paise coins
= Rs. 0.50 × 6 = Rs. 3
The value of twenty five paise coins
= Rs. 0.25 × 8 = Rs. 2
Total value = 5 + 3 + 2 = Rs. 10
If the total value is Rs. 10
there are 6 coins of fifty paise
if the total value is Rs. 210, then the number of 50 coins = × 6 = 126

9. If a sum of Rs.3150 were distributed among Ravi, Vijay and Suresh in the ratio of 12:9:14 respectively, then find the share of Vijay.
Ans: Rs.810
Sol: Vijay's Share = 9/35 × 3150 = Rs.810

10. The total number of students in a school is 2850. If the number of boys in the school is 1650, then what is the respective ratio of the total number of boys to the total number of girls in the school?
Ans: 11:8
Sol: Total number of students = 2850
Number of boys = 1650
Number of girls = 2850-1650 = 1200
Ratio between boys and girls
=1650 : 1200 = 11 : 8

11. A sum of money is divided among A, B, C and D in the ratio of 5 : 6 : 12 : 15 respectively. If the share of C is Rs. 4092, then what is the total amount of money?
Ans: Rs. 12958
Sol: Let the share of A, B, C and D be Rs. 5x, 6x, 12x and 15x respectively. C's share is Rs.4092
⇒ 12x = 4092 ⇒ x= 4092/12 = 341
∴ Total money = 38x = 38 × 341= Rs.12958

12. Asum of Rs. 2820 has been distributed among A, B and C in the ratio respectively. What is the share of B?
Ans: Rs. 900
Sol: LCM of 3, 4 and 5 is 60
∴ ratio =60/3:60/4:60/5  = 20 : 15 : 12
B's share = 15/20+15+32 × 2820
= 15/47  × 2820 = Rs. 900

13. A, B and C divide an amount of Rs. 6300 amongst themselves in the ratio of 7:6:8 respectively. If an amount of Rs.300 is added to each of their shares, what will be the new respective ratio of their shares of amount?
Ans: 8 : 7 : 9
Sol: Total shares = 7 + 6 + 8 = 21
∴ 21 parts = 6300
each part = 6300/21 = 300
∴  Their shares are

7 × 300, 6 × 300 and 8 × 300

⇒ 2100, 1800 and 2400

If 300 is added to each of them then their shares are 2400, 2100 and 2700

Their ratio = 2400 : 2100 : 2700

= 8 : 7 : 9

14.  Find out the two quantities whose difference is 30 and the ratio between them is 5/11.
Sol: The difference of quantities, which are in the ratio 5:11, is 6. To make the difference 30, we should Multiply them by 5.
Therefore 5:11=5*5:11*5=25:25

15. A factory employs skilled workers, unskilled workers and clerks in the ratio 8:5:1 and the wages of a skilled worker, an unskilled worker and a clerk are in the ratio 5:2:3 when 20 unskilled workers are employed the total daily wages fall amount to Rs. 318. Find out the daily wages paid to each category of employees.
Sol: Number of skilled worker: unskilled worker: clerks = 8:5:1 and the ratio of their respective Wages = 5:2:3

Hence the amount will be paid in the ratio 8 × 5 : 5 × 2 : 3 × 1 = 40 : 10 : 3
Hence total amount distributed among unskilled workers
(318/40+10+3)*10 =60
But the number of unskilled workers is 20, so the daily wages of unskilled worker

60/20=3
The wages of a skilled worker, an unskilled worker and a clerk are in the ratio = 5:2:3
Multiplying the ratio by  5/2 and 3/2  we get = 7.50 : 3 : 4.50
So, if an unskilled worker gets Rs.3 a day then a skilled worker gets Rs. 7.50 per day a clerks Rs. 4.50 a day

16. Two numbers are in the ratio of 11:13. If 12 be subtracted from each, the remainders are in the ratio of 7:9 Find out the numbers.
Sol: Since the numbers are in the ratio of 11:13. Let the numbers be 11x and 13x. Now if 2 is subtracted from each, the numbers become (11x -12) and (13x-12).  As they are in the ratio of 7:9              (11x-12): (13x-12):: 7: 9
⇒   (11x – 12) 9 = (13x – 12) 7
⇒  99x – 108 = 91x – 84
⇒   9x = 24 or x = 3

11*3=33, 13*3=39

17. In what ratio the two kinds of tea must be mixed together one at Rs. 48 per kg. and another at Rs. 32 per kg. So that the mixture may cost Rs. 36 per kg. ?
Sol:   Mohan and Sohan = 5:6
Sohan and Rakesh = 8:5
Mohan and Rakesh = 5/6*8/5=4/3
= 4 : 3

18. What should be subtracted from 15, 28, 20 and 38 so that the remaining numbers may be proportional? 19. If Rs. 279 were distributed among Ram, Mohan and Sohan in the ratio of 15:10:6 respectively, then how many rupees did Mohan obtain?
Sol: Ratio in which Ram, Mohan and Sohan got  = 15 :10 : 6
∴ Sum of ratios  = 15 + 10 + 6 = 31
∴ Share of Mohan  = Rs. 90

20. A bag contains of one rupee, 50 paise and 25 paise coins. If these coins are in the ratio of 2:3:10, and the total amount of coins is Rs288, find the number of 25 paise coins in the bag.
Sol:   Ratio of one rupee, 50 paise and 25 paise coins
= 2:3:10
∴ Ratio of their values  = 8:6:10 = 4:3:5
And sum of the ratios of their values = 4 + 3 + 5 = 12
∴  Value of 25 paise coins = 5*288/12= Rs. 120
∴ No. of 25 paise coins   = 120 × 4 = 480

Posted Date : 10-02-2021

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