1. Quantitative Methods Chapter 3 – RATIO & PROPORTION Session 3
Quantitative Methods Chapter 3 – RATIO & PROPORTION Session Pranjoy Arup Das

2. Ratio is the quantitative relationship between two or more quantities of the same kind which combine to form a single unit.
Vice Versa, when a single unit / amount is divided into two or more parts of the same kind, the quantitative relationship between the quantities of the parts is expressed in the form of a ratio.
Eg. Suppose 10 boys and 6 girls get together to form a club, the total no. of members of the club will be 16 and the ratio of male members : female members is…
 Ratio of males to females = 5:3
This means that in every 8 member group, there are 5 males and 3 females.
The first term of a ratio, which is 5 in our case, is called the antecedent and the second term, which is 3 in our case, is called the consequent. And 8 is called Sum of the ratio terms.
A RATIO IS ALWAYS EXPRESSED AS A VULGAR FRACTION NOT IN DECIMALS

3. Eg 2. The total no. of members of a club is 20 and the ratio of male members : female members = 3 : 2. We have to find the no. of male and female members.
Solution 1,
Since Males : Females = 3 : 2
No. of Males / No. of females= 3/2
No. of Males = (3/2) * No. of females
Since total members is 20
Males + Females = 20 => (3/2)* Females + Females = 20
Females = 8, Males = 3/2 * 8 = 12
Solution 2 – Let the no. of male members be 3x and no. of female members be 2x, where x is a non-zero number.
 3x + 2x = 20
=> x = 20/5 = 4
So the no. of male members = 3 * 4 = 12
the no. of female members = 2 * 4 = 8.

4. Solution 3 –
Since the ratio of boys to girls is 3:2, this means that out of 5 members, 3 are boys & 2 are girls.
Then out of 20 there are (3/5) * 20 = 12 boys
And out of 20, there are (2/5) * 20 = 8 girls
Point to note:
If P comprises of A & B which exist in the ratio a:b, then
Quantity or Value of A = a / (a + b) * P
Quantity or Value of B = b / (a + b) * P  
(a+b) is called the sum of the ratio terms.
So in the above case, P = 20, a = 3, b = 2
No. of boys = {3/(3+2)} * 20 = (3/5) * 20 = 12
No. of girls = {2/(3+2)} * 20 = (2/5) * 20 = 8

5. Eg 3. The ratio of male members to female members of a club is 4 : 5
Eg 3. The ratio of male members to female members of a club is 4 : 5. There are 40 females. We need to find the no. of male members.
Solution 1 – Let the no. of male members be 4x and no. of female members be 5x, where x is a non-zero number.
 5x = 40
=> x = 8
So the no. of male members = 4 * 8 = 32.
Solution 2 – Point to note:
If A & B are the ratio a:b, it means that
Quantity or Value of A / Quantity of B = a/b
Quantity or Value of A = (a /b) * Quantity of B
And Quantity or Value of B = (b/ a) * Quantity of A
So in the above case, a = 4, b = 5, B = 40
No. of male members = 4/5 * 40 = 32

6. Proportion
TWO MEANINGS:
1) Proportion is the ratio of a part of something to the whole of that something.
If A is a part of B, the ratio A : B will provide us the proportion of A (which is a part of B) to B (the whole).
Just as Percentage – out of 100, Proportion – out of 1.
Eg. In a class of 42 students, 24 students own a Samsung phone, 14 own a Nokia phone and 3 own a Micromax phone and 1 person owns a Karbonn phone. What is the proportion of students owning a samsung phone? What proportion and percentage of students own a Micromax phone?
Proportion =
= 24 / 42 = 4/7 = 0.55
Proportion is always between 0 and 1. It helps in Probability theory.

7. 2nd Meaning : Proportion also refers to the equality of two ratios.
If a:b = c:d, then a, b, c & d are in proportion.
Here b & c are called the means and a & d are called the extremes.
Product of means = product of extremes, i.e., b*c = a*d
If a:b=c:d, then d is the fourth proportional to a, b, & c.
If a:b = b:c, then c is the third proportional to a & b.

8. Three approaches to a ratio related problem :
If ratio of A to B = p : q, and the sum of A & B is M then to find the values of A & B, you may follow any of the 3 approaches to solve :
APPROACH 1:
Since A:B = p:q, this means
So, A = (p/q) * B
B = (q/p) * A
APPROACH 2 :
Value of A can be assumed as px and value of B can be assumed as qx. So the sum of A & B = px + qx.
Therefore, px + qx = M
APPROACH 3:
Value of A = {p/(p+q)} * M
Value of B = {q/ (p+q)} * M

9. Since , Meena’s age : Meera’s age = 4 : 3
RSA EX 12A, Pr no. 36 Page 254: The ratio of Meena’s age and Meera’s age is 4 : 3 and the sum of their ages is 28 years. What will be the ratio of their ages after 8 years?
APPROACH 1:
Since , Meena’s age : Meera’s age = 4 : 3
 Meena’s age / Meera’s age = 4 / 3
=> Meenas age = (4/3) * Meera’s age
And it is given that Meena’s age + Meera’s age = 28 years
 (4/3) * Meera’s age + Meera’s age = 28
=> Meera’s age = 28 * 3 / 7 = 12 years
Meena’s age = (4/3) * 12 = 16 years
Ratio of their ages after 8 years = ______ =______

10. APPROACH 2:
Let Meena’s age be 4x years and Meera’s age be 3x years
4x + 3x = 28 => x = 4
So, Meena’s age = 4*4 =16 years
Meera’s age = 3*4 = 12 years
Ratio of their ages after 8 years = ______ :______

11. APPROACH 3: Since meena :meera = 4:3, and Sum of their ages is 28 years Meena’s age = 4/(4+3)*28 = 4/7 * 28 = 16 years Meera’s age = 3/(4+3)*28 =3/7 * 28 = 12 years So the ratio of their ages after 8 years = _____:_____

12. Divide Rs. 420 among A,B,&C in the ratio 1/3 : 5/6 : 7/9
Ex. 7 Page 250 :
Divide Rs. 420 among A,B,&C in the ratio 1/3 : 5/6 : 7/9
Point to note: If a sum of money Rs. S is to be divided amongst A, B & C in the ratio a:b:c, then
A’s Share = a / (a+b+c) * S
B’s Share = b / (a+b+c) * S
C’s Share = c / (a+b+c) * S
(NOTE : a+b+c is referred to as sum of the ratio terms)
Since Rs. 420 is to be divided amongst A,B & C in the ratio 1/3 : 5/6 : 7/9
Sum of the ratio terms = (1/3) + (5/6) + (7/9)
= 35/18
A’s share = = = Rs.__________

13. Let the total sum of money be Rs. M
Exer. 12A, Pr no. 40 Page 254: A sum of money is divided among W,X,Y & Z in the ratio 3:7:9:13 respectively. If the shares of W & Y together are Rs , then what is the difference between the X’s share and Z’s share?
Solution :
Let the total sum of money be Rs. M
Rs. M is to be divided in the ratio W:X:Y:Z = 3:7:9:13
So W’s share = 3/( ) * M = Rs. (3/32) * M
& Y’s share = Rs. (9/ 32) * M
Given that
W ‘s share + Y’s share = Rs
(3/32)*M + (9/32)*M = 11172
M =
So now, X’s share = (7/32) * ________ = Rs. ________
Z ‘s share = (13/32) * ________ = Rs. ________
Difference between X & Z ‘s share = Rs. ____________ Rs

14. Solution : Ratio of milk to water = 5 :3 & Milk + Water = 20 Lts
Pr no. 47 Page 255: In a 20 litre mixture of milk and water, the ratio of milk and water is 5 : 3. If 4 litres of the mixture is taken out and 4 litres of milk is added, what will be the ratio of milk and water in the new mixture?
Solution : Ratio of milk to water = 5 :3 & Milk + Water = 20 Lts
Quantity of milk in 20 Ltrs of the mix = 5/8 * 20 = 12.5 Lts
Quantity of water in 20 Ltrs of the mix= 3/8 * 20 = 7.5 Lts
Given that 4 Ltrs of the mixture of milk and water is removed:
Quantity of Milk in 4 L of the mixture = 5 /8 * 4 = 2.5 Lts
So the total qnty of milk becomes = 12.5L – 2.5 L = 10 Lts
Qnty of water in 4 L of the mixture = 3/ 8 * 4 = 1.5 Lts
So the new qnty of water = 7.5 L – 1.5 L = 6 Lts
Now, if 4 ltrs of milk is added to the new mixture,
Quantity of milk becomes _______ Lts =
And qnty of water___________________________
So the new ratio of milk : water = _______ = ________
10 Lts
14 Lts
remains the same = 6 Lts

15. Solution : If A: B: C = 3:5:7, the A : B = 3:5 and B : C = 5 : 7
Pr no. 82 Page 257: The salaries of A,B & C are in the ratio 3:5:7. If these salaries are increased by 50%, 60% and 50% respectively, then what will be the ratio of the new salaries?
Solution : If A: B: C = 3:5:7, the A : B = 3:5 and B : C = 5 : 7
So A / B = 3 /5
That means,
So the ratio of A’s & B’s new salary = 9/2 : 8
And since B/C = 5/7
So the ratio of B’s & C’s new salary = 8 : 21/2
So the new ratio of A : B : C = =_________________

16. Practice session
Arithmetic, RS Aggarwal, Exercise 12A, Problem nos. 2, 9, 35, 37, 44, 48, 50, 53, 82, 85, 87, 95, &113.
RECAP: Three approaches to a ratio related problem :
If ratio of A to B = p : q, and the sum of A & B is M then to find the values of A & B, you may follow any of the 3 approaches to solve :
APPROACH 1:
Since A:B = p:q, this means
So, A = (p/q) * B
B = (q/p) * A
APPROACH 2 :
Value of A can be assumed as ‘px’ and value of B can be assumed as ‘qx’. So the sum of A & B = px + qx.
Therefore, px + qx = M
APPROACH 3:
Value of A = {p/(p+q)} * M
Value of B = {q/ (p+q)} * M