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

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

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.

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

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

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.

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.

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

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 = ______ =______

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 = ______ :______

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 = _____:_____

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.__________

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. 11172, 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/(3+7+9+13) * M = Rs. (3/32) * M & Y’s share = Rs. (9/ 32) * M Given that W ‘s share + Y’s share = Rs. 11172 (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. 29792

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 _______ + 4 Lts = And qnty of water___________________________ So the new ratio of milk : water = _______ = ________ 10 Lts 14 Lts remains the same = 6 Lts

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 = =_________________

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