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Now, we are going to discuss Ratio and Proportions and questions based on it in this blog. This is a concept that would be used in different other topics too so make sure your concepts are tightened up. Let’s first understand what is Ratio and Proportions, When two or more similar quantities are compared, then to represent this comparison, ratios are used.

The ratio between x and y can be represented as x : y, where x is called antecedent, and y is called the consequent. x : y or x/y A proportion is an expression which states that two ratios are equal e.g. 3/12 = 1/4. ⇒ 3 : 12 = 1 : 4

<h2>Ratio</h2>

As in ratios, two quantities are compared so quantities should be in the same unit and the ratio has no unit.

TIP: In ratios a : b is different from b : a.

Dividing a number in a Ratio:

Let ‘A’ be a given number. The given ratio is a1 : a2 , so ‘A’ has to be divided in the ratio of a1 : a2 , First part = a1 / (a-1 + a2 ) × A Second part = a2 / (a1 + a2 ) × A Since, ‘A’ has to be divided in the ratio so (first part + second part) = A

E.g. Dividing 3200 among P, Q, and R in the ratio of 5 : 2 : 9, find the amount received by R. Sol: Amount received by R = 9/(5 + 2 + 9) × 3200 = 9/16 × 3200 = 1800 (Ans.)

E.g. In a 40 liters mixture acetic acid and sodium acetate are in the ratio 3 : 1; find the amount of sodium acetate solution to be added to make the ratio 2 : 3. Sol: Sodium acetate in the solution = 1/(1 + 3) × 40 = 10 Rest = 40 - 10 = 30 Let ‘x’ amount be added of sodium acetate in the solution to make ratio 2 : 3 ⇒ 30 / (10 + x) = 2/3 ⇒ 90 = 20 + 2x ⇒ 2x = 70 ⇒ x = 35 (Ans).

(We will discuss these types of problems in the Mixtures and Alligations chapter.)

<h2>Comparsion of ratio</h2>

E.g. Which is greater, 5/8, or 3/8? Sol: We can easily compare both sides as the denominator is the same for both; ⇒ 5 > 3 so, 5/8 > 3/8

E.g. Which is greater, 4/7 or 5/9? Sol: As denominator of both fractions are different so we simply can’t compare, let’s simplify them, As 4/7 = 0.5714 and 5/9 = 0.5555 As 0.57 > 0.55 ⇒ 4/7 > 5/9 (Ans.)

<ul>
	<li> For comparing two fractions we can simply convert both ratios in such a way that both ratios have the same denominator, then compare their numerator, the fraction with greater numerator will be greater.</li>
	<li>For comparing two fractions we can simply convert both ratios in such a way that both ratios have the same numerator, then compare their denominator, the fraction with greater denominator will be lesser.</li>
</ul>

E.g. Which is greater, 4/7, or 5/9? Sol: Let’s try to make the numerator equal ⇒ (4 × 5)/(7 × 5) = 20/35; (5 × 4) / (9 × 4) = 20/36 As 35 < 36 so 4/7 > 5/9 (Ans.)

For solving questions related to ratios we need to compare two quantities, let’s discuss some properties of comparison; (i) a : b > c : d if ad > bc (ii) a : b < c : d if ad < bc (iii) a : b = c : d if ad = bc

E.g. Which is greater, 4/7, or 5/9? Sol: Let a/b = 4/7 and c/d = 5/9 ⇒ ad = 4 × 9, bc = 7 × 5 ⇒ ad = 36, bc = 35 ⇒ ad > bc, 4/7 > 5/9 (Ans.)

Without any cumbersome calculation, we arrived at our solution. 

<h2>Proportion</h2>

An equality or two ratios is called a proportion and we say that the four numbers are in proportion. If a / b = c / d or a : b :: c : d Here, a and d are called extremes (extreme terms) and b,c are called Means (middle terms) Or, a × d = b × c.

If a : b :: b : c, then these numbers a, b, c are said to be in

<h2>Continued proportion</h2>

⇒ a : b :: b : c ⇒ a : b = b : c ⇒ a × c = b × b ⇒ ac = b2 ⇒ c = b2/a

Where b is called Mean Proportional and c is called Third Proportional.

E.g. Mean proportional to ‘a’ and ‘b’. Sol: Let it be ‘x’ ⇒ a : x :: x : b ⇒ a × b = x × x ⇒ ab = x2 ⇒ x = ⇒ (ab) (Ans.)

E.g. The ratio of incomes of Raman and Gagan is 4 : 3 and the ratio of their expenditures is 3 : 2. If each person saves Rs 2500, then find their incomes and expenditures. Sol: Let the income of Raman be 4x and income of Gagan is 3x ⇒ Expenditures of Raman = 4x – 2500 ⇒ Expenditures of Gagan = 3x – 2500 According to question, (4x – 2500)/(3x – 2500) = 3/2 ⇒ 8x – 5000 = 9x – 7500 ⇒ x = 2500 Income of Gagan, 3x = 3 × 2500 = 7500 Income of Raman, 4x = 4 × 2500 = 10000 Expenditures of Gagan, 3x – 2500 = 7500 – 2500 = 5000 Expenditures of Raman, ⇒ 4x – 2500 = 10000 – 2500 = 7500. (Ans.)

This is a very common question in the quant section and it took a very long time to solve too, so let’s find a better alternative for this.

TIP: The incomes of two persons are in the ratio of a : b and their expenditures are in the ratio of c : d. If each of them saves X, then their incomes are given by, [X(d – c) / (ad – bc)] × a and [X(d – c) / (ad – bc)] × b, respectively. Their expenditures are given by, [X(b – a) / (ad – bc)] × c and [X(b – a) / (ad – bc)] × d, respectively.

Let’s try to solve the previous question by this method.

⇒ Income of Raman = [X(d – c) / (ad – bc)] × a = [2500(2 – 3) / (8 – 9)] × 4 = 10000 ⇒ Income of Gagan = [X(d – c)] / (ad – bc)] × b = [2500(2 – 3)/ (8 – 9)] × 3 = 7500 ⇒ Expenditures of Raman = [X(b – a) / (ad – bc)] × c = [2500(3 – 4) / (8 – 9)] × 3 = 7500 ⇒ Expenditures of Gagan = [X(b – a) / (ad – bc)] × d = [2500(3 – 4) / (8 – 9)] × 2 = 5000 (Ans.)

Yes, we got the solution and it’s fast if we remember the formula correctly.

<h2>Aged based problems</h2>

Problems based on ages generally consist of information of ages of two or more people and their correlation with each other in the past or present or future. Problems are very easy to solve if we understand them.

Basic Rules for problem based on ages:

Rule 1. If the ratio of present ages of A and B is x:y and n years ago, the ratio of their ages was p : q, then (kx - n)/(ky - n) = p/q, where k is a constant.

E.g. The ratio of the ages of A and B at present is 3 : 1. Four years earlier, the ratio was 4 : 1. Find the present age of A. Sol: Let the present ages of A and B be 3x and x years respectively. Now, Four years ago, the ages of A and B = (3x - 4) and (x - 4) years respectively. ⇒ (3x - 4)/(x - 4) = 4/1 Doing cross multiplication, (3x - 4) = 4 × (x - 4)  3x – 4 = 4x - 16 ⇒ 16 - 4 = 4x - 3x ⇒ x = 12 ⇒ Present age of A = 3x = 3 × 12 = 36 years (Ans).

E.g. The ratio between the ages of A and B is 3 : 5 and the sum of their ages are 56 years. The ratio between their ages 7 years ago was? Sol: Age of A = 3/(3 + 5) × 56 = 3/8 × 56 = 21 ⇒ Age of B = 56 – 21 = 35 ⇒ 7 years ago, age of A and B was = 21 – 7, 35 – 7 = 14, 28 ⇒ Ratio will be = 14/28 = 1/2 (Ans.)

We can do this calculation in mind too by taking less than a minute and save time for other questions.

Rule 2. If the ratio of present ages of A and B is x : y and after n years, the ratio of their ages will be p : q, then (kx + n)/(ky + n) = p/q, where k is a constant.

E.g. At present, the ratio of the ages of Maya and Chhaya is 6 : 5 and fifteen years from now, the ratio will change to 9 : 8. Find the present age of Maya. Sol: Let the present ages of Maya and Chhaya be 6x and 5x years, respectively. After 15 years, (6x + 15)/(5x + 15) = 9/8 ⇒ 8 × (6x + 15) = 9 × (5x + 15) ⇒ 48x + 120 = 45x + 135 ⇒ 48x - 45x = 135 -120 ⇒ 3x = 15 ⇒ x = 5 ⇒ Present age of Maya= 6x = 6 × 5 = 30 years (Ans).

The problems that we get in exams from this topic are a combination of both the rules that we have just studied. They 78 also involve some concepts and knowledge of how to solve linear equations (that we will discuss in algebra).

Now, that we have understood the rules, let’s solve some examples.

E.g. 4 years ago, the ratio of Vikah’s to Rahul’s age was 3 : 5. After 6 years, this ratio would become 4 : 5. Find the present age of Rahul. Sol: Let, the age of Vikah and Rahul four years ago be 3x and 5x. So, their present age will be (3x + 4) and (5x + 4) years respectively. Age of Vikah and Rahul after 6 years from present = (3x + 4 + 6) and (5x + 4 + 6) years respectively ⇒ (3x + 10) and (5x + 10) years respectively. Ratio of Vikah’s age to Rahul’s age after 6 years= 4 : 5 ⇒ (3x + 10) : (5x + 10) = 4 : 5 or ⇒ (3x + 10)/(5x + 10) = 4/5 Doing cross multiplication, 5 × (3x + 10) = 4 × (5x + 10) ⇒ 15x + 50 = 20x + 40 ⇒ 50 - 40 = 20x - 15x ⇒ x = 2 ⇒ Present age of Rahul = (5x + 4) = 5 × 2 + 4 = 14 years (Ans).

E.g. Six years ago, the ratio of ages of the two persons P and Q was 3 : 2. Four years hence the ratio of their ages will be 8 : 7. What is P’s age? Sol: Let the age of P and Q six years ago be 3x and 2x. So, their present age will be (3x + 6) and (2x + 6) years respectively. Ratio of P’s age to Q’s age after 4 years = 8 : 7 ⇒ (3x + 6 + 4) : (2x + 6 + 4) = 8 : 7 ⇒ (3x + 10) : (2x + 10) = 8 : 7 or ⇒ (3x + 10)/(2x + 10) = 8/7 Doing cross multiplication, 7 × (3x + 10) = 8 × (2x + 10) ⇒ 21x + 70 = 16x + 80 ⇒ 21x - 16x = 80 - 70 ⇒ x = 2 ⇒ Present age of P = (3x + 6) = 3 × 2 + 6 = 12 years (Ans).

This was all about Ratio, Proportion and Age Based Problems. Practice hard and stay connected with us.

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