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Ratio and Proportion, Let’s first understand what is Ratio and Proportion, 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

A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. 3/4 = 6/8 is an example of a proportion. When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion.

RATIO

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.

Amount received by R = 9 / (5 + 2 + 9) × 3200 = 9 / 16 × 3200  ⇒1800 (Ans.)

E.g. In a 40 litres 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.

Ans- Sodium acetate in the solution = 1 / (1 + 3) × 40 = 1 / 4 × 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).

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?

Ans- 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

Comparison of Ratios

E.g. Which is greater, 5 / 8, or 3 / 8?

Ans- 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?

Ans- 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.)

• 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.
• 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.

E.g. Which is greater, 4 / 7, or 5 / 9?

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

a:b < c:d if ad < bc

a:b = c:d if ad = bc

E.g. Which is greater, 4 / 7, or 5 / 9?

Ans- 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.

PROPORTION

An equality or tow 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 Continued Proportion.

⇒ 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. 8. Mean proportional of ‘a’ and ‘b.

Let it be ‘x’

⇒ a:x :: x:b

⇒ a × b = x × x

⇒ ab = x2

⇒ x = √(ab)

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 2500, then find their incomes and expenditures.

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 = 1000 – 2500 = 7500.(Ans.)

This is a very common question in the quant section and it took very much time for solving 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 Raman = 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.

E.g.  A and B are partners in a business. They invest in the ratio 5:6, at the end of 8 months B withdraws, if they receive profit at the end of the year in the ratio of 5:9 find how long A’s investment was used?

Let A’s investment was used for X months

Given, the ratio of invest (A: B) = 5:6

⇒ Ratio of time = X:8

⇒ Ratio of profit = 5X / 48 = 5 / 9

So,

⇒ X = 48 / 9 = 16 / 3 Months.