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Ratio and Proportion:

**What is ratio?**

The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as a : b. In the ratio, a : b, we call a as the first term or antecedent and b, the second term or consequent. For ex: The ratio 5 : 9 represents 5/9 with antecedent = 5, consequent = 9.

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

For ex: 4 : 5 = 8 : 10 = 12 : 15 etc. Also, 4 : 6 = 2 : 3

**What is proportion?**

The equality of two ratios is called proportion. If a : b = c : d, we write, a : b : : c : d, then d is called the fourth proportion. Here a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes.

Thus, a : b : : c : d <=> ( b* c) = (a * d)

- Fourth Proportional : If a : b = c : d, then d is called the fourth proportional to a, b, c
- Third Proportional: If a : b = b : c, then c is called the third proportional to a and b.
- Mean Proportional : Mean Proportional between a and b is √ab

**Comparison of Ratios:** We say that (a : b) > (c : d) <=> a/b > c/d

**Compounded Ratio:** The compounded ratio of the ratios (a : b), (c : d), (e : f) is (ace : bdf).

Duplicate ratio of (a : b) is (a^{2}:b^{2})

Sub Duplicate ratio of (a : b) is (√a : √b )

Triplicate ratio of (a : b) is (a^{2 }: b^{2})

Sub triplicate ratio of (a : b) is (a^{1/3 }: b^{1/3})

If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d) (componendo and dividendo)

**Variation:**

- We say that x is directly proportional to y, if x = ky for some constant k and we write, x ∝ y.
- We say that x is inversely proportional to y, if xy = k for some constant k and we write, x ∝ 1/y.

1. If A : B = 5 : 7 and B : C = 6 : 11, then A : B : C is:

A) 55: 77: 66

B) 30: 42: 77

C) 35: 49: 42

D) 22: 44: 33

E) None of these

Explanation: Answer is B

A : B = 5 : 7, B : C = 6 : 11 = (6* 7/6) : (11* 7/6) = 7 : 77/6

∴ A : B : C = 5 : 7 : 77/6 = 30: 42 : 77

2. If 2X = 3Y = 4Z, then X : Y : Z is:

A) 2 : 3: 4

B) 4: 3: 2

C) 6: 4 : 3

D) 20: 15 : 2

E) None of these

Explanation: Answer is C

Let 2X = 3Y = 4Z = k. Then, X = k/2, Y = k/3 and Z = k/4

X : Y : Z = k/2 : k/3 : k/4 = 6: 4: 3

3. If 15% of x = 20% of y, then x : y is:

A) 3 : 4

B) 4 : 3

C) 17 : 16

D) 16 : 17

E) None of these

Explanation: Answer is B

15% of x = 20% of y => 15x/100 = 20y/100 => x/y = (20/100* 100/15) = 4/3

x : y = 4 : 3

4. If (4x^{2} – 3y^{2}) : (2x^{2} + 5y^{2}) = 12 : 19, then (x : y) is:

A) 2 : 3

B) 1 : 2

C) 3 : 2

D) 1 : 4

E) None of these

Explanation: Answer is C

(4x^{2} – 3y^{2}) /(2x^{2} + 5y^{2}) = 12/19 <=> 19(4x^{2} – 3y^{2}) = 12(2x^{2} + 5y^{2})

52x^{2} = 117y^{2} <=> 4x^{2} = 9y^{2} x^{2}/y^{2} = 9/4 x/y = 3/2

∴ required ratio is 3 : 2

5. If 5x^{2} – 13xy + 6y^{2} = 0, then x : y is :

A) (2 : 1) only

B) (3 : 5) only

C) (5 : 3) or (1 : 2)

D) (3 : 5) or (2 : 1)

E) None of these

Explanation: Answer is D

5x^{2} – 13xy + 6y^{2} = 0 <=> 5x^{2} – 10xy – 3xy + 6y^{2} = 0

<=> 5x (x – 2y) – 3y (x – 2y) = 0 <=> (x – 2y) (5x – 3y) = 0

<=> x = 2y or 5x = 3y <=> x/y = 2/1 or x/y = 3/5

∴ (x : y) = (2 : 1) or (3 : 5)