## Ratio and Proportion

**Ratio:**

- The ratio of two quantities a and b of same units is the fraction x/y, where b ≠ 0

- The fraction x/y can be represented as x:y

**Different types of ratios are:**

**1) Duplicate ratio:**It is the ratio of squares of two numbers.

Duplicate ratio of the fraction | x | is given as: | x | = | x^{2} | or x : y = x^{2} : y^{2} |

y | y | y^{2} |

**2) Sub-duplicate ratio:**It is the ratio between square roots of two numbers.

Duplicate ratio of the fraction | x | is given as: | x | = | x | or x : y = x : y |

y | y | y |

**3) Triplicate ratio:**It is the ratio of cubes of two numbers.

Triplicate ratio of the fraction | x | is given as | x | = | x^{3} |

y | y | y^{3} |

**4) Sub- Triplicate ratio:**It is the ratio between cube roots of two numbers

Sub-Triplicate ratio of the fraction | x | is given as | x | = | x^{(1/3)} |

y | y | y^{(1/3)} |

**5) Compound ratio:**It is the ratio of product of first terms in every ratio to that of product of second term in every ratio.

For example:

Compound ratio of (a : x), (b : y), (c : z) is (abc : xyz)

**6) Inverse ratio:**The ratio formed by interchanging their old places in the ratio to new

The inverse ratio of 5 : 8 is 8 : 5.

**Proportion:**

**1)**Proportion is the equality of two ratios.

When (a : b = x : y) is represented as (a : b :: x : y), then a, b, x, y are said to be in proportion.

In (a : b :: x : y), a and y are called as extremes and b and x are called as mean terms.

Product of means = Product of extremes

**2)**Mean proportion: Mean proportion between x and y is given as xy

**3)**Third proportion: If p : q = q : s, then s is called as third proportional to p and q.

**4)**Fourth proportion: If u : v = x : y, then y is the fourth proportional of u, v and x.

**Quick Tips and Tricks**

**1) Comparison of ratios:**

If (x : y) > ( a : b) → | x | > | a |

y | b |

**2) Proportion**

**4) Variation:**

- If a = kb for some constant k, then we can say that a is directly proportional to b.

- If ba =k for some constant k, then we can say that a is inversely proportional to b.

**5)**If ratio between first and second quantity m : n = a : x, second and third quantity n : p = b : y, fourth and fifth quantity p : q = c : z, then m : n : p : q can be easily solved by using the trick shown below:

m : n : p : q = abc : xbc: xyc : xyz

**6)**If a number a is divided in the ratio x : y,

1) First part: | ax |

(x + y) |

2) Second part: | ay |

(x + y) |

**Question Variety**

Type 1: Proportion

**Examples:**

**Q 1.**If a : b = 2 : 3 and b : c = 5 : 7, then find a : b : c.

a. 12 : 15 : 9

b. 10 : 15 : 21

c. 14 : 12 : 21

d. 2 : 15 : 7

View solution

Correct option : (b)

Here, to find a : b : c, use the trick discussed in quick tips and tricks.

a : b : c = (2 x 5) : ( 3 x 5) : (3 x 7) = (10) : (15) : (21)

**Q 2.**If A : B : C = 3 : 4 : 7, then what is the ratio of (A / B) : (B / C) : (C / A)?

a. 63 : 48 : 196

b. 66 : 49 : 190

c. 56 : 40 : 186

d. 46 : 38 : 160

View solution

Correct option : (a)**Hint:** If a = kb for some constant k, then we can say that a is directly proportional to b.

A : B : C = 3 : 4 : 7

Assume, A = 3 k, B = 4 k, C = 7 k

Therefore,

A | = | (3k) | , | B | = | (4k) | , | C | = | (7k) |

B | (4k) | C | (7k) | A | (3k) |

A | = | (3) | , | B | = | (4) | , | C | = | (7) |

B | (4) | C | (7) | A | (3) |

L.C.M of 3, 4, 7 is 84

(3 x 84) / 4 = 63

(4 x 84) / 7 = 48

(7 x 84) / 3 = 196

Ratio of (A/B) : (B/C) : (C/A) = 63:48:196