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Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal. Here, a and b are any two integers. The ratio and proportion are the two important concepts, and it is the foundation to understand the various concepts in mathematics as well as in science.

Ratio and proportions are said to be faces of the same coin. When two ratios are equal in value, then they are said to be in proportion. In simple words, it compares two ratios. **Proportions** are denoted by the symbol **‘::’ or ‘=’**.

**Ratio**

- The ratio is the comparison between similar types of quantities, it is an abstract quantity and does not have any units.
- 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 the first term or
**antecedent**and b, the second term or**consequent.****Example:**The ratio 5 : 9 represents 5/9 with antecedent = 5, consequent = 9.

Note:The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.Example:4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3

__Let's see how questions appear from this chapter in CAT 2019:__

Try yourself:PYQ CAT 2019: The salaries of Ramesh, Ganesh and Rajesh were in the ratio 6:5:7 in 2010, and in the ratio 3:4:3 in 2015. If Ramesh’s salary increased by 25% during 2010-2015, then the percentage increase in Rajesh’s salary during this period is closest to?

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- The ratio should exist between the quantities of the same kind.
- While comparing two things, the units should be similar.
- There should be significant order of terms.
- The comparison of two ratios can be performed, if the ratios are equivalent like the fractions.

Try yourself:**A and B together have Rs. 1210. If (4 / 15) of A's amount is equal to (2 / 5) of B's amount, how much amount does B have?**

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**Proportion**

- The equality of two ratios is called proportion i.e. If a/b = c/d, then a, b, c, d are said to be in proportion.
- If a: b = c : d, we write a: b
**::**c : d and saying that a, b, c, d are in 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 x c) = (a x d)**

Try yourself:**Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:**

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**Fourth Proportional**

If a: b = c : d, then d is called the fourth proportional to a, b, c.**Third Proportional**

If a : b = c : d, then c is called the third proportion to a and b.**Mean Proportional:**

Mean proportional between a and b is

**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)

Try yourself:**In a mixture of 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quantity of water to be further added is:**

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Try yourself:**Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is Sumit's salary?**

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**➤ Duplicate Ratios**

- The 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
^{3 }: b^{3}) - 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]

**Variations**

- 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.

**(i)** **Direct Variation: **If A is in direct variation with B, then an increase or decrease in A will lead to a proportionate increase or decrease in B.

- A ∝ B
- A = KB

**(ii)** **Indirect Variation: **If A is in inverse variation with B, then an increase in A will lead to a Proportionate decrease in B and vice versa.

- A ∝ 1/B
- A = K/B

Try yourself:**If ‘x’ and ‘y’ are in a direct proportion then which of the following is correct?**

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**(iii)** **Joint Variation**

- Let us consider the area of a triangle, which is dependent on both, the height as well as the base of the rectangle.
- When both the dimension of the triangle changes, then the area also changes.

When the area of the triangle varies with the change in the base of the triangle.

A∝ b

- When the area of the triangle varies with the change in the height of the triangle.

A∝ h

- This is called the joint variation of the area of the triangle with respect to its base and height.

A∝ 1/2 x b.h

**Simple Method**

The LCM process gets very cumbersome when we have to find the ratio out of multiple ratios.

We have the following simple method for that for a chain of ratios of any length.

**Suppose you have the ratio train as follows**

► A : B = 1 : 2

► B : C = 2 : 3

► C : D = 5 : 6

► D : E = 7 : 8

If we were to find A : B : C : D : E, then the LCM method would have taken quite a long time which is infeasible in examinations of limited hours.

__The short cut is as follows: __

A : B : C : D : E can be written directly as:

► 1 × 2 × 5 × 7 : 2 × 2 × 5 × 7 : 2 × 3 × 5 × 7 : 2 × 3 × 6 × 7 : 2 × 3 × 6 × 8

► 70 : 140 : 210 : 252 : 288

__The thought algorithm for this case goes as:__To get the combined ratio of A : B : C : D : E, from A : B, B : C, C : D, and D : E

In the combined ratio of A : B : C : D : E.

- A will correspond to the product of all numerators (1 × 2 × 5 × 7).
- B will take the first denominator and the last 3 numerators (2 × 2 × 5 × 7).
- C, on the other hand, takes the first two denominators and the last 2 numerators (2 × 3 × 5 × 7).
- D takes the first 3 denominators and the last numerator (2 × 3 × 6 × 7) and E take all the four denominators (2 × 3 × 6 × 8).

**Product of Proportions**

If a:b = c:d is a proportion, then:

- Product of extremes = product of means i.e.,
**ad = bc** - Denominator addition/subtraction:
**a:a+b = c:c+d**and**a:a-b = c:c-d** - a, b, c, d,.... are in continued proportion means,
**a:b = b:c = c:d**= .... - a:b = b:c then b is called mean proportional and
**2b = ac** - The third proportional of two numbers, a and b, is c, such that, a:b = b:c.

d is fourth proportional to numbers a, b, c if a:b = c:d.

**Variations**

- If a ∝ b, provided c is constant and a ∝ c, provided b is constant, then a ∝ b ∝ c, if all three of them are varying.
- If A and B are in a business for the same time, then Profit distribution ∝ investment (Time is constant).
- If A and B are in a business with the same investment, then:

Profit distribution ∝ Time of investment (Investment is constant)

Profit Distribution ∝ Investment × Time

**Ratio and Proportion Tricks**

- If u/v = x/y, then uy = vx
- If u/v = x/y, then u/x = v/y
- If u/v = x/y, then v/u = y/x
- If u/v = x/y, then (u+v)/v = (x+y)/y
- If u/v = x/y, then (u-v)/v = (x-y)/y
- If u/v = x/y, then (u+v)/ (u-v) = (x+y)/(x-y), which is known as Componendo-Dividendo Rule
- If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then a =b = c

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