## Arithmetic Chapter 13 Fundamental Concept of Ratio And Proportion

**Arithmetic Chapter 13 What is meant by the ratio**

- In our daily life, we divide different materials in some definite ratio. For example, let your father divide some amount of money between you and your brother in the ratio 2 : 3. So what actually is meant by this 2 : 3? In the present chapter, we shall discuss a fundamental theory regarding this.
- In general,
**Ratio is the comparative relationship of two quantities of the same kind expressed in the same unit. The ratio is without any unit.** - While comparing similar kinds of quantities, sometimes we take the difference of them i.e., one quantity is less or more than the other by how much, or sometimes, we find how many times or parts is one of the other quantities and we always express it in ratio. Generally this last type of comparison between two quantities we call ratio.
- So by the ratio 2 : 3, we mean that if the first number is 2, then the second number will be 3. Similarly, if the first quantity is Rs. 2 or 2 gm or 2 kg or 2 km or
- liters or 2 m etc., then the second quantity will be Rs 3 or 3 gm or 3 kg or 3 km or
- liters of 3 m etc.

**Example:**

- If the ratio of Ram’s money and Shyam’s money is 4: 5, then it means that Ram has Rs 4 or a multiple of Rs 4, then Shyam has Rs 5 or the same multiple of Rs. 5.
- If you have 7 balls and your brother has 9 balls, then the ratio of balls = is 7:9 (writing your ball numbers first). If we write your brother’s ball numbers first then the ratio becomes 9: 7.
- If the first quantity of the two quantities is and the second quantity is b, then their ratio will be-a:
- But its converse may not be correct i.e. if the ratio of two quantities is a: b then it is not necessary that the first quantity is a and the second quantity is b.
- Now a question arises if a ratio is 2 : 3, then what are the values of the first quantity and how many are its numbers? Similarly, what are the values of the second quantity and how many are its numbers?
- Now the first quantity may be:2 x 1 = 2, 2 x 2 = ,2 x 3 = 6, 2 x 4 = 8 etc.
- Under this conditions the second quantity will be respectively 3×1=3, 3×2 = 6, 3 x 3 = 9, 3 x 4 = 12, etc.
- Similarly, the 1st quantity may be any one of the numbers 2/1, 2/2, 2/3, 2/4, 2/5, 2/6, …………., etc.
- Under this condition, the second quantity may be any one of the numbers 3/1, 3/2, 3/3, 3/4, 3/5, 3/6, ……., etc.
- Again the first quantity 2 is a real number. Now the question is that “Is the first number any one of variables values a or b or c or y or z?
- Then the answer is that it is sure that it may be. So what is the condition?
- The condition is that if the first quantity be a or x, then the second quantity will be \(\frac{3}{2} \times a=\frac{3 a}{2}\)
- or, \(\frac{3}{2} \times x=\frac{3 x}{2}\).
- Because, \(a: \frac{3 a}{2}=1: \frac{3}{2}=2: 3\)
- or, \(x: \frac{3 x}{2}=1: \frac{3}{2}=2: 3\)
- So the first quantity may be any real number of any quantity and its number is infinite; the second quantity only follows the first quantity under the condition that its number is also infinite.
- Here the first quantity is only open and the second quantity is closed under the condition.
- Again the same thing conversely holds for the second quantity and in that case, the second quantity is open and the first quantity is closed under the condition.
- So the ratio is a multiplicative relation with respect to a condition of an indefinite number or quantity with another number or quantity of the same kind.

## Arithmetic Chapter 13 Characteristics of Ratio

- The ratio is the quotient of two quantities of the same kind expressed in the same unit.
- The ratio may be between two or among more quantities of the same kind expressed in the same unit.
- When the ratio is between two quantities of the same class expressed in the same unit, the
is called the**first quantity**and the**Antecedent**is called the**second quantity****consequent.** - If the ratio is among more than two quantities or numbers, then the quantities or numbers are called respectively the
, etc.**first element, second element, third element, fourth element,** - The value of any given ratio will be different when any real number is added to or subtracted from the elements of the given ratio.
- For example, 2 : 3 ≠ (2 + 1) : (3 + 1) or, 3 : 4
- 2 : 3 ≠ (2 – 1) : (3 – 1) or, 1 : 2.
- The value of any given ratio will remain unchanged when the elements of the given ratio is multiplied or divided by any real numbers other than zero.
- For example, 2 : 3 = (2 x 2) : (3 x 2) = (2 x 3 : 3 x 2) = ………………. etc.
- 2: 3 = 2/2 : 3/2

= 2/3 : 3/2

= 2/4 : 3/4 = ……………. etc. - Generally, the ratio may be between two or among more than two numbers or quantities of the same kind or class while expressing in the same unit only.
- The original value or quantity of the elements in a given ratio may or may not be indicated by the individual values of the ratios.
- For example, if the given ratio is 2 : 3, then the values of the Antecedent and Consequent may or may not be 2 and 3.
- A given ratio is only a pure number, it has no unit.
- A given ratio can be expressed into a vulgar fraction and a vulgar fraction can also be expressed into a ratio.
- A given ratio can equivalently be expressed into another ratio.
- For example, the ratio 2 : 3 is also equivalently expressed into 10: 15.

## Arithmetic Chapter 13 Conversion Of A Given Ratio Into A Vulgar Fraction And A Vulgar Fraction Into A Ratio

- We express a given ration as a vulgar fraction.
- suppose we are a ratio 4: 7
- If we express it into a vulgar fraction, then it will be 4/7.
- So we have \(4: 7=\frac{4}{7}\)
- Similarly, we get, \(5: 6=\frac{5}{6}\)

\(8: 9=\frac{8}{9}\)

\(a: b=\frac{a}{b}\)

\(x: y=\frac{x}{y}\),………. etc. - Conversely, \(\frac{1}{2}=1: 2\),

\(\frac{2}{3}=2: 3\)

\(\frac{7}{8}=7: 8\)

\(\frac{p}{q}=p: q\)

\(\frac{m}{n}=m: n\) etc. - Therefore, we can express a vulgar fraction into a ratio.

## Arithmetic Chapter 13 To Express in Lowest Form Of The Rato

- If it is possible to divide each of the antecedent and consequent of a given ratio by an integral real number (other than zero), then we say that the ratio is not in the lowest form.
- Then we divide each of the antecedent and consequent by that integral real number, then the ratio formed is said to be expressed in the lowest form.
**For example**12: 15 = 4: 5 (Divide by 3)

20: 25 = 4: 5 (Divide by 5)

27: 30 = 9: 10 (divide by 3)

a^{2}: ab = a: b (Divide by a)

xy : xz = y : z (Divide by x)a

^{2}bc: ab^{2}c = a: b (Divide by abc)

## Arithmetic Chapter 13 Classification Of Ratio

- The different types of ratios are given below
**Simple Ratio:**The ratio of two quantities of the same kind expressed in the same unit is called a Simple Ratio.- So the ratio whose two terms (antecedent and consequent) are simple quantities of the same kind is called Simple Ratio.
**For example**₹ 4: ₹ 9 = 4: 9

5 m: 6 m = 5: 6

7 km: 11 km = 7:11 etc.**Simple Ratios are of three types:**

1. The ratio of greater inequality

2. Ratio of lesser inequality, and

3. Ratio of equality.**The ratio of greater inequality:**A ratio in which the antecedent is greater than the consequent is called a ratio of greater inequality.- The ratio a: b is said to be a ratio of greater inequality if a > b.
**For example**9: 8 (9 > 8); 13: 7 (13 > 7) etc. are the ratio of greater inequality.**The ratio of lesser inequality:**A ratio in which the antecedent is less than the consequent is called a ratio of lesser inequality.- The ratio a: b is said to be a ratio of lesser inequality if a < b.
**For example**6: 11 (6 < 11), 8: 15 (8 < 15), etc. are the ratio of lesser inequality.A ratio in which the antecedent and consequent are equal to each other is called a ratio of equality.**The ratio of equalit**y:- The ratio
:*a*is said to be a ratio of equality if*b*= b.*a* **For example**4: 4, 7: 7, 10: 10, etc. are the ratio of equality.**Compound ratio:**The ratio whose antecedent is obtained by the continued production of the antecedents of the given two or more ratios and the consequent is obtained by the continued production of the consequents of the aforesaid ratios is called the compound ratio of the given ratios.**For example**4: 5 and 6: 7 is 4 x 6: 5 x 7 = 24: 35.- Similarly, the compound ratio of 2 : 3, 4: 7, 8: 11, 10: 13 is (2 x 4 x 8 x 10): (3 x 7 x 11 x 13) = 640: 3003.
- The compound ratio of. a: x, b: y, c: z is (a x b x c) : (x x y x z) = abc : xyz.
**Inverse ratio:**If two ratios are such that the antecedent and consequent of one are respectively the consequent and antecedent of the other, then they are said to be the Inverse ratio of one another.- For example, the inverse ratio of 6: 7 is 7: 6; the inverse ratio of 10: 11 is 11: 10; the inverse ratio of a: b is b: a; the inverse ratio is x: y is y: x, etc.
**Duplicate Ratio:**A ratio, which is obtained in such a way that its antecedent is obtained by the square of the antecedent of a given ratio and its consequent is obtained by the square of the consequent of the given ratio, is called the Duplicate ratio of the given ratio.- A ratio x
^{2}: y^{2}is the duplicate ratio of the ratio x: y. **For example**the duplicate ratio of 2: 5 is 2^{2}: 5^{2}= 4: 25- The duplicate ratio of 5: 7 is 5
^{2}: 7^{2}= 25: 49 - The duplicate ratio of a: b is a
^{2}: b^{2}etc. A ratio, whose antecedent and consequent are obtained by the square root of the antecedent and consequent respectively of a given ratio, is called the sub-duplicate ratio of the given ratio.**Sub-duplicate Ratio:**- The sub-duplicate ratio of the ratio x: y is √x: √y
- For example The sub-duplicate ratio of 4: 9 = √4: √9 = 2 : 3
- The sub-duplicate ratio of 16:25= √16 : √25 =4:5
- The sub-duplicate ratio of a
^{2}: b^{2}= a: b etc. **Triplicate Ratio:**A ratio whose antecedent and consequent are obtained by the cube of the antecedent and consequent respectively of a given ratio, is called the Triplicate ratio of the given ratio.- The triplicate ratio of x: y is x
^{3}: y^{3}. **For example**the triplicate ratio of 3: 4 is 3^{3}: 4^{3}= 27: 64- The triplicate ratio of 1: 7 is 1
^{3}: 7^{3}= 1: 343 - The triplicate ratio of a: b is a
^{3}: b^{3}etc. A ratio, whose antecedent and consequent are obtained by the cube roots of the antecedent and consequent respectively of a given ratio, is called the sub-triplicate ratio of the given ratio.**Sub-triplicate Ratio:**- The sub-triplicate ratio of x: is 3√x: 3√y.
- For example the sub-triplicate ratio of 1: 27 is 3√1; 3√27 =1:3
- The sub-triplicate ratio of 8: 125 is 3√8: 3√125 = 2: 5
- The sub-triplicate ratio of x
^{3}: y^{3}is 3√x³ : 3√y³= x : y; etc.

## Arithmetic Chapter 13 Proportion

- When the values of two ratios, expressed in the lowest term, are equal, they are said to be in proportion and one is called proportional to the other.
- For example 4: 6 and 10: 15 be two given ratios and they are equal in their lowest terms.
- These two ratios are said to be in proportion and one is called proportional to the other and we write as 4: 6:: 10: 15.
- Similarly, 8: 12: : 14: 21; T 10: ₹ 15:: 6 meters: 9 meters, etc. are examples of proportions.
- When four quantities are so related that the ratio between the first and the second quantities is equal to the ratio between the third and the fourth quantities, then the four quantities are said to be in proportion.
- Here it is necessary to be mentioned that the first and second quantities are of the same kind, the third and fourth quantities may not be the previous same kind but may be different types of same kind quantities.
**One important formula :**- If four quantities are in proportion, then we have
- So,
- First quality = \(=\frac{Second quantity \times Third quantity}{Fourth quantity}\)
- Second quantity = \(=\frac{First quantity \times Fourth quantity}{Third quantity}\)
- Third quantity = \(=\frac{First quantity \times Fourth quantity}{Second quantity}\)
- Fourth quantity = \(=\frac{Second quantity \times Third quantity}{First quantity}\)

- If any three terms of a proportion of four terms are known then the remaining term(the unknown term) can be determined by the above rule

## Arithmetic Chapter 13 Different Types Of Proportion

**There are three types of proportions:****Simple proportion (or Direct proportion):**

**1. Definition:**Two quantities are so mutually related that the increase (or decrease) of the values of one results in the increase (or decrease) of the values of the other and the ratio between the two values of the first quantity is equal to the ratio between the corresponding two values of the second quantity

2. Then it is said that the two ratios are in a simple proportion (or Direct proportion).**Inverse (or Reciprocal) Proportion:**1. Two mutually related quantities are such that the increase (or decrease) of the values of one results in the decrease (or increase) of the values of the other, then the ratio between the two values of the first quantity is equal to the inverse or reciprocal ratio between the values of the second quantity, then it is said that either of the two ratios is in inverse (or reciprocal) proportion to the other.

**2. Definition:**If two ratios are such that one ratio is equal to the reciprocal of the other, then either of them is said to be in inverse or reciprocal proportion of the other.**Continued Proportion:**

**1. Definition:**If three quantities are such that the first quantity: Second quantity = Second quantity: Third quantity, then they are said to be in continued proportion.

2. The second quantity is called the Mean proportional between the first and third.

3. Here the second quantity or the Mean proportional**=***√(***First quantity****x**4. First quantity = (Second quantity)**Third quantity )**

^{2}÷ third quantity;

5. Third quantity = (Second quantity)^{2}÷ first quantity.