 # Operations on Rational Numbers

Every one of you already knows what rational numbers are. Do you know there are some operations that you can carry out with these rational numbers? Let us now study in detail about the operations on rational numbers. We will be studying addition, multiplication, subtraction, and division of these rational numbers examples.

### Suggested Videos        ## Operations on Rational Numbers

A rational number is a number that can be written in the form of a numerator upon a denominator. Here the denominator should not be equal to 0. The numerator and the denominator will be integers. A rational number is of the form

$$\frac{p}{q}$$

p = numerator, q= denominator, where p and q are integers and q ≠0

Examples: $$\frac{3}{5}$$, $$\frac{-3}{10}$$, $$\frac{11}{-15}$$. Here we can see that all the numerators and denominators are integers and even the denominators should be non-zero. To represent rationals, we divide the distance between two consecutive part into ‘n’ number of parts.

For example, to represent $$\frac{a}{3}$$ divide the distance between 0 and 1, 1 and 2 and so on in three equal parts and that parts are represented as $$\frac{1}{3}$$, $$\frac{2}{3}$$, $$\frac{3}{3}$$ and so on. Let us now see the operations of the rational numbers examples – addition, subtraction, multiplication, and division of rational numbers. ### Addition of the Rational Numbers

1) Adding rational numbers  witth same denominators: Let the two rational numbers be $$\frac{7}{3}$$ and $$\frac{-5}{3}$$. Adding these rational numbers,

$$\frac{7}{3}$$ + $$\frac{-5}{3}$$
= $$\frac{2}{3}$$

2) Adding rational numbers with different denominators: Let the two rational numbers be $$\frac{4}{3}$$ and $$\frac{3}{6}$$. Adding these rational numbers,

$$\frac{4}{3}$$ + $$\frac{3}{6}$$
Take the LCM of denominators. LCM of 3 and 6 is 6,
$$\frac{4×2}{3 ×2}$$ + $$\frac{3}{6}$$
= $$\frac{11}{6}$$

### Subtraction of the Rational Numbers

1) Subtraction of  rational numbers witth same denominators: Let the two rational numbers be $$\frac{5}{6}$$ and $$\frac{2}{6}$$. Subtracting these rational numbers,

$$\frac{5}{6}$$ – $$\frac{2}{6}$$
= $$\frac{3}{6}$$
= $$\frac{1}{2}$$

2) Subtracting rational numbers with different denominators: Let the two rational numbers be $$\frac{-12}{5}$$ and $$\frac{2}{6}$$. Subtarcting these rational numbers,

$$\frac{-12}{5}$$ – $$\frac{2}{6}$$
Take the LCM of denominators. LCM of 12 and 6 is 12,
$$\frac{-12}{5}$$ + $$\frac{2×2}{6 ×2}$$
=  $$\frac{-5 + 4 }{12}$$
= $$\frac{-1}{2}$$

### Multiplication of the Rational Numbers

When two rational numbers are multiplied, the numbers are known as  factors and result is the product. Let the two rational numbers be $$\frac{5}{4}$$ and $$\frac{10}{5}$$. Multiplying these rational numbers,

= $$\frac{50}{20}$$
= $$\frac{5}{2}$$

### Division of the Rational Numbers

To divide one rational number by the another rational number, we multiply the rationl numbers by the reciprocal of each other. Let the two rational numbers be $$\frac{1}{4}$$ and $$\frac{5}{4}$$. Diving these rational numbers,

$$\frac{1}{4}$$ ÷ $$\frac{5}{4}$$
= $$\frac{1}{4}$$ × $$\frac{5}{4}$$
= $$\frac{5}{16}$$

## Rational Numbers Examples for You

Question 1: Difference of greatest three digits number and smallest five digits number is:

1. 19001
2. 9011
3. 9001
4. 9091

Answer : C. Greatest three digit number is 999 and Smallest five digit number is 10000. Difference = 10000- 999 = 90001.

Question 2: Which of the following statements do not accurately describe the multiplicative inverse?

1. If a given fraction with numerator 1, the multiplicative inverse will be its denominator.
2. If a given whole number, the multiplicative inverse will be a fraction containing the number as numerator and 1  as the denominator.
3. In a statement root 3 ×= 1, x is the multiplicative inverse.
4. One pair of the number when multiplied together gives the number 1.

Answer : B. The multiplicative inverse will include 1 as the numerator.

Question 3: What is meant by a rational number?

Answer: A rational number refers to a number whose expression can take place as the fraction or quotient p/q of two integers, a non-zero denominator q and a numerator p.

Question 4: Can we say that 5 is a rational number?

Answer: Every integer happens to be a rational number. Furthermore, one can write each integer n in the form n/1. So, 5 = 5/1 and hence 5 is a rational number.

Question 5: Explain the irrational number with example?

Answer: The irrational numbers refer to all the real numbers which do not fall in the category of rational numbers. Rational numbers are those whose construction takes place from factions or ratios of integers. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π. It cannot be represented by any finite number of digits and nor does its repetition takes place.

Question 6: Can we say that the square root of 16 is a rational number?

Answer: Yes, the square root of 16 happens to be a rational number. This is because the square root of 16 happens to be 4. 4 happen to be an integer and a rational number.

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