How to Add, Subtract, Multiply, and Divide Rational Numbers
We have learned that when we divide an integer by another integer we might get an integer or a fraction. In all these cases the number is written in the form of a/b. For example, 10/ 5, 5 /2, 6/5, etc such numbers are called rational numbers. Remember that the denominator cannot be zero in any case.
A rational number may include any positive integer, a negative integer, a whole number, a decimal or a fraction.
Now let us learn different arithmetic operations like addition, subtraction, multiplication, division on rational numbers.
Rational Number Definition
A rational number a/b is said to be in its standard form if a and b have no common factors other than 1. i.e., a and b are co-primes, where b is 0. For example, 4/5,6/7,2/5 etc., are in the standard form. We can also say that fractions are examples of rational numbers.
A number like 5/10 is not in the standard form of rational numbers, it can be reduced to 1/2. Now ½ is the standard form of a rational number.
To identify if a number is rational or not, it should satisfy the following conditions.
A number should be represented in the form of a/b, where b ≠ 0.
The ratio a/b can be further represented in decimal form.
Arithmetic Operations with Rational Numbers
We have carried out arithmetic operations like addition, subtraction, multiplication, and division on integers and fractions. Similarly, we can carry out these operations with rational numbers. Arithmetic operations on rational numbers with the same denominators are easy to calculate but in the case of rational numbers with different denominators, we have to operate after making the denominators the same. Now let us study different arithmetic operations with rational numbers.
Addition Operation on Rational Numbers
Addition of rational numbers has two possibilities.
Consider rational numbers having the same denominator
For addition of rational numbers we can directly add the numerators.
For example Add 5/7 to 3/7
Here the denominator of both the rational number is 7. To add these numbers, we just keep the denominator the same and add the numerators.
\[= \frac{5}{7} + \frac{3}{7} \]
\[= \frac{5+3}{7} \]
\[= \frac{8}{7} \]
Consider Rational Numbers with Different Denominators
For the addition of rational numbers with different denominators first, we have to convert them into rational numbers with the same denominator.
To convert rational numbers with different denominators to the same denominators. We have to find the LCM of rational numbers.
Take the LCM of the denominators, then multiply both rational numbers by the LCM. we will get the rational numbers with the same denominator. Then add it.
For example Add \[= \frac{5}{6}\] and \[\frac{3}{5} \]
Solution: To evaluate \[= \frac{5}{6}\] and \[\frac{3}{5} \]
Convert the rational numbers with the same denominators.
Find LCM of 6 and 5 is 30
Multiply 30 by both rational numbers
We get,
\[ \frac{\frac{5}{6}\times{30} + \frac{3}{5}\times{30}}{30} \]
\[ = \frac{25 + 18}{30} \]
\[ = \frac{43}{30} \]
Subtraction Operation on Rational Numbers
Subtracting rational numbers is similar to in addition.
Consider rational numbers having the same denominator
For subtracting rational numbers with the same denominator we have to simply subtract the numerator.
For example Subtract \[ \frac{5}{7} \] to \[ \frac{3}{7} \]
Here the denominator of both the rational number is 7. To add these numbers, we just keep the denominator the same and add the numerators.
\[ = \frac{5}{7} - \frac{3}{7} \]
\[ = \frac{5-3}{7} \]
\[ = \frac{2}{7} \]
Consider Rational Numbers with Different Denominators
For subtracting rational numbers with different denominators first we have to convert them into rational numbers with the same denominator.
To convert rational numbers with different denominators to the same denominators we have to find the LCM of rational numbers.
Take the LCM of the denominators, then multiply both rational numbers by the LCM. we will get the rational numbers with the same denominator. Then subtract it.
For example, subtract \[= \frac{5}{6}\] and \[\frac{3}{5} \]
Solution: To evaluate \[ \frac{5}{6} - \frac{3}{5} \]
Convert the rational numbers with the same denominators.
LCM of 6 and 5 is 30
Multiply 30 by both rational numbers
We get,
\[ \frac{\frac{5}{6} \times{30} - \frac{3}{5} \times{30}}{30} \]
\[ = \frac{25 - 18}{30} \]
\[ = \frac{7}{30} \]
Multiplication Operation on Rational Number
For carrying our multiplication of rational numbers we don’t have to convert the different denominators into the same denominators.
Multiplication of rational numbers is equal to the product of numerators divided by the product of denominators.
Product of rational numbers = product of numerators/ product of denominator
For example, Multiply \[ \frac{2}{8} \] and \[ \frac{(-5)}{6} \]
Solution: For finding multiplication of rational numbers, multiply the numerators by the multiplication of denominators
(\[ \frac{2}{8} \times \frac{(-5)}{6} \])
\[ \frac{2\times (-5)}{8\times6} \]
\[ \frac{-10}{48} \]
Division Operation on Rational Number
For carrying out division on rational numbers we have to multiply the first rational number with the reciprocal of the second rational number.
Reciprocal of a rational number means taking the inverse of the number that is taking the numerator in place of the denominator and the denominator in place of the numerator. For example, the reciprocal of \[\frac{5}{6}\] is \[\frac{6}{5}\].
Example of division: Divide \[\frac{9}{2}\]by \[\frac{2}{3}\]
Solution: take the reciprocal of a second rational number and multiply i.e \[\frac{2}{3}\] is \[\frac{3}{2}\]
\[ = \frac{9}{2} \times \frac{2}{3} \]
\[ = \frac{{27}}{4} \]
Solved Examples
Example 1: Evaluate
\[ = \frac{5}{3} \times \frac{3}{4}\]
Solution:
\[ = \frac{5}{3} \times \frac{3}{4}\]
\[ = \frac{{5 \times 3}}{{2 \times 4}}\]
\[ = \frac{{15}}{8}\]
Example 2:
\[\frac{{13}}{3} - \left( { - \frac{{24}}{9}} \right) + \frac{{17}}{6}\]
Solution:
\[\frac{{13}}{3} + \frac{{24}}{9} + \frac{{17}}{6}\]
LCM of rational numbers = 18
\[= \frac{{13 \times 6}}{{3 \times 6}} + \frac{{24 \times 2}}{{9 \times 2}} + \frac{{17 \times 3}}{{6 \times 3}} \]
\[ = \frac{{78 + 48 + 51}}{{18}} \]
\[= \frac{{177}}{{18}} \]
Quiz Time
Simplify \[\frac{9}{4}\left( {\frac{1}{3} - \frac{5}{6} + \frac{1}{2}} \right) \div 5\]
Divide \[\left( {\frac{{28}}{5}} \right) \div \left( {\frac{{ - 30}}{7}} \right)\]
Properties of Rational Numbers
In general, rational numbers are those that can be written as p/q, where both p and q are integers and q is not zero. The following are the qualities of rational numbers:
Closure Property
Commutative Property
Associative Property
Distributive Property
Identity Property
Inverse Property
Closure Property
The outcomes of addition, subtraction, and multiplication operations for two rational integers say x and y, yield a rational number. When it comes to addition, subtraction, and multiplication, we may claim that they are closed under rational numbers. Consider the following example:
47/30 = (7/6)+(2/5)
Commutative Property
Addition and multiplication satisfy commutative property for rational numbers.
The commutative law of addition states that a+b = b+a.
ab = ba is a commutative multiplication law.
Associative Property
For addition and multiplication, rational numbers follow the associative property.
If x, y, and z are all rational, then the addition is as follows: x+(y+z)=(x+y)+z
x(yz)=(xy)z is the multiplication equation.
Identity Property
For rational numbers, 0 is an additive identity, whereas 1 is a multiplicative identity.
Examples:
1/2 + 0 equals 1/2 (Additive Identity)
1 x 1/2 = 1/2 (Identity Multiplication)
Inverse Property
The additive inverse of the rational number x/y is -x/y, while the multiplicative inverse is y/x.
Examples:
-1/4 is the additive inverse of 1/3. As a result, 1/4 + (-1/4) = 0.
1/3's multiplicative inverse is 3. As a result, 1/3 x 3 = 1
FAQs on Operations on Rational Numbers Made Easy
1. What are the basic arithmetic operations on rational numbers as per the CBSE syllabus?
The four basic arithmetic operations on rational numbers are addition, subtraction, multiplication, and division. These operations allow for the calculation and manipulation of numbers expressed as fractions (p/q form). Each operation follows specific rules, especially concerning common denominators and signs, which are fundamental concepts in the NCERT curriculum.
2. How do you add or subtract two rational numbers with different denominators?
To add or subtract rational numbers that have different denominators, you must first find a common denominator. The standard method involves these steps:
- Find the Least Common Multiple (LCM) of the denominators.
- Convert each rational number into an equivalent fraction with the LCM as the new, common denominator.
- Once the denominators are the same, simply add or subtract the numerators.
- Place the result over the common denominator and simplify the fraction if necessary.
3. What is the process for multiplying two rational numbers?
Multiplying rational numbers is a direct process. You do not need a common denominator. For any two rational numbers, say a/b and c/d, you simply multiply the numerators together and the denominators together. The formula is (a × c) / (b × d). The resulting fraction should be simplified to its lowest terms if possible.
4. How is division performed with rational numbers?
To divide one rational number by another, you multiply the first number by the reciprocal (or multiplicative inverse) of the second number. The reciprocal of a non-zero rational number c/d is d/c. Therefore, the operation (a/b) ÷ (c/d) is performed by calculating (a/b) × (d/c). It's important to remember that division by zero is undefined.
5. What is the correct order of operations to follow when solving a complex expression with rational numbers?
The correct order of operations for rational numbers is the same as for integers and is defined by the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction). This ensures that expressions are evaluated consistently:
- First, solve any expressions inside Brackets.
- Next, evaluate any Orders (powers or square roots).
- Then, perform Division and Multiplication as they appear from left to right.
- Finally, perform Addition and Subtraction as they appear from left to right.
6. Why is finding a common denominator essential for addition but not for multiplication of rational numbers?
A common denominator is essential for addition and subtraction because these operations involve combining quantities. The denominator represents the size of the parts of a whole, and you can only combine parts of the same size. For instance, you cannot directly add 'halves' and 'thirds'. By converting them to 'sixths' (a common denominator), you are working with uniform parts. Multiplication, in contrast, is about scaling or finding a fraction of another fraction, a process that doesn't require the parts to be of the same size initially.
7. What is the importance of the additive and multiplicative inverses in rational number operations?
Inverses are crucial for solving equations. The additive inverse of a number a/b is -a/b, because their sum is 0. It is fundamental to the concept of subtraction and isolating variables. The multiplicative inverse (or reciprocal) of a non-zero number a/b is b/a, because their product is 1. It is the key concept behind division, allowing us to convert a division problem into a multiplication problem.
8. How do properties like commutativity and associativity help in simplifying calculations with rational numbers?
The properties of rational numbers are powerful tools for simplification.
- The Commutative Property (a + b = b + a; a × b = b × a) allows you to reorder numbers in addition and multiplication. This can help group compatible numbers together.
- The Associative Property ((a + b) + c = a + (b + c)) allows you to regroup numbers in a long chain of additions or multiplications, making calculations easier.
9. Can you provide a real-world example where operations on rational numbers are used?
Operations on rational numbers are used frequently in daily life. For example, in financial planning, if you own 1/3 of a company's stock and decide to sell 1/2 of your holding, you would multiply 1/3 by 1/2 to find that you sold 1/6 of the total company stock. Another example is in construction or carpentry, where measurements like 5 ½ inches must be added to 3 ¾ inches, requiring addition of rational numbers.
