Operations on Rational Numbers: Techniques & Examples

Techniques followed in the arithmetic operations is quite similar to operations on rational numbers as well. Addition, subtraction, multiplication and division are the basic operations that are used on both fractions and integers. It is important for students to understand what rational numbers are before they apply different operations on them. This article aims to discuss the basics associated with operations on rational numbers and other details associated with the same.

A set of rational numbers is represented using the letter \(”Q”.\) Like real numbers, arithmetic operations, such as addition, subtraction, multiplication, and division, can be applied to rational numbers also. The operations on rational numbers can be performed in two ways. In this article, we will study the different arithmetic operations on rational numbers with solved examples.

Definition of Rational Numbers

A number which is in the form of \(\frac{p}{q},\) where \(p\) and \(q\) are co-prime integers and \(q \ne 0,\) is called a rational number.

Learn the Concepts on Rational Numbers

Examples:
a) Every whole number and natural numbers are rational numbers, as we can write, \(0 = \frac{0}{1},8 = \frac{8}{1},3 = \frac{3}{1},284 = \frac{{284}}{1}.\)
b) Every integer is a rational number, \( – 85 = \frac{{ – 85}}{1},13 = \frac{{13}}{1}.\)
c) Every fraction is a rational number because fractions are represented as \(\frac{a}{b},\) where \(a\) and \(b\) are co-prime integers and \(b \ne 0.\)

Positive Rational Numbers

A rational number is called a positive rational number if both its numerator and denominator are positive or negative.
Example: \(\frac{{ – 3}}{{ – 8}},\frac{{21}}{{67}},\frac{{ – 9}}{{ – 211}},\frac{{43}}{{29}}\)

Negative Rational Numbers

A rational number is called a negative rational number if either its numerator or denominator (but not both) is a negative integer.
Example: \(\frac{{ – 7}}{2},\frac{6}{{ – 23}},\frac{{ – 58}}{{1195}},\frac{{38}}{{ – 3}}\)

General Rules Relating to Rational Numbers

Rule 1: If \(\frac{p}{q}\) is a rational number and \(m\) is a non-zero integer, then \(\frac{p}{q} = \frac{{p \times m}}{{q \times m}}\)
Example: \(\frac{{ – 2}}{3} = \frac{{ – 2 \times 2}}{{3 \times 2}} = \frac{{ – 2 \times 3}}{{3 \times 3}} = \frac{{ – 2 \times 4}}{{3 \times 4}} \ldots .\)
Thus, \(\frac{{ – 2}}{3} = \frac{{ – 4}}{6} = \frac{{ – 6}}{9} = \frac{{ – 8}}{{12}} \ldots \)

Rule 2: If \(\frac{p}{q}\) is a rational number and \(m\) is a common divisor of \(p\) and \(q,\) then \(\frac{p}{q} = \frac{{p \div m}}{{q \div m}}\)
Example: \(\frac{{27}}{{30}} = \frac{{27 \div 3}}{{30 \div 3}} = \frac{9}{{10}}\) (Note HCF of \(27,30\) is \(3\))
\(\frac{{ – 8}}{{12}} = \frac{{ – 8 \div 4}}{{12 \div 4}} = \frac{{ – 2}}{3}\) (Note HCF of \(8,12\) is \(4\))

Rule 3: If the denominator of a given rational number is negative, we multiply the numerator and denominator by \( – 1\) to get an equivalent rational number with a positive denominator.

Standard form of a rational number: A rational number \(\frac{p}{q}\) is said to be in standard (or simplest) form if:
1. the denominator \(\left( q \right)\) is a positive integer.
2. \(p\) and \(q\) are co-primes. This means \(p\) and \(q\) have no common factor between them other than \(1\)
Method: To express a given rational number in standard form, we first convert it into a rational number with a positive denominator and divide its numerator and denominator by their HCF.

Rule 4: \(\frac{a}{b} = \frac{c}{d} \Leftrightarrow a \times d = c \times b\) is true.

Rule-\(1\) and Rule-\(2\) described above give us equivalent rational numbers.

Equivalent Rational Numbers

Two rational numbers are said to be equivalent rational numbers if one can be obtained from the other by multiplying or dividing its numerator and denominator by the same non-zero number.

Example:

i) \(\frac{3}{2} = \frac{6}{4} = \frac{9}{6} = \frac{{12}}{8} = \frac{{15}}{{10}}\) are the equivalent rational numbers
ii) \(\frac{{24}}{{30}} = \frac{{12}}{{15}} = \frac{4}{5}\) etc. are the equivalent rational numbers

Arithmetic Operations on Rational Numbers

The basic arithmetic operations performed on rational numbers are:

1. Addition of rational numbers
2. Subtraction of rational numbers
3. Multiplication of rational numbers
4. Division of rational numbers

Addition of Rational Numbers

Case 1: When the denominators are the same: When the denominators of the given rational numbers are the same, we add the numerators and keep the denominator the same.

Let \(\frac{a}{b}\) and \(\frac{c}{b}\) be any two rational numbers.
Then, \(\frac{a}{b} + \frac{c}{b} = \frac{{a + c}}{b}\)

Example: Add \(\frac{5}{{11}}\) and \(\frac{{ – 7}}{{11}}\)
We have, \(\frac{5}{{11}} + \frac{{ – 7}}{{11}} = \frac{{5 + \left({ – 7} \right)}}{{11}} = \frac{{ – 2}}{{11}}\)

Case 2: When one denominator is a multiple of the other denominator: When one denominator is a multiple of the other denominator, we multiply the numerator and denominator by the same number so that the denominators of both rational numbers will be the same.

Let \(\frac{a}{b}\) and \(\frac{d}{{bc}}\) be two rational number
Then, \(\frac{a}{b} + \frac{d}{{bc}} = \frac{{a \times c}}{{b \times c}} + \frac{d}{{bc}} = \frac{{ac + d}}{{bc}}\)

Example: Add \(\frac{{ – 11}}{8}\) and \(\frac{{15}}{{16}}\)
\(\frac{{ – 11}}{8} + \frac{{15}}{{16}} = \frac{{ – 11 \times 2}}{{8 \times 2}} + \frac{{15}}{{16}} = \frac{{ – 22 + 15}}{{16}} = \frac{{ – 7}}{{16}}\) (\(16\) is a multiple of \(8\))

Case 3: When the denominators are co-primes: When the denominators are co-primes, we multiply the numerator of the first rational number with a denominator of the second rational number and the numerator of the second rational number by the denominator of the first rational number.
The new numerator will be the sum of these two products. Then, multiply both the denominators of the given rational numbers to obtain the denominator of the sum.

Let \(\frac{a}{b}\) and \(\frac{c}{d}\) be two rational numbers
Then, \(\frac{a}{b} + \frac{c}{d} = \frac{{ad + cb}}{{bd}}\)

Example: Add \(\frac{3}{5} + \frac{1}{3} = \frac{{3 \times 3 + 1 \times 5}}{{5 \times 3}} = \frac{{14}}{{15}}\)

Case 4: When the given rational numbers have different denominators: When the given rational numbers have different denominators, we find the sum as given below:

Step 1: We first find the LCM of the denominators of the given rational numbers
Step 2: Convert each of the given rational numbers such their LCM is the denominator for both.
Step 3: Now add the rational numbers with the same denominators as discussed above

Example: Add \(\frac{{ – 3}}{6}\) and \(\frac{{ 5}}{4}\)
LCM of the denominators, LCM \(\left({6,4} \right) = 2 \times 2 \times 3 = 12\)
Now, \(\frac{{ – 3}}{6} = \frac{{ – 3 \times 2}}{{6 \times 2}} = \frac{{ – 6}}{{12}},\frac{5}{4} = \frac{{5 \times 3}}{{4 \times 3}} = \frac{{15}}{{12}}\)
Now, \(\frac{{ – 3}}{6} + \frac{5}{4} = \frac{{ – 6}}{{12}} + \frac{{15}}{{12}} = \frac{{ – 6 + 15}}{{12}} = \frac{9}{{12}} = \frac{3}{4}\)

Subtraction of Rational Numbers

Additive inverse of a rational number: An additive inverse of a rational number is the number that gives zero on adding with the original number. If \(x\) is the original number, then \( – x\) is its additive inverse.

Example:\(\frac{{ – 3}}{5}\) is the additive inverse of \(\frac{{3}}{5}\)

Subtraction

Subtraction is the inverse process of addition. One rational number can be subtracted from another rational number by adding its additive inverse to it.

Let \(\frac{p}{q}\) and \(\frac{r}{s}\) be two rational numbers
Then, \(\frac{p}{q} – \frac{r}{s} = \frac{p}{q} + \frac{{ – r}}{s},\) where \(\frac{{ – r}}{s}\) is the additive inverse of \(\frac{r}{s}.\)

Example: Subtract \(\frac{4}{8}\) from \(\frac{{ – 5}}{{12}}\)
Additive inverse of \(\frac{4}{8}\) is \(\frac{{ – 4}}{{8}}\)
\(\frac{{ – 5}}{{12}} – \frac{4}{8} = \frac{{ – 5}}{{12}} + \frac{{ – 4}}{8}\)
\( = \frac{{ – 5 \times 2}}{{12 \times 2}} + \frac{{ – 4 \times 3}}{{8 \times 3}}\)
\( = \frac{{ – 10 – 12}}{{24}}\)
\( = \frac{{ – 22}}{{24}}\)
\( = \frac{{ – 11}}{{12}}\)

Multiplication of Rational Numbers

The product of the two or more rational numbers is the rational number whose numerator is the product of the two or more numerators and whose denominator is the product of two or more denominators.

For two rational numbers \(\frac{a}{c}\) and \(\frac{b}{d}\) we define: \(\frac{a}{c} \times \frac{b}{d} = \frac{{a \times b}}{{c \times d}} = \frac{{{\text{product}}\,{\text{of}}\,{\text{numerators}}}}{{{\text{product}}\,{\text{of}}\,{\text{denominators}}}}\)

Example: Find the product of \(\frac{{ – 5}}{7}\) and \(\frac{{ 9}}{11}\)
\(\frac{{ – 5}}{7} \times \frac{9}{{11}} = \frac{{ – 5 \times 9}}{{7 \times 11}}\)
\( = \frac{{ – 45}}{{77}}\)

Division of Rational Numbers

Multiplicative inverse of a rational number: For every non-zero rational number \(\frac{p}{q}\) there exists a rational number \(\frac{q}{p}\) such that \(\frac{p}{q} \times \frac{q}{p} = 1 = \frac{q}{p} \times \frac{p}{q}\)
The rational number \(\frac{q}{p}\) is called the multiplicative inverse or reciprocal of \(\frac{p}{q}.\)

Example: \(\frac{3}{5}\) is the multiplicative inverse of \(\frac{5}{3}\)

Division

The division is the inverse of multiplication. If \(a\) and \(b\) are two integers, then \(a \div b = a \times \frac{1}{b}\). It means we multiply the dividend by the multiplicative inverse of the divisor. We apply the same rule for the division of rational numbers.
If \(\frac{p}{r}\) and \(\frac{q}{s}\) are two rational numbers then,
\(\frac{p}{r} \div \frac{q}{s} = \frac{p}{r} \times \frac{s}{q}\left({\frac{q}{s} \ne 0} \right)\)
Or \(\frac{p}{r} \div \frac{q}{s} = \frac{{p \times s}}{{r \times q}}\)

Example: Divide \(\frac{2}{5}\) and \(\frac{3}{7}\)
\(\frac{2}{5} \div \frac{3}{7} = \frac{2}{5} \times \frac{7}{3}\)
\( = \frac{{2 \times 7}}{{5 \times 3}}\)
\( = \frac{{14}}{{15}}\)

Solved Examples – Operations on Rational Numbers

Q.1. Find the sum of \(\frac{{ – 4}}{{12}} + \frac{{ – 5}}{9} + \frac{{13}}{{18}}\)
Ans: LCM of denominators \( = \) LCM \(\left({12,9,18} \right) = 2 \times 3 \times 2 \times 3 = 36\)
We have, \(\frac{{ – 4}}{{12}} = \frac{{ – 4 \times 3}}{{12 \times 3}} = \frac{{ – 12}}{{36}},\frac{{ – 5}}{9} = \frac{{ – 5 \times 4}}{{9 \times 4}} = \frac{{ – 20}}{{36}},\frac{{13}}{{18}} = \frac{{13 \times 2}}{{18 \times 2}} = \frac{{26}}{{36}}\)
Now, \(\frac{{ – 4}}{{12}} + \frac{{ – 5}}{9} + \frac{{13}}{{18}} = \frac{{ – 12}}{{36}} + \frac{{ – 20}}{{36}} + \frac{{26}}{{36}}\)
\( = \frac{{ – 12 – 20 + 26}}{{36}}\)
\( = \frac{{ – 6}}{{36}}\)
\( = \frac{{ – 1}}{6}\)
Therefore, the sum of \(\frac{{ – 4}}{{12}} + \frac{{ – 5}}{9} + \frac{{13}}{{18}} = \frac{{ – 1}}{6}.\)

Q.2. Find the additive inverse of \(\frac{{ – 13}}{8}.\)
Ans: Additive inverse of a rational number \(\frac{p}{q}\) is \(\frac{-p}{q}\)
\(\therefore \) The additive inverse of \(\frac{{ – 13}}{8}.\) is \(\frac{{ 13}}{8}.\)

Q.3. Find the product of \(\frac{{ – 2}}{6}\) and \(\frac{{21}}{7}\)
Ans: As we know, the product of two or more rational numbers \( = \frac{{{\text{product}}\,{\text{of}}\,{\text{numerators}}}}{{{\text{product}}\,{\text{of}}\,{\text{denominators}}}}\)
Hence, the product of given rational number \( = \frac{{ – 2}}{6} \times \frac{{21}}{{ – 7}} = \frac{{ – 2 \times 21}}{{6x – 7}}\)
\( = 1\)
Thus, the product of given rational numbers is \(1.\)

Q.4. The cost of 15 pens is \(₹67\frac{1}{2}.\) Find the cost of each pen.
Ans: Given, the cost of \(15\) pens \( =₹ 67\frac{1}{2} = ₹\frac{{135}}{2}\)
Let us assume the cost of \(1\) pen \( =₹ x\)
Now, \(15 \times x = \frac{{135}}{2}\)
\( \Rightarrow x = \frac{{135}}{2} \div 15\)
\( = \frac{{135}}{2} \times \frac{1}{{15}} = \frac{{135 \times 1}}{{2 \times 15}} = \frac{9}{2} = 4\frac{1}{2}.\)
Thus, the cost of each pen is \(₹4\frac{1}{2}.\)

Q.5. Ram and Raj went to a pizza parlour to celebrate Ram’s birthday. They ordered a pizza. Ram ate \(\frac{1}{5}\) of the pizza, and Raj ate \(\frac{1}{4}\) of the remaining pizza. What portion of the pizza is left?
Ans: Total pizza ordered \(= 1\)
The portion of pizza eaten by Ram \( = \frac{1}{5}\)
The portion of pizza eaten by Raj \( = \frac{1}{4}\)
Total pizza was eaten by Ram and Raj \( = \frac{1}{5} + \frac{1}{4} = \frac{{4 + 5}}{{20}} = \frac{9}{{20}}\)
The portion of pizza left \( = 1 – \) pizza eaten by Ram and Raj
\( = 1 – \left({\frac{9}{{20}}} \right)\)
\( = \left({\frac{{20 – 9}}{{20}}} \right)\)
\( = \frac{{11}}{{20}}\)
Thus, the portion of pizza left is \( = \frac{{11}}{{20}}.\)

Summary

In this article, we studied the definition of rational numbers and the standard form of rational numbers. Then we discussed different arithmetic operations on the rational number: the addition, subtraction, multiplication and division of rational numbers. Few solved examples have been provided for a better understanding of the operations on the rational numbers.

Know the Properties of Rational Numbers

Frequently Asked Questions (FAQs)

Frequently asked questions related to operations of rational numbers is listed as follows:

Q.1. What are the four operations of rational numbers?
Ans: There are four basic arithmetic operations on rational numbers: addition, subtraction, multiplication, and division.

Q.2. How do you use rational numbers?
Ans: Uses of rational numbers in real-life:
1. When you share a pizza or anything.
2. When you completed homework half portion, you say that you completed \(50\% \) i.e. \(\frac{1}{2}\)
3. Interest rates on loans and mortgages.
4. Interest in savings accounts.

Q.3. Write down the examples of rational numbers.
Ans: Any number in the form of \(\frac{p}{q},\) where \(p\) and \(q\) are co-prime integers, and \(q\) is not equal to \(0\) is a rational number. Examples of rational numbers are \(\frac{1}{3},\frac{{ – 3}}{2},\frac{2}{{20}}, – 4.\)

Q.4. How do you multiply rational numbers?
Ans: Product of two or more rational numbers \( = \frac{{{\text{product}}\,{\text{of}}\,{\text{numerators}}}}{{{\text{product}}\,{\text{of}}\,{\text{denominators}}}}\)
Example, \(\frac{2}{3} \times \frac{1}{5} = \frac{{2 \times 1}}{{3 \times 5}} = \frac{2}{{15}}\)

Q.5. What are the rules for rational numbers?
Ans: There are four general rules for rational numbers that are briefly explained below:
1) If \(\frac{p}{q}\) is a rational number and \(m\) is a non-zero integer, then \(\frac{p}{q} = \frac{{p \times m}}{{q \times m}}\)
2) If \(\frac{p}{q}\) is a rational number and \(m\) is a common divisor of \(p\) and \(q,\) then \(\frac{p}{q} = \frac{{p \div m}}{{q \div m}}\)
3) If the denominator of a given rational number is negative, we multiply the numerator and denominator by \( – 1\) to get an equivalent rational number with a positive denominator.
4) \(\frac{a}{b} = \frac{c}{d} \Leftrightarrow a \times d = c \times b\)

Q.6. Write the correct order of operations for rational numbers?
Ans: Step 1: Use brackets as per the BODMAS rule.
Step 2: Simplify exponential expressions.
Step 3: Perform multiplication and division as they occur from left to right.
Step 4: Perform addition and subtraction as they occur from left to right.