What Are the Operations on Rational Numbers?
Operations on rational numbers refer to the arithmetic operations given by addition, subtraction, multiplication, and division. Rational numbers are numbers that can be written in the form $\frac{p}{q}$, where p and q are integers and $q \neq 0$. The set of rational numbers is represented by the symbol ℚ.
Arithmetic operations on rational numbers refer to the mathematical operations carried out on two or more rational numbers.
The basic arithmetic operations performed on rational numbers are:
- Addition of Rational Numbers (with same denominators and with different denominators)
- Subtraction of Rational Numbers (with same denominators and with different denominators)
- Multiplication of Rational Numbers
- Division of Rational Numbers
Let’s study each of them in detail with the steps and examples.
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Addition of Rational Numbers
There are two cases possible when adding two or more rational numbers.
Case 1: When the denominators of the given rational numbers are equal.
When denominators are equal, add the numerators and keep the same denominator.
| Addition of Rational Numbers (When denominators are equal.) |
|---|
| RULE: Add the numerators. Keep the common denominator. $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$ |
| Examples: $\bullet\;\frac{2}{8} + \frac{3}{8} = \frac{2 + 3}{8} = \frac{5}{8}$ $\bullet\;\frac{4}{6} + \frac{1}{6} = \frac{4 + 1}{6} = \frac{5}{6}$ $\bullet\;\frac{7}{16} + \frac{9}{16} = \frac{7 + 9}{16} = \frac{16}{16} = 1$ |
Case 2: When the denominators of given rational numbers are different
When the denominators are not equal, we first need to find a common denominator. Let’s understand this with an example.
Example: Add the rational numbers $\frac{2}{5}$ and $\frac{3}{4}$.
Step 1: Find the LCM of the denominators of the given rational numbers.
Here, the LCM of 4 and 5 is 20.
Step 2: Change the denominator of each rational number to 20 by multiplying both numerator and denominator by an appropriate factor.
$\frac{2 \times 4}{5 \times 4} = \frac{8}{20}$ and $\frac{3 \times 5}{4 \times 5} = \frac{15}{20}$
Step 3: For these new rational numbers (having a common denominator), add the numerators and keep the common denominator.
$\frac{2}{5} + \frac{3}{4} = \frac{23}{20}$
| Addition of Rational Numbers (When denominators are different) |
|---|
| RULE: Find the LCM of denominators and solve to find the common denominator. |
| Examples: $\bullet\;\frac{2}{8} + \frac{1}{3} = \frac{6}{24} + \frac{8}{24} = \frac{6 + 8}{24} = \frac{14}{24} = \frac{7}{12}$ $\bullet\;\frac{1}{6} + \frac{5}{2} = \frac{1}{6} + \frac{15}{6} = \frac{15 + 1}{6} = \frac{16}{6} = \frac{8}{3}$ |
Subtraction of Rational Numbers
We will discuss the same cases for subtraction of rational numbers as well.
Case 1: When the denominators of the given rational numbers are equal:
Subtract the numerators and keep the denominator the same.
| Subtraction of Rational Numbers (When denominators are equal.) |
|---|
| RULE: Subtract the numerators. Keep the common denominator. $\frac{a}{c} \;-\; \frac{b}{c} = \frac{a \;-\; b}{c}$ |
| Examples: $\bullet\;\frac{5}{8} \;-\; \frac{2}{8} = \frac{3}{8}$ $\bullet\;\frac{2}{8} \;-\; \frac{3}{8} = \frac{2 \;-\; 3}{8} = \frac{-1}{8}$ $\bullet\;\frac{4}{6} \;-\; \frac{1}{6} = \frac{4 \;-\; 1}{6} = \frac{3}{6} = \frac{1}{2}$ $\bullet\;\frac{7}{16} \;-\; \frac{9}{16} = \frac{7 \;-\; 9}{16} = \frac{-2}{16} = \frac{-1}{8}$ |
Case 2: When the denominators of the given numbers are unequal:
Here, we first make the denominators equal using the LCM method.
| Subtraction of Rational Numbers (When denominators are different) |
|---|
| RULE: Find the LCM of denominators and solve to find the common denominator. |
| Examples: $\bullet\;\frac{2}{8} \;-\; \frac{1}{4} = \frac{2}{8} \;-\; \frac{2}{8} = 0$ $\bullet\;\frac{1}{5} \;-\; \frac{5}{3} = \frac{3}{15} \;-\; \frac{25}{15} = \frac{-22}{15}$ |
Example: subtract $\frac{1}{3}$ from $\frac{1}{2}$.
Step 1: Find the LCM of the denominators of the given rational numbers.
In this case, the LCM of 2 and 3 is 6.
Step 2: Convert each rational number into an equivalent rational number with the LCM as the new denominator.
$\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$ and $1 \times 2}{3 \times 2 = \frac{2}{6}$
Step 3: Subtract the numerators. Keep the common denominator.
$\frac{3}{6}\;-\; \frac{2}{6} = \frac{1}{6}$
Therefore, $\frac{1}{2} \;-\; \frac{1}{3} = \frac{1}{6}$
Multiplication of Rational Numbers
It is very easy to multiply rational numbers.
Step 1: Multiply the numerators. Write the product as the numerator of the answer.
Step 2: Multiply the denominators. Write the product as the denominator of the answer.
Step 3: Reduce the final answer to its lowest form.
| Multiplication of Rational Numbers |
|---|
| RULE: Multiply the numerators. Multiply the denominators. $\frac{a}{c} \times \frac{b}{d} = \frac{a \times b}{c \times d}$ |
| Examples: $\bullet\;\frac{2}{5} \times \frac{3}{4} = \frac{2 \times 3}{5 \times 4} = \frac{6}{20}$ $\bullet\;\frac{2}{8}\frac{\;-\;3}{8} = \frac{2 (\;-\;3)}{8 \times 8} = \frac{-6}{64} = \frac{-3}{32}$ $\bullet\;\frac{(\;-\;4)}{6} \times \frac{1}{4} = \frac{(\;-\;4) \times 1}{6\times 4} = \frac{\;-\;4}{24} = \frac{\;-\;1}{6}$ $\bullet\;\frac{7}{16} \times \frac{9}{10} = \frac{7 \times 9}{16 \times 10} = \frac{63}{160}$ |
Division of Rational Numbers
To divide a rational number by another rational number, we multiply the first rational number (dividend) by the reciprocal of the second rational number (divisor).
| Division of Rational Numbers |
|---|
| RULE: Multiply the dividend with the reciprocal of the divisor. $\frac{a}{c} \div \frac{b}{d} = \frac{a}{c}\times\frac{d}{b}$ |
| Examples: $\bullet\;\frac{2}{5} \div \frac{3}{4} = \frac{2}{5} \times \frac{4}{3} = \frac{8}{15}$ $\bullet\;\frac{7}{4} \div \frac{2}{7} = \frac{7}{4} \times \frac{7}{2} = \frac{49}{8}$ |
Example: Find $\frac{2}{3} \div \frac{1}{5}$.
Step 1: Find the reciprocal of the divisor.
Reciprocal of $\frac{1}{5} = \frac{5}{1}$.
Step 2: Multiply the dividend with the reciprocal of the divisor.
$\frac{2}{3} \times \frac{5}{1} = = \frac{2 \times 5}{3 \times 1} = \frac{10}{3}$
Multiplication or division of integers with the same signs produces a positive result.
Multiplication or division of two integers with unlike signs results in a negative answer.
These rules can be applied to the multiplication and division of rational numbers as well.
Order of Operations with Rational Numbers
Order of operations with rational numbers is no different from the order of operations you have used so far for. Order of operations tells us the correct sequence in which a mathematical expression should be evaluated. We use the PEMDAS rule to remember the order.
P – Parentheses
E – Exponents
M – Multiplication
D – Division
A – Addition
S – Subtraction
Operations on Rational Numbers with Negative Signs
A rational number is said to be negative if the numerator and denominator have opposite signs. Rational number operations with negatives follow the same rules as integers.
| Operation | Rule |
|---|---|
| Addition & Subtraction | If the numbers have SAME SIGN, then ADD the numbers and KEEP THE SIGN. Examples: $9 + 7 = 16$ $(\;-\;9) + (\;-\;7) = (\;-\;16)$ |
| If the numbers have DIFFERENT SIGNS, then SUBTRACT the numbers and take the SIGN OF THE GREATER NUMBER. Examples: $(-9) + 7 = \;-\;2$ $9 + (\;-\;7) = 2$ Special case: $a \;-\; (\;-\;b) = a + b$ $9 \;-\;(\;-\;7) = 9 + 7 = 16$ | |
| Multiplication &Division | SAME SIGN: The answer will be POSITIVE. $(+) (+) = (+)$ $(-) (-) = (+)$ Examples: $5 \times 4 = 20$ $(\;-\;2) \times (\;-\;1) = 2$ $(\;-\;20) \div (\;-\;5) = 4$ DIFFERENT SIGNS: The answer will be NEGATIVE. $(\;-\;) (+) = (\;-\;)$ Examples: $(\;-\;5) \times (\;-\;4) = 20$ $(2) \times (\;-\;1) = \;-\;2$ $(\;-\;20) \div 5 = \;-\;4$ |
Properties of Operations on Rational Numbers
Properties of rational numbers make it easy to perform different mathematical operations on them. These properties come handy when simplifying expressions or solving equations.
Closure Property of Rational Numbers
For two rational numbers, the addition, subtraction, and multiplication always results in a rational number. Thus, rational numbers are closed under addition, subtraction, and multiplication. The closure property isn’t applicable for the division of rational numbers as division by zero isn’t defined.
Associative Property of Rational Numbers
Rational numbers obey the associative property for addition and multiplication.
Thus, for any three rational numbers x, y, and z, we have
$x + (y + z) = (x + y) + z$
$x \times (y \times z) = (x \times y) \times z$
Examples:
- $\frac{1}{3} + (\frac{1}{4} + \frac{3}{3}) = (\frac{1}{3} + \frac{1}{4}) + \frac{3}{3}$
- $\frac{1}{3} \times (\frac{1}{4}\times \frac{3}{3}) = (\frac{1}{3} \times \frac{1}{4}) \times \frac{3}{3}$
Commutative Property of Rational Numbers
The addition and multiplication of rational numbers is always commutative. Subtraction of rational numbers doesn’t obey commutative property.
Commutative Law of Addition: $x + y = y + x$
Example: $\frac{1}{3} + \frac{2}{3} = \frac{2}{3} + \frac{1}{3} = \frac{3}{3}$
Commutative Law of Multiplication: $x \times y = y \times x$
Example: $\frac{1}{2} \times \frac{2}{3} = \frac{2}{3} \times \frac{1}{2} = \frac{2}{6}$
Distributive Property of Rational Numbers
Rational numbers follow distributive property over addition and subtraction.
For rational numbers A, B, and C, we have
- $A \times (B \;-\; C) = (A \times B) \;-\; (A \times C)$
- $A \times (B + C) = (A \times B) + (A \times C)$
Example: $\frac{1}{3} \times (\frac{1}{4} + \frac{2}{5}) = (\frac{1}{3} \times \frac{1}{4})$ + (\frac{1}{3} \times \frac{2}{5})$
L.H.S. $= \frac{1}{3} \times (\frac{1}{4} + \frac{2}{5}) = \frac{1}{3} \times (\frac{17}{20}) = \frac{17}{60}$
R.H.S. $= (\frac{1}{3} \times \frac{1}{4}) + (\frac{1}{3} \times \frac{2}{5}) = \frac{1}{12} + \frac{2}{10} = \frac{17}{60}$
Additive Identity and Multiplicative Identity
0 is the additive identity of any rational number. When we add 0 to any rational number, the resultant is the number itself.
$x + 0 = x$ …for any rational number x
1 is the multiplicative identity of any rational number. When we multiply 1 with any rational number, the result is the number itself.
$x \times 1 = x$ …for any rational number x
Additive Inverse and Multiplicative Inverse
For any rational number $\frac{x}{y}$, the additive inverse is given by $(\;-\;\frac{x}{y})$.The addition of a rational number and its additive inverse is always 0.
For any rational number $\frac{x}{y}$, the multiplicative inverse or reciprocal is given by $\frac{y}{x}$ . The product of a rational number and its reciprocal is always 1.
Rational Numbers Operations Anchor Chart
An anchor chart displays the important information from a given lesson. Let’s recall and summarize a few important points.
| Addition of Rational Numbers | Subtraction of Rational Numbers |
|---|---|
| $\bullet\;$Same denominators: $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$ $\bullet\;$Different denominators: Use the LCM method. | $\bullet\;$Same denominators: $\frac{a}{c} \;-\; \frac{b}{c} = \frac{a \;-\; b}{c}$ $\bullet\;$Different denominators: Use the LCM method. |
| Multiplication of Rational Numbers $\frac{a}{c} \times \frac{b}{d} = \frac{a \times b}{c \times d}$ | Division of Rational Numbers $\frac{a}{c}\div \frac{b}{d} = \frac{a}{ c}\times \frac{d}{b}$ |
| Adding and Subtracting Rational Numbers $\bullet\;$with SAME signs: ADD and KEEP $\bullet\;$with DIFFERENT signs: SUBTRACT and take the SIGN OF THE LARGER NUMBER. $\bullet\;$Special case: $a \;-\; (\;-\;b) = a + b$ | |
| Multiplying and Dividing Rational Numbers $\bullet\;$with SAME signs: Answer is POSITIVE $\bullet\;$with DIFFERENT signs: Answer is NEGATIVE | |
Facts about Operations on Rational Numbers
- Rational numbers are denoted by the letter ℚ.
- Pythagoras and his disciples believed that all numbers were rational. It is said that the discovery of irrational numbers shocked them!
- Identity property holds true for addition and multiplication. It does not hold true for subtraction and division of rational numbers.
- Subtraction of rational numbers doesn’t obey commutative property.
- The inverse property does not apply to the division and subtraction of rational numbers.
Conclusion
In this article, we have learned about operations on rational numbers. We learned that the techniques followed in the arithmetic operations are quite similar to operations on rational numbers as well. Now, let’s apply rational number operations and properties to solve a few examples and practice problems.
Solved Examples on Operations on Rational Numbers
1. Find the additive inverse of $\frac{5}{12}$.
Solution:
Additive inverse of rational number $\frac{x}{y}$ is $(\frac{-x}{y})$.
Thus, additive inverse of $\frac{5}{12}$ is $\frac{-5}{12}$.
2. Solve the following equation using the distributive property: $6 \times (20 + 5)$.
Solution:
According to the distributive property over addition,
$A (B + C) = (A \times B) + (A \times C)$
$6 \times (20 + 5) = (6 \times 20) + (6 \times 5)$
$= 120 + 30$
$= 150$
Thus, $6 \times (20 + 5) = 150$
3. From a rope 252 ft long, a part from end measuring $\frac{62}{8}$ ft is cut off. Find the length of the remaining rope.
Solution:
Total length of the rope $= \frac{25}{2}$ ft
Length of the rope cut off $= \frac{62}{8}$ ft
The length of the remaining rope $= \frac{25}{2} \;-\; \frac{62}{8}$
The LCM of 8 and 2 is 8.
$\frac{25}{2} \;-\; \frac{62}{8} = \frac{25\times 4}{2 \times 4} \;-\; \frac{62 \times 1}{8 \times 1}$
$= \frac{100}{8} \;-\; \frac{62}{8}$
$= \frac{38}{8}$
$= \frac{19}{4}$
The length of the remaining rope is $\frac{19}{4}$ ft
4. Divide the rational numbers $\frac{2}{5} \div \frac{3}{6}$.
Solution:
Reciprocal of the divisor $\frac{3}{6}$ is $\frac{6}{3}$.
Multiply the dividend by the reciprocal of the divisor.
$\frac{2}{5} \div \frac{3}{6} = \frac{2}{5} \times \frac{6}{3}$
$= \frac{2 \times 6}{5 \times 3}$
$= \frac{12}{15}$
$= \frac{4}{5}$
Thus, $\frac{2}{5} \div \frac{3}{6} = \frac{4}{5}$
5. Find the multiplicative inverse of $\frac{2}{3} + \frac{3}{2}$.
Solution:
To find the reciprocal, we need to simplify the expression first.
$\frac{2}{3} + \frac{3}{2} = \frac{2\times 2}{3\times 2} + \frac{3 \times 3}{2\times 3}$
$ = \frac{4}{6} +\frac{9}{6}$
$ = \frac{13}{6}$
Thus, the multiplicative inverse of $\frac{13}{6}$ is $\frac{6}{13}$.
Practice Problems on Operations on Rational Numbers
Operations on Rational Numbers - Methods, Steps, Facts, Examples
Rational numbers are closed under the operations of _______.
Rational numbers are numbers that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
$\frac{1}{2} + \frac{2}{4} =$ _______.
$\frac{1}{2} + \frac{2}{4} = \frac{1 \times 2}{2 \times 2} + \frac{2}{4} = \frac{2}{4} + \frac{2}{4} = \frac{4}{4} = 1$
The value of $\frac{1}{4} \div \frac{2}{5}$ is _______.
$\frac{1}{4} \div 25= \frac{1}{4} \times \frac{5}{2} = \frac{5}{8}$
The commutative property of rational numbers is not applicable for the _______.
Subtraction and division of rational numbers is not commutative.
According to distributive property, $\frac{1}{3} \times (\frac{1}{4} + \frac{2}{5}) =$ _______.
According to the distributive property,
$A \times (B + C) = (A \times B) + (A \times C)$
Thus, $\frac{1}{3} \times (\frac{1}{4} + \frac{2}{5}) = (\frac{1}{3} \times \frac{1}{4}) + (\frac{1}{3} \times \frac{2}{5})$
Frequently Asked Questions on Operations on Rational Numbers
How to find the decimal form of a rational number?
Divide the numerator by the denominator using the long division method.
$\frac{5}{8} = 5\div8 = 0.625$
What are irrational numbers?
Irrational numbers are the type of real numbers that cannot be expressed in the rational form $\frac{p}{q}$, where p, q are integers and $q \neq 0$. In simple words, all the real numbers that are not rational numbers are irrational.
How many irrational numbers are between two rational numbers?
There are infinite irrational numbers between any two rational numbers.
Can the sum of two irrational numbers be a rational number?
Whenever you add or subtract two irrational numbers, you may get a rational number or an irrational number.Example: $(2 + \sqrt{5}) + (3\;-\;\sqrt{5}) = 5$
What is the sum of a rational number and an irrational number?
Adding a rational number and an irrational number always results in an irrational number.
