Class 7 — Mathematics (NCERT)
Rational Numbers
Class 7
- ✓By the end of this lesson students will be able to define rational numbers and identify them.
- ✓By the end of this lesson students will be able to represent rational numbers on a number line.
- ✓By the end of this lesson students will be able to understand and find equivalent rational numbers.
- ✓By the end of this lesson students will be able to perform addition, subtraction, multiplication, and division on rational numbers.
- ✓By the end of this lesson students will be able to compare and order rational numbers.
Key concepts
A rational number is a number that can be expressed in the form p/q, where p and q are integers and q is not equal to zero. All integers and fractions are rational numbers. For example, 1/2, -3/4, 5 (which can be written as 5/1), and 0 (which can be written as 0/1) are rational numbers.
A rational number is said to be positive if both its numerator and denominator are either positive or both are negative. For example, 2/3 and -4/-5 are positive rational numbers. A rational number is said to be negative if one of its numerator or denominator is positive and the other is negative. For example, -2/3 and 4/-5 are negative rational numbers.
Rational numbers are said to be equivalent if they represent the same value. We can obtain equivalent rational numbers by multiplying or dividing both the numerator and the denominator of a rational number by the same non-zero integer. For example, 1/2, 2/4, 3/6, and -5/-10 are all equivalent rational numbers.
To represent a rational number p/q on a number line, we divide the unit length between two consecutive integers into 'q' equal parts. Then, we mark the 'p'th part from the starting integer. For positive rational numbers, we move to the right of 0, and for negative rational numbers, we move to the left of 0.
A rational number p/q is said to be in its standard form if its denominator q is a positive integer and the numerator p and the denominator q have no common factor other than 1 (i.e., their H.C.F. is 1). For example, 3/5 is in standard form, but 6/10 is not (as H.C.F. of 6 and 10 is 2). To reduce to standard form, divide both numerator and denominator by their H.C.F.
To compare two rational numbers, we first convert them into equivalent rational numbers with the same positive denominator (usually the L.C.M. of the denominators). Then, we compare their numerators. The rational number with the greater numerator is the greater rational number.
If the denominators are the same, we add the numerators and keep the denominator common: (p/q) + (r/q) = (p+r)/q. If the denominators are different, we find the L.C.M. of the denominators, convert the rational numbers to equivalent rational numbers with the L.C.M. as the common denominator, and then add their numerators.
If the denominators are the same, we subtract the numerators and keep the denominator common: (p/q) - (r/q) = (p-r)/q. If the denominators are different, we find the L.C.M. of the denominators, convert the rational numbers to equivalent rational numbers with the L.C.M. as the common denominator, and then subtract their numerators.
To multiply two rational numbers, we multiply their numerators and multiply their denominators.
To divide a rational number by another non-zero rational number, we multiply the first rational number by the reciprocal of the second rational number.
Key facts to remember
- 1A rational number can be written as p/q, where p and q are integers and q ≠ 0.
- 2Every integer and every fraction is a rational number.
- 3Zero is a rational number (e.g., 0/1).
- 4There are infinitely many rational numbers between any two given rational numbers.
- 5Equivalent rational numbers are obtained by multiplying or dividing both numerator and denominator by the same non-zero integer.
- 6A rational number is in standard form if its denominator is positive and the H.C.F. of its numerator and denominator is 1.
- 7To add or subtract rational numbers with different denominators, find their L.C.M. and convert to equivalent rational numbers.
- 8To divide rational numbers, multiply the first rational number by the reciprocal of the second.
Worked examples
Example 1
Represent -3/4 on a number line.
Answer
The point representing -3/4 is located at the third division to the left of 0, when the unit length between 0 and -1 is divided into 4 equal parts.
Always ensure the denominator is positive before representing on a number line.
Example 2
Simplify: (2/3) + (-5/6)
Answer
-1/6
Remember to simplify the final answer to its standard form.
Example 3
Find the value of (-4/5) ÷ (3/7).
Answer
-28/15
Ensure the divisor is not zero. Always find the reciprocal of the second rational number.
Common mistakes
- ✗Forgetting that the denominator 'q' in p/q cannot be zero.
- ✗Incorrectly finding the L.C.M. of denominators when adding or subtracting rational numbers.
- ✗Confusing the operation of division with multiplication by simply dividing numerators and denominators.
- ✗Not reducing rational numbers to their standard form when required.
- ✗Making errors in applying sign rules (e.g., multiplying/dividing negative numbers).
Exam tips
- ★Always simplify your final answer to its standard form unless specified otherwise.
- ★Pay careful attention to the signs of rational numbers during all operations.
- ★When comparing or adding/subtracting rational numbers, ensure their denominators are positive.
- ★Practice representing both positive and negative rational numbers on a number line accurately.
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