CBSE Class 7 Maths: Rational Numbers Ex 9.1 Explained

What Exactly Are Rational Numbers?

A rational number is any number that can be expressed in the form p/q, where 'p' and 'q' are integers, and 'q' is not equal to zero. Think of it as a fancy way to include both fractions and integers under one big umbrella!

Let's break down the definition:

1. p and q are integers: This means 'p' (the numerator) and 'q' (the denominator) can be positive whole numbers (1, 2, 3...), negative whole numbers (-1, -2, -3...), or zero (but only 'p' can be zero, not 'q').
2. q ≠ 0: The denominator cannot be zero. Why? Because division by zero is undefined in mathematics. Imagine trying to divide a pizza into zero slices – it just doesn't make sense!

How are they different from fractions?
Fractions (like 1/2, 3/4) always have a positive whole number as both numerator and denominator. Rational numbers are more general; their numerators and denominators can be negative integers too. For example, -3/5 is a rational number, but not a fraction. Also, all integers (like 5, -7, 0) are rational numbers because they can be written as 5/1, -7/1, or 0/1. This makes the set of rational numbers much broader than fractions or integers alone. They help us describe quantities like temperature below zero, debts, or precise measurements.

Identifying Rational Numbers: Quick Check!

• Example 1: Is 7 a rational number? Yes, because 7 can be written as 7/1. Here, p=7 and q=1. Both are integers, and q is not zero.
• Example 2: Is -2/5 a rational number? Yes, because p=-2 and q=5. Both -2 and 5 are integers, and 5 is not zero.
• Example 3: Is 0 a rational number? Yes, because 0 can be written as 0/1, 0/2, 0/-3, etc. Here, p=0 (an integer) and q is any non-zero integer. So, 0 is a rational number.
• Example 4: Is 3/0 a rational number? No, because the denominator 'q' is 0. Division by zero is undefined, so 3/0 is not a rational number.

Finding Equivalent Rational Numbers

1. Understanding Equivalence — Equivalent rational numbers represent the same value, even though they look different. Just like 1/2 is the same as 2/4 or 5/10. Think of it as cutting a cake into more slices, but still having the same amount.
2. Method 1: Multiplication — To find an equivalent rational number, multiply both the numerator (p) and the denominator (q) by the same non-zero integer. Example: Find three equivalent rational numbers for 3/4. Multiply by 2: (3 2) / (4 2) = 6/8 Multiply by 3: (3 3) / (4 3) = 9/12 Multiply by -5: (3 -5) / (4 * -5) = -15/-20 (which is also 15/20)
3. Method 2: Division (Simplifying to Standard Form) — If the numerator and denominator have common factors (other than 1), you can divide both by that common factor to get an equivalent rational number. This process is often used to reduce a rational number to its standard form (or simplest form), where the only common factor between p and q is 1, and the denominator is always positive. Example: Express -18/24 in standard form. Identify common factors: Both 18 and 24 are divisible by 2, 3, 6. Divide by the greatest common factor (GCF), which is 6: (-18 ÷ 6) / (24 ÷ 6) = -3/4. * Since -3 and 4 have no common factors other than 1, and the denominator is positive, -3/4 is the standard form.

Comparing Rational Numbers: Which is Greater?

1. Step 1: Ensure Positive Denominators — If any rational number has a negative denominator, convert it to an equivalent rational number with a positive denominator. For example, 2/-3 becomes -2/3. (Multiply numerator and denominator by -1).
2. Step 2: Find the LCM of Denominators — Find the Least Common Multiple (LCM) of the denominators of the rational numbers you want to compare. This helps us get a common base for comparison.
3. Step 3: Convert to Equivalent Rational Numbers — Convert each rational number into an equivalent rational number with the LCM as its new denominator. To do this, multiply the numerator and denominator by the factor needed to change the original denominator into the LCM.
4. Step 4: Compare the Numerators — Once both rational numbers have the same positive denominator, simply compare their numerators. The rational number with the larger numerator is the greater one. Example: Compare 2/3 and 3/5. 1. Denominators (3 and 5) are already positive. 2. LCM of 3 and 5 is 15. 3. Convert: 2/3 = (25)/(35) = 10/15. And 3/5 = (33)/(53) = 9/15. 4. Compare numerators: 10 > 9. So, 10/15 > 9/15, which means 2/3 > 3/5.
5. Special Case: Comparing Negative Rational Numbers — When comparing negative rational numbers, remember that the number closer to zero on the number line is greater. For example, -1/2 is greater than -3/4. After converting to common denominators, say -2/4 and -3/4, since -2 > -3, then -2/4 > -3/4. Visualizing on a number line helps a lot!

Exam Tip: Handling Negative Rational Numbers Carefully!

One of the most common mistakes students make when working with rational numbers, especially in exams, is mismanaging negative signs. Always remember these points:

1. Standard Form: A rational number is in standard form if its denominator is positive. If you have 5/-7, always rewrite it as -5/7 by multiplying both numerator and denominator by -1. This makes comparisons and calculations much easier and less prone to errors.
2. Number Line for Comparison: When comparing two negative rational numbers, visualize them on a number line. The number that is to the right is always greater. For example, -1/2 vs -1/3. On a number line, -1/3 (which is about -0.33) is to the right of -1/2 (which is -0.5). So, -1/3 > -1/2. Don't just look at the absolute value; the sign matters a lot!
3. Operations with Negatives: When performing addition, subtraction, multiplication, or division, strictly follow the rules for integer operations (e.g., negative times negative is positive, etc.). A small sign error can lead to a completely wrong answer.

Practice Questions with Solutions

• Q: 1. Identify which of the following are rational numbers: (a) -5, (b) 4/9, (c) 7/0, (d) 0.
• A: Step 1: Recall the definition of a rational number: p/q where p, q are integers and q ≠ 0. Step 2: Check each option. (a) -5: Can be written as -5/1. Here p=-5, q=1. Both are integers, q≠0. So, -5 is a rational number. (b) 4/9: Here p=4, q=9. Both are integers, q≠0. So, 4/9 is a rational number. (c) 7/0: Here q=0. Division by zero is undefined. So, 7/0 is not a rational number. (d) 0: Can be written as 0/1. Here p=0, q=1. Both are integers, q≠0. So, 0 is a rational number. Final answer: -5, 4/9, and 0 are rational numbers.
• Q: 2. Write three equivalent rational numbers for -2/7.
• A: Step 1: To find equivalent rational numbers, multiply both the numerator and the denominator by the same non-zero integer. Step 2: Multiply by 2: (-2 2) / (7 2) = -4/14. Step 3: Multiply by 3: (-2 3) / (7 3) = -6/21. Step 4: Multiply by -1: (-2 -1) / (7 -1) = 2/-7 (which is also -2/7, as standard form requires a positive denominator). Final answer: Three equivalent rational numbers for -2/7 are -4/14, -6/21, and 2/-7 (or -2/7 itself).
• Q: 3. Express the rational number 15/-25 in standard form.
• A: Step 1: First, ensure the denominator is positive. Multiply numerator and denominator by -1: (15 -1) / (-25 -1) = -15/25. Step 2: Find the greatest common factor (GCF) of the absolute values of the numerator (15) and the denominator (25). The common factors are 1, 5. The GCF is 5. Step 3: Divide both the numerator and the denominator by their GCF: (-15 ÷ 5) / (25 ÷ 5) = -3/5. Final answer: The standard form of 15/-25 is -3/5.
• Q: 4. Compare the rational numbers -4/5 and -3/4. Which one is greater?
• A: Step 1: Ensure denominators are positive. Both 5 and 4 are positive. Step 2: Find the LCM of the denominators (5 and 4). The LCM is 20. Step 3: Convert both rational numbers to equivalent fractions with denominator 20. For -4/5: (-4 4) / (5 4) = -16/20. For -3/4: (-3 5) / (4 5) = -15/20. Step 4: Compare the numerators: -16 and -15. On a number line, -15 is to the right of -16, so -15 > -16. Final answer: Since -15/20 > -16/20, then -3/4 > -4/5. So, -3/4 is greater.

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