CBSE Class 8 Maths: Rational Numbers Exercise 1.2 – Number Line & Between Two Numbers

Understanding Rational Numbers and Their Properties

Before we jump into Exercise 1.2, let's quickly recap what rational numbers are. 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 them as special kinds of fractions! For example, 1/2, -3/4, 5 (which can be written as 5/1), and 0 (which can be written as 0/1) are all rational numbers. All natural numbers, whole numbers, and integers are also rational numbers because they can be written with a denominator of 1.

The number line, which you've used for integers, can be extended to include rational numbers. Unlike integers, which have distinct gaps, rational numbers are densely packed. This means between any two rational numbers, no matter how close, there are infinitely many other rational numbers. This property is crucial for understanding how to find rational numbers between two given numbers, a key part of this exercise.

How to Represent Rational Numbers on a Number Line

1. Step 1: Identify the Type of Rational Number — First, determine if the rational number is positive or negative, and whether it's a proper fraction (numerator < denominator) or an improper fraction (numerator >= denominator). Proper Fractions (e.g., 3/5, -2/7): These lie between 0 and 1 (if positive) or 0 and -1 (if negative). Improper Fractions (e.g., 7/4, -9/2): Convert them into mixed fractions first (e.g., 7/4 = 1 3/4). This tells you between which two consecutive integers the number lies (e.g., 1 3/4 is between 1 and 2).
2. Step 2: Draw the Number Line — Draw a straight line and mark the integer points (..., -2, -1, 0, 1, 2, ...). Make sure to include the integers between which your rational number lies. For example, if you need to plot 1 3/4, ensure 1 and 2 are clearly marked.
3. Step 3: Divide the Relevant Segment — Look at the denominator of your rational number. This tells you into how many equal parts you need to divide the segment between the two integers you identified in Step 1. For example, to represent 3/4, divide the segment between 0 and 1 into 4 equal parts. If representing 1 3/4, divide the segment between 1 and 2 into 4 equal parts.
4. Step 4: Count and Mark the Point — Starting from the integer point closer to 0 (e.g., 0 for 3/4, or 1 for 1 3/4), count the number of parts indicated by the numerator. Move to the right for positive numbers and to the left for negative numbers. Mark this final point with a clear dot and label it with the rational number.

Finding Rational Numbers Between Two Given Rational Numbers

• Example 1: Find 5 rational numbers between 1/4 and 1/2. Method 1: Common Denominator (and increasing space) Step 1: Find the LCM of the denominators (4 and 2), which is 4. Step 2: Convert both fractions to equivalent fractions with denominator 4: 1/4 (already has denominator 4) 1/2 = (1 2) / (2 2) = 2/4 Step 3: We need to find 5 numbers between 1/4 and 2/4. There's not enough 'space' yet. Multiply the numerator and denominator of both fractions by a suitable number, say 10 (or any number greater than the number of rational numbers required + 1, e.g., 5+1=6). 1/4 = (1 10) / (4 10) = 10/40 2/4 = (2 10) / (4 10) = 20/40 Step 4: Now, we can easily pick 5 rational numbers between 10/40 and 20/40. These could be: 11/40, 12/40, 13/40, 14/40, 15/40 Final Answer: Five rational numbers between 1/4 and 1/2 are 11/40, 12/40, 13/40, 14/40, and 15/40. (Many other answers are possible.)
• Example 2: Find a rational number between 1/3 and 1/5 using the Mean Method. Method 2: Mean Method Step 1: The mean (average) of two rational numbers is always a rational number that lies between them. Step 2: Add the two given rational numbers and divide by 2. (1/3 + 1/5) / 2 Step 3: Find a common denominator for 1/3 and 1/5, which is 15. (5/15 + 3/15) / 2 Step 4: Add the numerators: (8/15) / 2 Step 5: Divide by 2 (which is the same as multiplying by 1/2): (8/15) * (1/2) = 8/30 Step 6: Simplify the fraction: 8/30 = 4/15 Final Answer: A rational number between 1/3 and 1/5 is 4/15.

YoLearn's Exam Tip: Avoid Common Mistakes!

When working with rational numbers, especially on number lines or finding numbers between them, students often make a few common errors. Be careful:

1. Sign Errors: Always pay close attention to whether a rational number is positive or negative. This determines whether you move to the right (positive) or left (negative) from zero on the number line. For example, -3/4 is very different from 3/4!
2. Improper Fractions: Don't forget to convert improper fractions to mixed fractions (e.g., 9/2 = 4 1/2) before trying to locate them on the number line. This makes it much easier to identify the correct integer segment.
3. Insufficient 'Space': When finding rational numbers between two given numbers, if the initial common denominator doesn't give you enough integer steps in the numerator, multiply both the numerator and denominator by a larger factor (like 10, 20, or even 100) to create more 'space'.
4. Simplification: Always simplify your final rational number to its lowest terms unless otherwise specified. For instance, 12/40 can be simplified to 3/10.

Practice Questions with Solutions

• Q: Represent 7/4 on the number line. A: Step 1: Convert the improper fraction to a mixed fraction: 7/4 = 1 and 3/4. This means the number lies between 1 and 2. Step 2: Draw a number line and mark integers 0, 1, 2. Step 3: Divide the segment between 1 and 2 into 4 equal parts (since the denominator is 4). Step 4: Count 3 parts from 1 towards 2. Mark this point. This point represents 7/4. Final answer: The point marked at 1 and 3/4 between 1 and 2 represents 7/4.
• Q: Represent -5/6 on the number line. A: Step 1: Recognize that -5/6 is a proper fraction and negative, so it lies between -1 and 0. Step 2: Draw a number line and mark integers 0 and -1. Step 3: Divide the segment between 0 and -1 into 6 equal parts (since the denominator is 6). Step 4: Count 5 parts from 0 towards -1. Mark this point. This point represents -5/6. Final answer: The point marked at 5/6 distance from 0 towards -1 represents -5/6.
• Q: Find three rational numbers between -2 and 0. A: Step 1: Write the integers as rational numbers with a common denominator. We can write -2 as -20/10 and 0 as 0/10. Step 2: Now we need to find three rational numbers between -20/10 and 0/10. Step 3: Choose any three numerators between -20 and 0, maintaining the denominator. For example, -19, -10, -1. Step 4: The rational numbers are -19/10, -10/10 (which is -1), and -1/10. Final answer: Three rational numbers between -2 and 0 are -19/10, -1, and -1/10. (Other valid answers exist).
• Q: Find five rational numbers between 2/3 and 4/5. A: Step 1: Find the LCM of the denominators (3 and 5), which is 15. Step 2: Convert both fractions to equivalent fractions with denominator 15: 2/3 = (2 5) / (3 5) = 10/15 4/5 = (4 3) / (5 3) = 12/15 Step 3: Currently, there's only 11/15 between 10/15 and 12/15. We need more space. Multiply both fractions' numerators and denominators by a larger number, say 10: 10/15 = (10 10) / (15 10) = 100/150 12/15 = (12 10) / (15 10) = 120/150 Step 4: Now, we can easily find five rational numbers between 100/150 and 120/150. Step 5: For example, 101/150, 105/150, 110/150, 115/150, 119/150. Final answer: Five rational numbers between 2/3 and 4/5 are 101/150, 105/150, 110/150, 115/150, and 119/150.

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