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Rational Numbers – CBSE NCERT Study Resources

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7th

7th - Mathematics

Rational Numbers

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Overview of the Chapter: Rational Numbers

This chapter introduces students to the concept of Rational Numbers, their properties, and operations. Rational numbers are an essential part of the number system, bridging the gap between integers and fractions. The chapter covers definitions, representations, and arithmetic operations involving rational numbers.

Rational Number: A number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.

Key Topics Covered

  • Definition and examples of rational numbers
  • Positive and negative rational numbers
  • Standard form of rational numbers
  • Comparison of rational numbers
  • Operations on rational numbers (addition, subtraction, multiplication, division)
  • Representation of rational numbers on the number line

Detailed Explanation

1. Definition of Rational Numbers

Rational numbers include all integers, fractions, and decimals that terminate or repeat. They can be positive, negative, or zero.

Example: 3/4, -5/2, 0, 6, -2.75 are all rational numbers.

2. Positive and Negative Rational Numbers

A rational number is positive if both numerator and denominator have the same sign. It is negative if they have opposite signs.

Example: 2/3 (positive), -4/5 (negative).

3. Standard Form of Rational Numbers

A rational number is in its standard form when its denominator is a positive integer and the numerator and denominator have no common factors other than 1.

Example: The standard form of 6/-8 is -3/4.

4. Comparison of Rational Numbers

Rational numbers can be compared by cross-multiplying or converting them to a common denominator.

Example: To compare 3/4 and 5/6, find a common denominator (12): 9/12 < 10/12.

5. Operations on Rational Numbers

Addition and Subtraction

Rational numbers are added or subtracted by finding a common denominator and then performing the operation on the numerators.

Example: 1/2 + 1/3 = (3 + 2)/6 = 5/6.

Multiplication and Division

Multiply numerators and denominators directly for multiplication. For division, multiply by the reciprocal of the divisor.

Example: (2/3) × (4/5) = 8/15; (2/3) ÷ (4/5) = (2/3) × (5/4) = 10/12 = 5/6.

6. Representation on the Number Line

Rational numbers can be plotted on the number line by dividing the space between integers into equal parts based on the denominator.

Example: To represent 3/4, divide the space between 0 and 1 into 4 equal parts and mark the third point.

Summary

This chapter provides a foundational understanding of rational numbers, their properties, and operations. Mastery of these concepts is crucial for advanced topics in mathematics.

All Question Types with Solutions – CBSE Exam Pattern

Explore a complete set of CBSE-style questions with detailed solutions, categorized by marks and question types. Ideal for exam preparation, revision and practice.

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

These are 1-mark questions requiring direct, concise answers. Ideal for quick recall and concept clarity.

Question 1:
What is the additive inverse of 5/7?
Answer:
-5/7
Question 2:
Find the reciprocal of -3/4.
Answer:
-4/3
Question 3:
Simplify: 2/3 + (-4/5).
Answer:
-2/15
Question 4:
Is 0 a rational number? Justify.
Answer:

Yes, as 0 = 0/1.

Question 5:
Write a rational number between 1/2 and 3/4.
Answer:
5/8
Question 6:
What is the multiplicative inverse of -1?
Answer:
-1
Question 7:
Solve: (7/8) × (-16/21).
Answer:
-2/3
Question 8:
Express -4/5 as a rational number with denominator 15.
Answer:
-12/15
Question 9:
Find the sum of 3/7 and its additive inverse.
Answer:
0
Question 10:
Which is greater: -5/6 or -3/4?
Answer:
-3/4
Question 11:
Subtract 1/6 from -1/2.
Answer:
-2/3
Question 12:
Divide: (-9/10) ÷ (3/5).
Answer:
-3/2
Question 13:
What is the additive inverse of 5/7?
Answer:

The additive inverse of 5/7 is -5/7 because their sum is zero.

Question 14:
Find a rational number between 1/4 and 1/2.
Answer:

One rational number between 1/4 and 1/2 is 3/8.
Calculation:
(1/4 + 1/2) ÷ 2 = (2/8 + 4/8) ÷ 2 = (6/8) ÷ 2 = 3/8.

Question 15:
Is 0 a rational number? Justify.
Answer:

Yes, 0 is a rational number because it can be written as 0/1, where both numerator and denominator are integers and denominator ≠ 0.

Question 16:
Express -3/5 as a rational number with denominator 20.
Answer:

To express -3/5 with denominator 20:
-3/5 × 4/4 = -12/20.

Question 17:
What is the reciprocal of -8/9?
Answer:

The reciprocal of -8/9 is -9/8 because their product is 1.

Question 18:
Simplify: (3/4) + (-5/6).
Answer:

Simplifying (3/4) + (-5/6):
Find LCM of 4 and 6 = 12.
Convert to like fractions: 9/12 + (-10/12) = -1/12.

Question 19:
Which property is illustrated by (2/3) × (4/5) = (4/5) × (2/3)?
Answer:

This illustrates the commutative property of multiplication for rational numbers.

Question 20:
Find the value of (-7/12) ÷ (14/-15).
Answer:

Solving (-7/12) ÷ (14/-15):
Reciprocal of divisor: -15/14.
Multiply: (-7/12) × (-15/14) = 105/168 = 5/8 (simplified).

Question 21:
Write the standard form of 18/-24.
Answer:

The standard form of 18/-24 is -3/4.
Divide numerator and denominator by their HCF (6).

Question 22:
Verify: 1/2 + (1/3 + 1/4) = (1/2 + 1/3) + 1/4.
Answer:

Verification:
Left side: 1/2 + (7/12) = 13/12.
Right side: (5/6) + 1/4 = 13/12.
Both sides are equal, confirming the associative property of addition.

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

These 2-mark questions test key concepts in a brief format. Answers are expected to be accurate and slightly descriptive.

Question 1:
What is the additive inverse of -5/7?
Answer:

The additive inverse of -5/7 is 5/7 because their sum is zero.
-5/7 + 5/7 = 0.

Question 2:
Find the reciprocal of 3/4.
Answer:

The reciprocal of 3/4 is 4/3 because multiplying them gives 1.
3/4 × 4/3 = 1.

Question 3:
Is 0 a rational number? Justify your answer.
Answer:

Yes, 0 is a rational number because it can be expressed as 0/1, where the denominator is not zero.

Question 4:
Simplify: (-2/3) + (4/5).
Answer:

First, find the LCM of denominators (3 and 5), which is 15.
Convert fractions: (-2/3) = -10/15, (4/5) = 12/15.
Now, add them: -10/15 + 12/15 = 2/15.

Question 5:
What is the multiplicative identity for rational numbers?
Answer:

The multiplicative identity for rational numbers is 1 because any number multiplied by 1 remains unchanged.
Example: 5/7 × 1 = 5/7.

Question 6:
Express -8/11 as a rational number with denominator 22.
Answer:

Multiply numerator and denominator by 2 to get denominator 22.
-8/11 = (-8 × 2)/(11 × 2) = -16/22.

Question 7:
Find the sum of 7/12 and its additive inverse.
Answer:

The additive inverse of 7/12 is -7/12.
Sum: 7/12 + (-7/12) = 0.

Question 8:
Which rational number has no reciprocal?
Answer:

0 has no reciprocal because division by zero is undefined.

Question 9:
Subtract 1/6 from -1/2.
Answer:

First, convert fractions to common denominator (6).
-1/2 = -3/6.
Now, subtract: -3/6 - 1/6 = -4/6 = -2/3 (simplified).

Question 10:
Are all integers rational numbers? Give an example.
Answer:

Yes, all integers are rational numbers because they can be written as fractions with denominator 1.
Example: 5 = 5/1.

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

These 3-mark questions require brief explanations and help assess understanding and application of concepts.

Question 1:
Explain why every integer is a rational number with an example.
Answer:

Every integer is a rational number because it can be expressed as a fraction with denominator 1.
For example, the integer 5 can be written as 5/1, which satisfies the definition of a rational number (a number of the form p/q, where p and q are integers and q ≠ 0).

Question 2:
Find three rational numbers between 1/4 and 1/2.
Answer:

To find three rational numbers between 1/4 and 1/2, we can convert them to equivalent fractions with a common denominator.
Step 1: Convert 1/4 and 1/2 to have denominator 8.
1/4 = 2/8
1/2 = 4/8
Step 2: Now, the numbers between 2/8 and 4/8 are 3/8, 5/16, and 7/16 (by further conversion).

Question 3:
Simplify: (3/5) + (-4/7) - (2/3).
Answer:

Step 1: Find the LCM of denominators 5, 7, and 3, which is 105.
Step 2: Convert each fraction:
3/5 = 63/105
-4/7 = -60/105
-2/3 = -70/105
Step 3: Perform the operations:
63/105 - 60/105 - 70/105 = (63 - 60 - 70)/105 = -67/105.

Question 4:
Represent the rational number -7/4 on a number line.
Answer:

To represent -7/4 on a number line:
Step 1: Convert -7/4 to a mixed number: -1 3/4.
Step 2: Locate -2 and -1 on the number line.
Step 3: Divide the segment between -2 and -1 into 4 equal parts.
Step 4: Mark the point 3/4 units to the left of -1, which is -7/4.

Question 5:
Verify whether the rational numbers 15/-20 and -9/12 are equal.
Answer:

To verify equality, simplify both fractions:
Step 1: Simplify 15/-20:
Divide numerator and denominator by 5: -3/4.
Step 2: Simplify -9/12:
Divide numerator and denominator by 3: -3/4.
Since both simplify to -3/4, they are equal.

Question 6:
Define rational numbers and give two examples.
Answer:

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.


Examples:
1. 3/4 (where p = 3, q = 4)
2. -5/2 (where p = -5, q = 2)
Question 7:
Compare the rational numbers 2/3 and 4/5 using the concept of equivalent fractions.
Answer:

To compare 2/3 and 4/5, we find their equivalent fractions with the same denominator (LCM of 3 and 5 is 15).


Step 1: Convert 2/3 to denominator 15.
2/3 = (2 × 5)/(3 × 5) = 10/15
Step 2: Convert 4/5 to denominator 15.
4/5 = (4 × 3)/(5 × 3) = 12/15
Step 3: Compare numerators.
Since 10 < 12, 2/3 < 4/5
Question 8:
Find three rational numbers between -1/2 and 1/2.
Answer:

To find rational numbers between -1/2 and 1/2, we can take their average or convert them to higher equivalent fractions.


Method: Convert to denominator 8.
-1/2 = -4/8
1/2 = 4/8
Three rational numbers between them are:
1. -3/8
2. 0 (which is 0/8)
3. 2/8 (simplifies to 1/4)
Question 9:
Explain why the sum of two rational numbers is always a rational number with an example.
Answer:

The sum of two rational numbers is always rational because when we add two fractions a/b and c/d, the result is (ad + bc)/bd, which is still in the form of p/q (where p and q are integers and q ≠ 0).


Example:
Let’s add 1/2 and 1/3:
1/2 + 1/3 = (3 + 2)/6 = 5/6
Here, 5 and 6 are integers, and 6 ≠ 0, so 5/6 is rational.
Question 10:
Represent the rational number -7/4 on a number line and describe the steps.
Answer:

To represent -7/4 on a number line:


Step 1: Draw a number line with 0 at the center.
Step 2: Mark points to the left for negative numbers.
Step 3: Since -7/4 = -1.75, it lies between -2 and -1.
Step 4: Divide the segment between -2 and -1 into 4 equal parts.
Step 5: Count 3 parts from -1 towards -2 (as -7/4 is -1 and -3/4).
Result: The point representing -7/4 is 3/4 units to the left of -1.

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

These 5-mark questions are descriptive and require detailed, structured answers with proper explanation and examples.

Question 1:
Explain how to add two rational numbers with different denominators using the LCM method. Give an example from our textbook.
Answer:
Introduction

We studied that rational numbers can be added by finding a common denominator. The LCM method helps simplify this process.


Argument 1
  • First, find the LCM of the denominators.
  • Convert both fractions to equivalent fractions with the LCM as the denominator.

Argument 2

For example, to add 1/4 and 2/3, the LCM of 4 and 3 is 12. We rewrite them as 3/12 and 8/12, then add to get 11/12.


Conclusion

This method ensures accuracy and is used in real-life measurements like splitting ingredients in recipes.

Question 2:
Describe the commutative property of rational numbers under addition with an example. How is it useful in daily life?
Answer:
Introduction

We learned that the commutative property states that changing the order of numbers does not affect the result.


Argument 1
  • For rational numbers, a/b + c/d = c/d + a/b.
  • Example: 2/5 + 1/3 = 1/3 + 2/5 = 11/15.

Argument 2

This property is useful in daily life, like splitting bills where the order of adding amounts doesn’t matter.


Conclusion

Our textbook shows this property simplifies calculations and ensures consistency in results.

Question 3:
Compare the multiplicative inverse of two rational numbers using an example. Why is it important?
Answer:
Introduction

The multiplicative inverse of a rational number a/b is b/a, such that their product is 1.


Argument 1
  • For 3/4, the inverse is 4/3 because (3/4) × (4/3) = 1.
  • For -5/7, the inverse is -7/5.

Argument 2

This concept is vital in solving equations and dividing fractions, like scaling recipes in cooking.


Conclusion

Our textbook demonstrates its role in simplifying complex problems efficiently.

Question 4:
Explain how to add two rational numbers with different denominators using an example from our textbook.
Answer:
Introduction

We studied that rational numbers can be added by finding a common denominator. Our textbook shows an example of adding 1/2 and 1/3.


Argument 1
  • First, find the LCM of denominators 2 and 3, which is 6.
  • Convert both fractions: 1/2 becomes 3/6, and 1/3 becomes 2/6.

Argument 2

Now, add the numerators: 3/6 + 2/6 = 5/6. This is the simplest form.


Conclusion

Thus, adding rational numbers with different denominators requires a common denominator, as shown in NCERT examples.

Question 5:
Describe the commutative property of rational numbers under addition with a real-life example.
Answer:
Introduction

We learned that the commutative property means changing the order of numbers does not affect the result. For rational numbers, a + b = b + a.


Argument 1
  • Example: 2/5 + 3/5 = 5/5 = 1, and 3/5 + 2/5 also equals 1.

Argument 2

In real life, if you add 1/2 kg of sugar to 1/4 kg of flour, the total is the same as adding 1/4 kg flour to 1/2 kg sugar.


Conclusion

This property simplifies calculations, as shown in NCERT exercises.

Question 6:
Compare the rational numbers 4/7 and 5/8 using cross-multiplication. Show steps as per NCERT method.
Answer:
Introduction

To compare 4/7 and 5/8, we use cross-multiplication, as taught in our textbook.


Argument 1
  • Multiply numerator of first by denominator of second: 4 × 8 = 32.
  • Multiply numerator of second by denominator of first: 5 × 7 = 35.

Argument 2

Since 32 < 35, 4/7 < 5/8. This matches NCERT’s step-by-step method.


Conclusion

Cross-multiplication is a reliable way to compare rational numbers, as demonstrated in class.

Question 7:
Explain how to add two rational numbers with different denominators using the example 2/3 + (-5/4). Verify your answer.
Answer:
Introduction

We studied that rational numbers can be added by finding a common denominator. Our textbook shows this using LCM.


Argument 1
  • Find LCM of 3 and 4, which is 12.
  • Convert fractions: 2/3 = 8/12 and -5/4 = -15/12.

Argument 2

Now, add them: 8/12 + (-15/12) = -7/12. Verification: -7/12 lies between -1 and 0, which makes sense.


Conclusion

Thus, the sum is -7/12. This method ensures accuracy in calculations.

Question 8:
Show that rational numbers are closed under multiplication using the example (-3/5) × (4/7). Explain the steps.
Answer:
Introduction

Our textbook defines closure property: if two rational numbers are multiplied, the result is also rational.


Argument 1
  • Multiply numerators: -3 × 4 = -12.
  • Multiply denominators: 5 × 7 = 35.

Argument 2

The product is -12/35, which is a rational number. This confirms closure property.


Conclusion

Since the result is rational, the property holds true for all rational numbers.

Question 9:
A rope of length 7/4 meters is cut into 5 equal pieces. Find the length of each piece and express it as a rational number.
Answer:
Introduction

We learned that division of rational numbers is similar to fractions. Here, we divide 7/4 by 5.


Argument 1
  • Rewrite 5 as 5/1.
  • Multiply 7/4 by reciprocal of 5/1: 7/4 × 1/5 = 7/20.

Argument 2

Each piece is 7/20 meters long. This is a rational number in simplest form.


Conclusion

Thus, the length of each piece is 7/20 meters, proving division works for rational numbers.

Question 10:
Describe the commutative property of rational numbers under addition with real-life examples.
Answer:
Introduction

We learned that the commutative property states that changing the order of numbers does not affect the result.


Argument 1
  • For rational numbers, a + b = b + a.
  • Example: 2/3 + 1/4 = 1/4 + 2/3 = 11/12.

Argument 2

In real life, if you add 1.5 kg of apples to 0.5 kg of oranges, the total is the same as adding 0.5 kg of oranges to 1.5 kg of apples.


Conclusion

This property simplifies calculations and is useful in daily life.

Question 11:
Compare the multiplicative inverse of two rational numbers using NCERT examples.
Answer:
Introduction

The multiplicative inverse of a rational number a/b is b/a, where a ≠ 0 and b ≠ 0.


Argument 1
  • Example from NCERT: The inverse of 5/7 is 7/5.
  • For -3/4, it is -4/3.

Argument 2

When multiplied, a number and its inverse always give 1. For instance, (5/7) × (7/5) = 1.


Conclusion

Understanding inverses helps in solving equations and simplifying expressions.

Question 12:
Explain the concept of rational numbers with examples. How are they different from whole numbers and integers? Provide a real-life scenario where rational numbers are used.
Answer:

Rational numbers are numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Examples include 1/2, -3/4, and 5/1 (which is also an integer).


Difference from whole numbers and integers:

  • Whole numbers are non-negative integers (0, 1, 2, ...) and do not include fractions or negatives.
  • Integers include all whole numbers and their negatives (-1, -2, ...) but still exclude fractions.
  • Rational numbers include all integers and fractions, making them a broader category.


Real-life application: Measuring ingredients for a recipe, like using 1/2 cup of sugar or 3/4 teaspoon of salt, involves rational numbers.

Question 13:
Solve the following and simplify to the lowest form: (3/4) + (-5/6) + (7/12). Show each step clearly and explain the concept of lowest common denominator (LCD) used in the solution.
Answer:

Step 1: Find the LCD of denominators 4, 6, and 12.
The multiples of 4: 4, 8, 12, 16...
The multiples of 6: 6, 12, 18...
The multiples of 12: 12, 24...
LCD = 12 (smallest common multiple).


Step 2: Convert each fraction to have denominator 12.
3/4 = (3×3)/(4×3) = 9/12
-5/6 = (-5×2)/(6×2) = -10/12
7/12 remains 7/12.


Step 3: Add the fractions.
9/12 + (-10/12) + 7/12 = (9 - 10 + 7)/12 = 6/12.


Step 4: Simplify to lowest form.
6/12 ÷ 6/6 = 1/2.


Explanation of LCD: The lowest common denominator is the smallest number that all denominators divide into evenly, allowing fractions to be added or subtracted easily.

Question 14:
Explain the concept of rational numbers with examples. How are they different from integers? Provide a real-life scenario where rational numbers are used.
Answer:

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Examples include 1/2, -3/4, and 5/1 (which is also an integer).


Rational numbers differ from integers because integers are whole numbers (positive, negative, or zero) and do not include fractions or decimals. For example, 2 is an integer, but 2/3 is a rational number.


In real life, rational numbers are used in measurements, such as dividing a pizza into equal parts. If a pizza is cut into 8 slices and you eat 3 slices, you have consumed 3/8 of the pizza, which is a rational number.

Question 15:
Compare and contrast the properties of addition and multiplication of rational numbers. Verify these properties using the numbers 2/3 and -1/4.
Answer:

Properties of Addition:

  • Commutative Property: a + b = b + a
  • Associative Property: (a + b) + c = a + (b + c)
  • Additive Identity: a + 0 = a


Properties of Multiplication:

  • Commutative Property: a × b = b × a
  • Associative Property: (a × b) × c = a × (b × c)
  • Multiplicative Identity: a × 1 = a


Verification:
For 2/3 and -1/4:
Addition: 2/3 + (-1/4) = 5/12 and -1/4 + 2/3 = 5/12 (Commutative verified).
Multiplication: 2/3 × (-1/4) = -2/12 = -1/6 and -1/4 × 2/3 = -2/12 = -1/6 (Commutative verified).

Question 16:
Explain the concept of rational numbers with examples. How are they different from integers? Provide a real-life situation where rational numbers are used.
Answer:

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Examples include 1/2, -3/4, and 5/1 (which is also an integer).


Rational numbers differ from integers because integers are whole numbers (positive, negative, or zero) and do not include fractions or decimals. For example, 2 is an integer, but 2/3 is a rational number.


A real-life application of rational numbers is in measuring ingredients for cooking. For instance, if a recipe requires 3/4 cup of flour, this is a rational number representing a precise quantity.

Question 17:
Explain the concept of rational numbers with examples. How are they different from integers? Discuss their representation on the number line.
Answer:

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Examples include 1/2, -3/4, and 5/1 (which is also an integer).

Difference from integers: While all integers are rational numbers (since they can be written as p/1), not all rational numbers are integers. For example, 3/2 is rational but not an integer.

Representation on the number line:
1. Divide the distance between two integers into equal parts based on the denominator.
2. For 1/2, split the 0 to 1 gap into 2 parts; the first mark represents 1/2.
3. Negative rationals like -3/4 are plotted similarly in the opposite direction.

Rational numbers fill the gaps between integers, making the number line dense.

Question 18:
Solve: (3/5 + -2/7) - (1/14). Show each step and verify your answer using the commutative property of addition for rational numbers.
Answer:

Step 1: Find LCM of denominators (5,7) = 35
(3/5) = (3×7)/(5×7) = 21/35
(-2/7) = (-2×5)/(7×5) = -10/35
Step 2: Add fractions
21/35 + (-10/35) = (21-10)/35 = 11/35
Step 3: Subtract 1/14
LCM of 35 and 14 = 70
11/35 = 22/70
1/14 = 5/70
22/70 - 5/70 = 17/70 (Final Answer)

Verification using commutative property:
The property states a + b = b + a.
Let’s rearrange: (-2/7 + 3/5) - 1/14
= (-10/35 + 21/35) - 5/70
= 11/35 - 5/70 = 17/70 (Same result)

This confirms our answer is correct as per the property.

Question 19:
Compare the properties of addition and multiplication of rational numbers using examples. Verify the commutative property for both operations with two rational numbers of your choice.
Answer:

The addition and multiplication of rational numbers follow specific properties:

  • Commutative Property: The order of numbers does not change the result.
    For addition: a + b = b + a
    For multiplication: a × b = b × a
  • Associative Property: Grouping does not affect the result.
  • Distributive Property: Multiplication distributes over addition.

Let’s verify the commutative property for 1/2 and 3/4:


Addition:
1/2 + 3/4 = (2 + 3)/4 = 5/4
3/4 + 1/2 = (3 + 2)/4 = 5/4
Hence, 1/2 + 3/4 = 3/4 + 1/2.


Multiplication:
1/2 × 3/4 = 3/8
3/4 × 1/2 = 3/8
Hence, 1/2 × 3/4 = 3/4 × 1/2.


Both operations satisfy the commutative property.

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)

These 4-mark case-based questions assess analytical skills through real-life scenarios. Answers must be based on the case study provided.

Question 1:
Rahul bought 3/4 kg of apples and 1/2 kg of oranges. His friend Riya bought 5/6 kg of apples and 2/3 kg of oranges. Compare the total weight of fruits bought by both using rational numbers.
Answer:
Problem Interpretation

We need to find and compare the total weight of fruits bought by Rahul and Riya.

Mathematical Modeling
  • Rahul: 3/4 + 1/2 = (3/4 + 2/4) = 5/4 kg
  • Riya: 5/6 + 2/3 = (5/6 + 4/6) = 9/6 = 3/2 kg
Solution

Converting to like terms: 5/4 = 1.25 kg, 3/2 = 1.5 kg. Riya bought more fruits (1.5 kg > 1.25 kg).

Question 2:
A pizza is divided into 8 equal slices. Priya ate 3 slices, and Raj ate 1/4 of the pizza. Represent their consumption as rational numbers and find who ate more.
Answer:
Problem Interpretation

We must represent pizza slices eaten as fractions and compare them.

Mathematical Modeling
  • Priya: 3 slices = 3/8 of pizza
  • Raj: 1/4 of pizza = 2/8 (converted to denominator 8)
Solution

Comparing 3/8 > 2/8, Priya ate more pizza. Our textbook shows such conversions using LCM.

Question 3:
Rahul bought 3/4 kg of apples and 1/2 kg of oranges. His friend Riya bought 5/6 kg of mangoes.
Problem Interpretation: Compare the total weight of fruits bought by Rahul and Riya using rational numbers.
Answer:
Problem Interpretation: We need to find and compare the total weight of fruits bought by Rahul and Riya.
Mathematical Modeling: Rahul's total = 3/4 + 1/2 = 3/4 + 2/4 = 5/4 kg. Riya's total = 5/6 kg.
Solution: Converting to like denominators, 5/4 = 15/12 and 5/6 = 10/12. Since 15/12 > 10/12, Rahul bought more fruits.
Question 4:
A water tank has 7/8 of its capacity filled. If 3/5 of the stored water is used, what fraction of the tank's total capacity remains?
Problem Interpretation: Find the remaining water after usage as a fraction of the total capacity.
Answer:
Problem Interpretation: We must calculate the remaining water in the tank after usage.
Mathematical Modeling: Water used = 3/5 × 7/8 = 21/40. Remaining water = 7/8 - 21/40.
Solution: Converting to like denominators, 7/8 = 35/40. So, 35/40 - 21/40 = 14/40 = 7/20. Thus, 7/20 of the tank remains.
Question 5:
Rahul bought 3/4 kg of apples and 1/2 kg of oranges. His friend Riya bought 5/6 kg of mangoes and 2/3 kg of bananas. Compare the total weight of fruits bought by Rahul and Riya using rational number operations.
Answer:
Problem Interpretation

We need to find and compare the total weight of fruits bought by Rahul and Riya using addition of rational numbers.

Mathematical Modeling
  • Rahul: 3/4 + 1/2 = (3/4 + 2/4) = 5/4 kg
  • Riya: 5/6 + 2/3 = (5/6 + 4/6) = 9/6 = 3/2 kg
Solution

Converting to like terms: 5/4 = 1.25 kg, 3/2 = 1.5 kg. Riya bought more fruits (1.5 kg > 1.25 kg).

Question 6:
A water tank has 7/8 of its capacity filled. After usage, 1/4 of the tank's water was consumed. Calculate the remaining water as a fraction of the tank's total capacity.
Answer:
Problem Interpretation

We must find the remaining water after subtracting the consumed fraction from the initial filled quantity.

Mathematical Modeling
  • Initial water: 7/8
  • Consumed: 1/4 of total capacity
Solution

Subtract consumed water: 7/8 - 1/4 = (7/8 - 2/8) = 5/8. The remaining water is 5/8 of the tank's capacity.

Question 7:
Rahul bought 3/4 kg of apples and 1/2 kg of oranges. His friend Riya bought 5/6 kg of mangoes.

(a) Who bought more fruits in total?
(b) What is the total weight of fruits bought by both?
Answer:
Problem Interpretation

We need to compare and add the weights of fruits bought by Rahul and Riya.


Mathematical Modeling
  • Rahul: 3/4 kg + 1/2 kg = 5/4 kg
  • Riya: 5/6 kg

Solution

(a) Rahul bought more (5/4 kg > 5/6 kg). (b) Total weight = 5/4 + 5/6 = 25/12 kg (≈2.08 kg).

Question 8:
A pizza is divided into 8 equal slices. Priya ate 3 slices, and Raj ate 1/4 of the pizza.

(a) What fraction did Priya eat?
(b) Who ate more pizza?
Answer:
Problem Interpretation

We must find the fractional consumption and compare Priya's and Raj's portions.


Mathematical Modeling
  • Priya: 3/8 of pizza
  • Raj: 1/4 (= 2/8) of pizza

Solution

(a) Priya ate 3/8. (b) Priya ate more (3/8 > 2/8). Our textbook shows similar problems with fractions.

Question 9:
Rahul bought 3/4 kg of apples and 1/2 kg of oranges. His friend Riya bought 5/6 kg of apples and 2/3 kg of oranges. Compare the total weight of fruits bought by both using rational numbers.
Answer:
Problem Interpretation

We need to find and compare the total weight of fruits Rahul and Riya bought.

Mathematical Modeling
  • Rahul's total = 3/4 + 1/2 = 5/4 kg
  • Riya's total = 5/6 + 2/3 = 9/6 kg (simplified to 3/2 kg)
Solution

Convert to like terms: 5/4 kg (1.25 kg) vs 3/2 kg (1.5 kg). Riya bought more fruits.

Question 10:
A pizza is divided into 8 equal slices. Priya ate 3 slices, and Raj ate 1/4 of the pizza. Represent their consumption as rational numbers and find who ate more.
Answer:
Problem Interpretation

We must compare Priya's and Raj's pizza consumption using fractions.

Mathematical Modeling
  • Priya ate 3/8 of the pizza.
  • Raj ate 1/4, which equals 2/8 when converted.
Solution

Since 3/8 > 2/8, Priya ate more pizza. Our textbook shows how to compare fractions using common denominators.

Question 11:

Rahul and Priya were given two rational numbers, 5/6 and 7/8, to compare. Rahul claimed that 5/6 is greater, while Priya argued that 7/8 is greater. Help them resolve the dispute by comparing the two numbers step-by-step.

Answer:

To compare 5/6 and 7/8, we need to find a common denominator since their denominators are different.


Step 1: Find the Least Common Multiple (LCM) of 6 and 8.


LCM of 6 and 8 = 24.


Step 2: Convert both fractions to equivalent fractions with denominator 24.


5/6 = (5 × 4)/(6 × 4) = 20/24.


7/8 = (7 × 3)/(8 × 3) = 21/24.


Step 3: Compare the numerators.


Since 21 > 20, 21/24 > 20/24.


Thus, 7/8 > 5/6. Priya is correct.

Question 12:

In a school event, students were asked to represent the rational number -3/4 on a number line. Explain the steps to plot -3/4 accurately, including how to divide the number line.

Answer:

To plot -3/4 on a number line, follow these steps:


Step 1: Draw a horizontal number line with 0 at the center, positive numbers to the right, and negative numbers to the left.


Step 2: Since the denominator is 4, divide the space between 0 and -1 into 4 equal parts.


Step 3: Label the divisions as -1/4, -2/4 (which simplifies to -1/2), -3/4, and -1.


Step 4: Locate -3/4 on the third division to the left of 0.


Visualization Tip: Imagine the space between 0 and -1 as a whole pizza cut into 4 slices. -3/4 represents 3 slices to the left of 0.

Question 13:

Rahul and Priya were comparing their scores in a math test. Rahul scored 7/12 and Priya scored 5/8 of the total marks. They wanted to find out who performed better. Help them by comparing their scores using the concept of rational numbers.

Answer:

To compare Rahul's score (7/12) and Priya's score (5/8), we need to find a common denominator.
The denominators are 12 and 8.
The LCM of 12 and 8 is 24.

Now, convert both fractions to have denominator 24:
7/12 = (7 × 2)/(12 × 2) = 14/24
5/8 = (5 × 3)/(8 × 3) = 15/24

Now, compare the numerators:
14 < 15, so 14/24 < 15/24.

Thus, 7/12 < 5/8, which means Priya performed better than Rahul.

Question 14:

Shyam bought 3/4 kg of apples and 2/5 kg of oranges. He wants to know the total weight of fruits he purchased. Help him by adding these rational numbers and express the answer in its simplest form.

Answer:

To find the total weight of fruits, we add 3/4 kg (apples) and 2/5 kg (oranges).
The denominators are 4 and 5.
The LCM of 4 and 5 is 20.

Convert both fractions to have denominator 20:
3/4 = (3 × 5)/(4 × 5) = 15/20
2/5 = (2 × 4)/(5 × 4) = 8/20

Now, add the fractions:
15/20 + 8/20 = 23/20 kg.

The fraction 23/20 is already in its simplest form because 23 and 20 have no common factors other than 1.

Thus, the total weight of fruits Shyam bought is 23/20 kg or 1 3/20 kg.

Question 15:

Rahul and Priya were given a problem to solve: "Find three rational numbers between 1/4 and 1/2." Rahul used the mean method, while Priya used the equivalent fractions method.

(a) Show both methods step-by-step.
(b) Which method is more efficient? Justify.

Answer:

(a) Methods:

Rahul's Mean Method:
Step 1: Find mean of 1/4 and 1/2 → (1/4 + 1/2)/2 = (3/4)/2 = 3/8
Step 2: Find mean of 1/4 and 3/8 → (1/4 + 3/8)/2 = (5/8)/2 = 5/16
Step 3: Find mean of 3/8 and 1/2 → (3/8 + 1/2)/2 = (7/8)/2 = 7/16

Priya's Equivalent Fractions Method:
Step 1: Convert to denominators 8 → 1/4 = 2/8, 1/2 = 4/8
Step 2: Write fractions between them → 3/8
Step 3: Convert to denominators 16 → 1/4 = 4/16, 1/2 = 8/16
Step 4: Write fractions between them → 5/16, 6/16, 7/16

(b) Efficiency:
Priya's method is faster as it generates multiple numbers in fewer steps. Mean method requires repeated calculations but ensures precise midpoints.

Question 16:

A pizza is divided into 12 equal slices. Riya ate 3/12, Sam ate 1/4, and Dia ate 0.25 of the pizza.

(a) Represent their consumption as rational numbers in standard form.
(b) Who ate the most? Show comparison using cross-multiplication.

Answer:

(a) Standard Form:
Riya: 3/12 = 1/4 (divided numerator/denominator by 3)
Sam: 1/4 is already standard
Dia: 0.25 = 25/100 = 1/4 (simplified)

(b) Comparison:
All ate equal amounts (1/4 each).
Verification:
Compare Riya (1/4) and Sam (1/4):
1×4 = 4 and 1×4 = 4 → Equal
Similarly, Dia's 1/4 matches others.

Question 17:

Rahul and Priya were given two rational numbers, 5/6 and 3/4, respectively. They were asked to find the sum of these numbers. Rahul added the numerators and denominators directly, getting 8/10, while Priya found the LCM of the denominators first and then added them correctly.

Based on this case:

  • Identify the mistake Rahul made.
  • Show the correct steps Priya followed to add the rational numbers.
  • What is the final sum?
Answer:

Mistake by Rahul: He added the numerators and denominators directly, which is incorrect for adding rational numbers. Rational numbers must have a common denominator before addition.


Correct steps by Priya:

  • Find the LCM of denominators 6 and 4, which is 12.
  • Convert both fractions to equivalent fractions with denominator 12:
    5/6 = (5 × 2)/(6 × 2) = 10/12
    3/4 = (3 × 3)/(4 × 3) = 9/12
  • Add the numerators: 10/12 + 9/12 = 19/12

Final sum: The correct sum of 5/6 and 3/4 is 19/12.

Question 18:

In a science experiment, a liquid's temperature was recorded as -7/5°C in the morning and 3/2°C in the afternoon.

Based on this case:

  • Calculate the difference between the afternoon and morning temperatures.
  • Is the result a positive or negative rational number? What does this indicate about the temperature change?
Answer:

Step 1: Find the difference:


Afternoon temperature - Morning temperature = 3/2 - (-7/5)
= 3/2 + 7/5 (Subtracting a negative is the same as adding)
= (3 × 5)/(2 × 5) + (7 × 2)/(5 × 2) (LCM of 2 and 5 is 10)
= 15/10 + 14/10
= 29/10°C


Step 2: Analyze the result:


The result is 29/10, a positive rational number. This indicates that the temperature increased from morning to afternoon.

Question 19:

Rahul and Priya are comparing their heights. Rahul's height is 5/2 meters, and Priya's height is 9/4 meters. They want to know who is taller and by how much. Help them solve this problem step-by-step using rational numbers.

Answer:

To compare Rahul and Priya's heights, we need to compare the rational numbers 5/2 and 9/4.


Step 1: Convert both fractions to have the same denominator (LCM of 2 and 4 is 4).
5/2 = (5 × 2)/(2 × 2) = 10/4
9/4 remains 9/4.


Step 2: Compare the numerators.
10/4 > 9/4 because 10 > 9.


Step 3: Calculate the difference.
10/4 - 9/4 = 1/4 meters.


Conclusion: Rahul is taller than Priya by 1/4 meters.

Question 20:

A pizza is divided into 8 equal slices. Ravi ate 3/8 of the pizza, and Sita ate 1/4 of the pizza. Find the total fraction of pizza eaten by them together. Also, represent the remaining pizza as a rational number.

Answer:

To find the total fraction of pizza eaten by Ravi and Sita, we add the rational numbers 3/8 and 1/4.


Step 1: Convert 1/4 to have the same denominator as 3/8 (LCM of 4 and 8 is 8).
1/4 = (1 × 2)/(4 × 2) = 2/8.


Step 2: Add the fractions.
3/8 + 2/8 = 5/8.


Step 3: Calculate the remaining pizza.
Total pizza = 8/8 (or 1 whole).
Remaining pizza = 8/8 - 5/8 = 3/8.


Conclusion: Ravi and Sita ate 5/8 of the pizza together, and 3/8 of the pizza is left.

Question 21:

Rahul and Priya were given two rational numbers, 5/6 and 3/4, respectively. They were asked to find the sum of these numbers and then express the result in its simplest form. Help them solve the problem step-by-step.

Answer:

To find the sum of 5/6 and 3/4, follow these steps:


Step 1: Find the Least Common Denominator (LCD) of the denominators 6 and 4.
The multiples of 6: 6, 12, 18, 24...
The multiples of 4: 4, 8, 12, 16...
LCD = 12

Step 2: Convert both fractions to equivalent fractions with the LCD as the denominator.
5/6 = (5 × 2)/(6 × 2) = 10/12
3/4 = (3 × 3)/(4 × 3) = 9/12

Step 3: Add the numerators while keeping the denominator the same.
10/12 + 9/12 = 19/12

Step 4: Simplify the fraction if possible.
19/12 is already in its simplest form since 19 and 12 have no common factors other than 1.

Thus, the sum of 5/6 and 3/4 is 19/12.

Question 22:

In a school, 2/5 of the students prefer cricket, while 1/3 prefer football. The rest prefer other sports. What fraction of students prefer other sports? Show your calculations clearly.

Answer:

To find the fraction of students who prefer other sports, follow these steps:


Step 1: Find the total fraction of students who prefer cricket and football.
2/5 (cricket) + 1/3 (football)

Step 2: Find the Least Common Denominator (LCD) of 5 and 3.
LCD = 15

Step 3: Convert the fractions to equivalent fractions with the LCD as the denominator.
2/5 = (2 × 3)/(5 × 3) = 6/15
1/3 = (1 × 5)/(3 × 5) = 5/15

Step 4: Add the fractions.
6/15 + 5/15 = 11/15

Step 5: Subtract the total from 1 (whole) to find the fraction preferring other sports.
1 - 11/15 = 15/15 - 11/15 = 4/15

Thus, 4/15 of the students prefer other sports.

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