Class 8 RD Sharma Solutions Chapter 1 Rational Numbers Exercise 1.8 (original) (raw)
Last Updated : 11 Sep, 2024
Exercise 1.8 in Chapter 1 of RD Sharma's Class 8 mathematics textbook represents the culmination of the study of rational numbers, offering a comprehensive and challenging set of problems that integrate all the concepts learned throughout the chapter. This exercise is designed to provide solutions and detailed explanations to complex problems involving rational numbers, serving as both a review and an extension of previously covered material. The solutions presented in this exercise demonstrate advanced problem-solving techniques, logical reasoning, and mathematical rigor, helping students to not only find the correct answers but also to understand the underlying principles and methodologies.In Class 8 Mathematics, Chapter 1 focuses on Rational Numbers a fundamental concept in arithmetic and algebra. Exercise 1.8 delves into practical problems and applications involving rational numbers helping students develop a deeper understanding of their properties and operations.
Rational Numbers
Rational numbers are numbers that can be expressed as the quotient or fraction p\q where p and q are integers and 𝑞≠0. They include integers, fractions, and finite or repeating decimals. Rational numbers are crucial in various mathematical operations and real-world applications as they help represent quantities that are not whole numbers.
**Question 1. Find a rational number between -3 and 1.
**Solution:
Between two rational numbers x and y
Rational number = (x + y) / 2
= (-3 + 1) / 2
= -2/2
= -1
So, the rational number between -3 and 1 is -1
**Question 2. Find any five rational numbers less than 2.
**Solution:
Five rational numbers less than 2 can be 0, 1, -2/5, 3/2, -4/5
**Question 3. Find two rational numbers between -2/9 and 5/9
**Solution:
The rational number between -2/9 and 5/9 is
= (-2/9 + 5/9)/2
= (3/9) / 2
3 is the common factor
= 1/3/2
= 1/3 * 1/2
= 1/6
Now the rational number between -2/9 and 1/6 is
= (-2/9 + 1/6) / 2
LCM is 18
= ((-2 × 2 + 1 × 3) /18) / 2
= (-4 + 3) / 36
= -1/36
Rational numbers between -2/9 and 5/9 are -1/36, 1/6
**Question 4. Find two rational numbers between 1/5 and 1/2
**Solution:
The rational number between 1/5 and 1/2 is
= (1/5 + 1/2)/2
LCM is 10
= ((1 × 2 + 1 × 5) / 10) / 2
= (2 + 5) / 20 = 7/20
Now the rational number between 1/5 and 7/20 is
= (1/5 + 7/20) / 2
LCM is 20
= ((1 × 4 + 7 × 1) / 20) / 2
= (4 + 7) / 40
= 11/40
Rational number between 1/5 and 1/2 are 7/20, 11/40
**Question 5. Find ten rational numbers between 1/4 and 1/2.
**Solution:
Other than average method, rational numbers can be found by converting the given rational numbers into equivalent rational numbers with same denominators.
The LCM for 4 and 2 is 4.
1/4 = 1/4
1/2 = (1 × 2) / 2 * 2= 2/4
1/4 = (1 × 20 / 4 × 20) = 20/80
2/4 = (2 × 20 / 4 × 20) = 40/80
So, we now know that 21, 22, 23,…39 are integers between numerators 20 and 40.
Rational numbers between 1/4 and 1/2 are 21/80, 22/80, 23/80, …., 39/80
**Question 6. Find ten rational numbers between -2/5 and 1/2.
**Solution:
Other than average method, rational numbers can be found by converting the given rational numbers into equivalent rational numbers with same denominators.
The LCM for 5 and 2 is 10.
-2/5 = (-2 × 2) / 10 = -4/10
1/2 = (1 × 5) / 10 = 5/10
-4/10 = (-4 × 2 / 10 × 2) = -8/20
5/10 = (5 × 2 / 10 × 2) = 10/20
So, we now know that -7, -6, -5,…10 are integers between numerators -8 and 10.
Rational numbers between -2/5 and 1/2 are -7/20, -6/20, -5/20, …., 9/20
**Question 7. Find ten rational numbers between 3/5 and 3/4.
**Solution:
Other than average method, rational numbers can be found by converting the given rational numbers into equivalent rational numbers with same denominators.
The LCM for 5 and 4 is 20.
3/5 = 3 × 4/5 × 4 = 12/20
3/4 = 3 × 5/4 × 5 = 15/20
Making the denominator 100
12/20 = 12 × 5/20 × 5 = 60/100
15/20 = 15 × 5/20 × 5 = 75/100
Rational numbers between 3/5 and 3/4 are 61/100, 62/100, 63/100, …., 74/100
Practice Questions with Soutions
1. Q: Simplify: (3/4 + 2/5) ÷ (3/4 - 2/5)
A: Step 1: Find a common denominator (20) for 3/4 and 2/5
3/4 = 15/20, 2/5 = 8/20
Step 2: Add and subtract
(15/20 + 8/20) ÷ (15/20 - 8/20) = 23/20 ÷ 7/20
Step 3: Divide fractions by multiplying by the reciprocal
(23/20) × (20/7) = 23/7
2. Q: If a = -2/3 and b = 3/4, find the value of (a² - b²) / (a - b)
A: Step 1: Calculate a² and b²
a² = (-2/3)² = 4/9, b² = (3/4)² = 9/16
Step 2: Calculate a - b
a - b = -2/3 - 3/4 = -8/12 - 9/12 = -17/12
Step 3: Substitute into the expression
(4/9 - 9/16) / (-17/12)
Step 4: Simplify
(64/144 - 81/144) / (-17/12) = -17/144 / (-17/12) = 1/12
3. Q: Prove that (√3 + √2)(√3 - √2) is a rational number.
A: Step 1: Multiply (√3 + √2)(√3 - √2)
= (√3)² - (√2)² = 3 - 2 = 1
Since 1 is a rational number, the expression is rational
4. Q: Solve for x: (x - 1)/(x + 1) + (x + 1)/(x - 1) = 5/2
A: Step 1: Find a common denominator
[(x - 1)² + (x + 1)²] / [(x + 1)(x - 1)] = 5/2
Step 2: Simplify
(2x² + 2) / (x² - 1) = 5/2
Step 3: Cross multiply
4x² + 4 = 5x² - 5
Step 4: Solve for x
x² = 9, so x = ±3
Check both solutions to ensure they work in the original equation.
5. Q: In a mixture of 80 liters, the ratio of milk to water is 3:5. How much milk should be added to make the ratio 5:3?
A: Step 1: Calculate current amounts
Milk: 3/8 × 80 = 30 liters, Water: 5/8 × 80 = 50 liters
Step 2: Let x be the amount of milk to be added
(30 + x) : 50 = 5 : 3
Step 3: Solve the equation
3(30 + x) = 5(50)
90 + 3x = 250
3x = 160
x = 160/3 ≈ 53.33 liters
6. Q: If the sum of three consecutive rational numbers is 15/4, find these numbers.
A: Step 1: Let the numbers be x - 1/4, x, and x + 1/4
Step 2: Form an equation
(x - 1/4) + x + (x + 1/4) = 15/4
Step 3: Solve for x
3x = 15/4
x = 5/4
Step 4: The three numbers are 1, 5/4, and 6/4
7. Q: Prove that √5 is an irrational number.
A: Proof by contradiction:
Assume √5 is rational. Then √5 = a/b where a and b are integers with no common factors.
5 = a²/b²
5b² = a²
a² is divisible by 5, so a must be divisible by 5. Let a = 5k.
5b² = 25k²
b² = 5k²
b² is divisible by 5, so b must be divisible by 5.
This contradicts our assumption that a and b have no common factors.
Therefore, √5 must be irrational.
8. Q: A shopkeeper marks up his goods by 25% and then gives a discount of 20%. Find his profit percentage.
A: Step 1: Calculate the marked-up price
If cost price is x, marked-up price is 1.25x
Step 2: Calculate the selling price after discount
Selling price = 1.25x × 0.8 = x
Step 3: Calculate profit percentage
Profit = (Selling price - Cost price) / Cost price × 100%
= (x - x) / x × 100% = 0%
The shopkeeper makes no profit.
9. Q: If a:b = 2:3 and b:c = 4:5, find the ratio a:b:c in its simplest form.
A: Step 1: From a:b = 2:3, let a = 2k and b = 3k
Step 2: From b:c = 4:5, let b = 4m and c = 5m
Step 3: Equate the two expressions for b
3k = 4m
k = 4m/3
Step 4: Substitute to find a, b, and c in terms of m
a = 2k = 2(4m/3) = 8m/3
b = 3k = 3(4m/3) = 4m
c = 5m
Step 5: Simplify the ratio
a:b:c = 8m/3 : 4m : 5m = 8:12:15'
10. Q: Solve the inequality: (2x - 1)/(x + 2) > (x + 1)/(2x - 3), x ≠ 3/2, x ≠ -2
A: Step 1: Cross multiply
(2x - 1)(2x - 3) > (x + 1)(x + 2)
Step 2: Expand
4x² - 6x - 2x + 3 > x² + 3x + 2
Step 3: Simplify
3x² - 8x + 3 > 0
Step 4: Factor
(3x - 1)(x - 3) > 0
Step 5: Solve the inequality
x < 1/3 or x > 3, x ≠ 3/2, x ≠ -2
Summary
Exercise 1.8 in Chapter 1 of RD Sharma's Class 8 mathematics textbook provides comprehensive solutions to a diverse set of problems involving rational numbers. The solutions cover a wide range of topics, including simplification of complex rational expressions, solving equations and inequalities with rational coefficients, proofs of irrationality, and practical applications of rational numbers in real-world scenarios. These solutions demonstrate various problem-solving techniques, emphasizing logical reasoning, algebraic manipulation, and step-by-step approaches to complex problems. By studying these solutions, students can gain a deeper understanding of rational numbers, develop advanced problem-solving skills, and prepare for more challenging mathematical concepts in future studies.