Summary of CBSE Class 8 Mathematics - Rational Numbers

CBSE Class 8 Mathematics - Rational Numbers includes the following topics:

1. Introduction to Rational Numbers
2. Properties of Rational Numbers
3. Representing Rational Numbers on the Number Line
4. Rational Numbers Between Two Rational Numbers
5. Operations on Rational Numbers
6. Decimal Expansion of Rational Numbers
7. Comparison of Rational Numbers

In summary, rational numbers are those which can be expressed as the ratio of two integers, where the denominator is not equal to 0. Rational numbers include fractions, terminating decimals, and repeating decimals.

Some of the important operations that can be performed on rational numbers include addition, subtraction, multiplication, and division. Decimal expansion of rational numbers is also an important aspect which can help in comparing them.

The 8th grade CBSE chapter on Rational Numbers is a comprehensive guide to understanding the properties and applications of rational numbers. The chapter begins by defining rational numbers and discussing their basic properties. It then moves on to explore positive and negative rational numbers and how they can be compared and represented on a number line. The chapter also covers operations on rational numbers, including addition, subtraction, multiplication, and division, as well as simplification. Students will also learn how to find rational numbers between two given rational numbers. Finally, the chapter concludes by providing practical applications of rational numbers in distance and speed problems, profit and loss problems, and simple interest problems.

Key Notes for CBSE Class 8 Mathematics - Rational Numbers

Key notes from the chapter on Rational Numbers in CBSE Class 8 Mathematics:

1. 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.
2. Rational numbers include integers and fractions. They can be positive, negative, or zero.
3. Rational numbers can be represented on the number line.
4. To simplify a rational number, divide both the numerator and denominator by their common factors until no further simplification is possible.
5. Adding and subtracting rational numbers requires finding a common denominator, which is the lowest multiple of the denominators of the given rational numbers.
6. To multiply rational numbers, multiply the numerators and denominators separately and then simplify the product.
7. To divide rational numbers, invert the second fraction and multiply it to the first fraction.
8. The decimal representation of a rational number can either be a terminating decimal or a repeating decimal.
9. To convert a terminating decimal to a fraction, we write the digits after the decimal point as the numerator and the place value of the last digit as the denominator.
10. To convert a repeating decimal to a fraction, we write the repeating digits as the numerator and append as many 9s in the denominator as there are repeating digits.
11. Rational numbers can be compared using cross-multiplication or by expressing them with a common denominator.
12. When working with rational numbers, it is important to avoid division by zero and to simplify fractions before performing operations.
13. Rational numbers have applications in various fields, such as finance, measurement, and engineering.
14. Irrational numbers, such as pi and root 2, cannot be expressed in the form of a ratio of two integers and do not have a repeating decimal representation.
15. Rational numbers are closed under addition, subtraction, multiplication, and division. This means that the result of any operation performed on rational numbers will always be a rational number.

Practice Questions for CBSE Class 8 Mathematics - Rational Numbers

1. Which of the following numbers are rational numbers?
a) 5
b) 1.8333333...
c) -4/7
d) √2

Answer: Options a and c are rational numbers. Options b and d are not.

2. What should be subtracted from (-7/3) to get (-5/6)?

Answer: We need to find the value that, when subtracted from (-7/3), gives us (-5/6). This can be found by subtracting the numerator of the second fraction from the numerator of the first fraction and subtracting the denominator of the second fraction from the denominator of the first fraction.
=(-7/3) - (-5/6) = (-7/3) + (5/6) = (-14/6) + (5/6) = (-9/6) = -3/2.

Therefore, subtracting (-3/2) from (-7/3) gives us (-5/6).

3. Which is greater, 5/8 or 3/4?

Answer: To compare 5/8 and 3/4, we need to convert both fractions to a common denominator. The least common multiple of 8 and 4 is 8. So, we can convert 5/8 to 5/8 x 1/1 = 5/8 and we can convert 3/4 to 6/8 by multiplying both the numerator and denominator by 2. So, we have: 5/8 < 6/8.

Therefore, 3/4 is greater than 5/8.

4. Write three rational numbers between -3/5 and -1/5.

Answer: We need to find three rational numbers between -3/5 and -1/5. One way to do this is to add a common fraction between -3/5 and -1/5, for example:

-2/5, -1/3, and -1/2.

Therefore, three rational numbers between -3/5 and -1/5 are -2/5, -1/3, and -1/2.

5. What is the product of 3/8 and (4/5 + 1/4)?

Answer: We need to find the product of 3/8 and (4/5 + 1/4). We can first simplify the second fraction by finding a common denominator between 5 and 4:

(4/5) + (1/4) = (16/20) + (5/20) = (21/20).

Then, we can multiply 3/8 by 21/20:

(3/8) x (21/20) = (3 x 21/8) x (21/20) = (63/160).

Therefore, the product of 3/8 and (4/5 + 1/4) is 63/160.

6. Arrange -3/4, -1/2, and -5/8 in descending order.

Answer: To arrange -3/4, -1/2, and -5/8 in descending order, we can convert all the fractions to a common denominator. The least common multiple of 4, 2, and 8 is 8. So:

-3/4 = -6/8,
-1/2 = -4/8, and
-5/8 = -5/8.

Arranging in descending order:

-6/8 > -5/8 > -4/8.

Therefore, the descending order is -3/4, -5/8, and -1/2.

7. Express the decimal 0.764 as a rational number.

Answer: We can express 0.764 as a rational number by converting the decimal to a fraction. Since the decimal has three digits after the decimal point, we can express it as 764/1000. We can simplify this fraction by dividing both the numerator and denominator by their common factor of 4, to get: 191/250.

Therefore, the rational number equivalent to the decimal 0.764 is 191/250.

8. Add -5/6 and 3/4.

Answer: To add -5/6 and 3/4, we need to find a common denominator between 6 and 4. The least common multiple of 6 and 4 is 12. So, we can convert -5/6 to -10/12 by multiplying both the numerator and denominator by 2 and we can convert 3/4 to 9/12 by multiplying both numerator and denominator by 3. Then, we can add:

-10/12 + 9/12 = -1/12.

Therefore, the sum of -5/6 and 3/4 is -1/12.

9. Which of these rational numbers is a terminating decimal: 3/5, 2/3, 7/8, 1/4?

Answer: A rational number is a terminating decimal if and only if its decimal representation ends after a finite number of digits. So, let's consider each of the given options:

- 3/5 = 0.6, which is a terminating decimal.
- 2/3 = 0.666666..., which is a non-terminating repeating decimal.
- 7/8 = 0.875, which is a terminating decimal.
- 1/4 = 0.25, which is a terminating decimal.

Therefore, the rational numbers that are terminating decimals are 3/5, 7/8, and 1/4.

10. Subtract 5/6 from 7/3.

Answer: To subtract 5/6 from 7/3, we need to find a common denominator. The least common multiple of 6 and 3 is 6. So, we can convert 7/3 to 14/6 by multiplying both the numerator and denominator by 2 and we can convert 5/6 to 5/6 x 1/1 = 5/6 by multiplying it by 1 (whose value is 6/6). Then, we can subtract:

14/6 - 5/6 = 9/6.

We can simplify 9/6 by dividing both the numerator and denominator by their common factor of 3, to get: 3/2.

Therefore, the difference between 7/3 and 5/6 is 3/2.

11. Write two rational numbers whose product is 3/4.

Answer: Let x and y be two rational numbers whose product is 3/4. Then, we have xy = 3/4. To find two such numbers, we can choose any two rational numbers whose product is 3/4. For example, x = 1/2 and y = 3/2. Then, xy = (1/2) × (3/2) = 3/4.

12. Subtract -5/8 from 3/4.

Answer: To subtract -5/8 from 3/4, we need to find a common denominator. The least common multiple of 4 and 8 is 8. So, we can write 3/4 as 6/8. Then, we can write -5/8 as -5/8 × 1/1 = -5/8. Now, we can subtract: 6/8 - 5/8 = 1/8. Therefore, 3/4 - (-5/8) = 1/8.

13. Find the value of 2/3 + 4/9 - 5/6.

Answer: To add and subtract these fractions, we need to find a common denominator. The least common multiple of 3, 9, and 6 is 18. So, we can write each fraction with a denominator of 18: 2/3 = 12/18, 4/9 = 8/18, and 5/6 = 15/18. Now, we can add and subtract: 12/18 + 8/18 - 15/18 = 5/18. Therefore, 2/3 + 4/9 - 5/6 = 5/18.

14. Find two rational numbers between -2 and 0.

Answer: To find two rational numbers between -2 and 0, we need to find two rational numbers x and y such that -2 < x < y < 0. One possible pair of rational numbers is -3/4 and -1/2. Another possible pair is -7/10 and -3/5.

15. Simplify (2/3) ÷ (4/5).

Answer: To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. So, (2/3) ÷ (4/5) = (2/3) × (5/4) = 10/12. To simplify this fraction, we can divide both the numerator and denominator by their greatest common factor, which is 2. So, 10/12 simplifies to 5/6. Therefore, (2/3) ÷ (4/5) = 5/6.

16. Find the value of -7/8 × (-5/6).

Answer: To multiply two fractions, we multiply the numerators together and multiply the denominators together. So, (-7/8) × (-5/6) = (7 × 5) / (8 × 6) = 35/48. Therefore, -7/8 × (-5/6) = 35/48.

17. Write 0.81 as a fraction in simplest form.

Answer: To write 0.81 as a fraction, we can write it as 81/100. To simplify this fraction, we can divide both the numerator and denominator by their greatest common factor, which is 9. So, 81/100 simplifies to 9/10. Therefore, 0

18. If x is a rational number such that 8x - 5 = 3x + 7, then what is the value of x?
Hint: Try to solve for x algebraically.

Answer: We can solve the equation 8x - 5 = 3x + 7 by subtracting 3x from both sides and adding 5 to both sides, which gives us 5x = 12. Therefore, x = 12/5.

19. Write a rational number that is halfway between (1/2) and (3/4). Hint: Find the average of the two numbers.

Answer: The average of (1/2) and (3/4) is (1/2 + 3/4)/2 = 5/8. So, 5/8 is halfway between (1/2) and (3/4).

20. Find two irrational numbers that are between 0.1 and 0.2.

Answer: One irrational number between 0.1 and 0.2 is √2/10. Another irrational number between 0.1 and 0.2 is √3/10. Note that both of these numbers have decimal representations that start with 0.1 and have non-repeating, non-terminating decimals.

21. Simplify the expression ((5/8)^2)/((2/5)*(10/9)). Hint: Simplify each fraction separately, then cancel out common factors.

Asnwer: We can simplify each fraction separately:

- (5/8)^2 = 25/64
- (2/5)*(10/9) = 4/9

Then, we can divide:

(25/64) ÷ (4/9) = (25/64) x (9/4) = 225/256.

Therefore, ((5/8)^2)/((2/5)*(10/9)) simplifies to 225/256.

22. Express the number 1.236486486... as a rational number. Hint: Notice the repeated decimal pattern and try to express it as a quotient of two integers.

Answer: The repeating decimal 1.236486486... can be expressed as a sum of a non-repeating decimal and a repeating decimal. Let x = 1.236. Then, 1000x = 1236.486486...

Subtracting x from 1000x gives:

1000x - x = 999x = 1235.25.

Solving for x gives:

x = 1235.25/999 = 247/200.

Therefore, 1.236486486... can be expressed as 247/200, which is a rational number.

23. Simplify: [(√3 + 2)/(√3 - 2)] x [(√2 - √5)/(√2 + √5)]. Hint: Use difference of squares to simplify.

Answer: We can simplify the expression [(√3 + 2)/(√3 - 2)] x [(√2 - √5)/(√2 + √5)] by using difference of squares to simplify the denominator. This gives:

[(√3 + 2)/(√3 - 2)] x [(√2 - √5)/(√2 + √5)]

= [(√3 + 2)/(√3 - 2)] x [(√2 - √5)/(√2 + √5)] x [(√3 + 2)/(√3 + 2)] x [(√2 - √5)/(√2 - √5)]

= [((√3)^2 - 2^2) x ((√2 - √5)^2)] / [((√3)^2 - 2^2) x ((

√2 + √5)^2]

= [(3 - 4)(2 - 2√10 + 5)] / [(3 - 4)(2 + 2√10 + 5)]

= (-7√10 - 1) / 9.

24. Prove that the square root of any non-perfect square is irrational. Hint: Prove the contrapositive statement by assuming that the square root of a non-perfect square is rational.

Answer: Assume that the square root of a non-perfect square is rational. Then, we can express it in the form a/b, where a and b are positive integers with no common factors. Squaring both sides gives a^2/b^2 = c, where c is a non-perfect square. But since c is not a perfect square, a^2 must have at least one prime factor in its prime factorization that has odd exponent. However, since a^2 divides b^2 and squares of primes have even exponents, b^2 must also have that prime factor in its prime factorization, which is a contradiction to the assumption that a and b have no common factors. Therefore, the square root of any non-perfect square is irrational.

25. Give an example of a rational number between (2/3) and (3/4) that has a repeating decimal representation.

Answer: We can express (2/3) and (3/4) with a common denominator of 12, which gives us (8/12) and (9/12), respectively. Then, we can find a fractional equivalent with a repeating decimal by expressing the fraction (8/12) as (6 + 2)/12 and (9/12) as (6 + 3)/12.

Thus, we get:

(6 + 2)/12 = 1/2 + 1/6 + 1/12 = 0.5(0.999...) + 0.1666... + 0.08333... = 0.7499...

(6 + 3)/12 = 1/2 + 1/4 + 1/12 = 0.5 + 0.25 + 0.08333... = 0.83333...

Therefore, a rational number between (2/3) and (3/4) that has a repeating decimal representation is 0.75.

26. What is the sum of the series: 1/3 + 1/7 + 1/11 + 1/15 + ... + 1/203?

Answer: The given series is an arithmetic sequence with a common difference of 1/4 and first term of 1/3. Using the partial sum formula for an arithmetic sequence, we get:

1/3 + 1/7 + 1/11 + 1/15 + ... + 1/203

= (n/2)((2a + (n-1)d)/2)

= (51/2)((2(1/3) + (51-1)(1/4))/2)

= (51/2)(104/12)

= 2217/68.

Therefore, the sum of the given series is 2217/68.

27. If x and y are rational numbers such that x + y = 4 and xy = 3, then what are the possible values of x and y?

Answer: Using Vieta's formulas, we have:

x + y = 4; xy = 3

Squaring the first equation gives us:

(x + y)^2 = 16

Expanding and simplifying, we get:

x^2 + 2xy + y^2 = 16

Substituting xy = 3, we get:

x^2 + 6 + y^2 = 16

x^2 + y^2 = 10

Now, squaring the equation x + y = 4 gives us:

x^2 + 2xy + y^2 + 2xy = 16

Substituting xy = 3, we get:

x^2 + 6 + y^2 + 6 = 16

x^2 + y^2 = 4

Subtracting these two equations, we get:

6 = 10 - 4

Therefore, (x - y)^2 = x^2 - 2xy + y^2 = (x^2 + y^2) - 2xy = 10 - 6 = 4.

Taking the square root of both sides, we get:

|x - y| = 2

So, either x - y = 2 or x - y = -2.

Solving for x and y using the system of equations x + y = 4 and xy = 3, we get:

x = 1 + √10, y = 1 - √10, or

x = 1 - √10, y = 1 + √10.

Therefore, the possible values of x and y are (1 + √10, 1 - √10) and (1 - √10, 1 + √10).

28. Express 0.626262... as a fraction in lowest terms. Hint: Use the geometric series formula to find an infinite sum with repeating decimals.

Answer: Let x = 0.626262... Then, 100x = 62.6262..., and subtracting 10x from 100x gives us:

90x = 62.

Therefore, x = 62/90 = 31/45.

To express x as a fraction in lowest terms, we need to simplify the fraction by dividing both numerator and denominator by their greatest common factor, which is 1 in this case. Hence, the decimal representation 0.626262... is equivalent to the fraction 31/45 in lowest terms.