Important questions on Rational numbers for Class 8 Maths
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Important questions - Class 8 Maths - Rational numbers - MCQ
1. What is the sum of -3/4 and 5/8?
a) 1/8
b) 1/2
c) 7/8
d) 1/4
Answer: c) 7/8
Explanation: To add these fractions, we need to first find a common denominator, which in this case is 8. -3/4 can be rewritten as -6/8, so the sum becomes (-6/8) + (5/8) = -1/8. However, we simplify this fraction to get the final answer of 7/8.
2. What is the product of -1/3 and -2/7?
a) 2/21
b) 2/3
c) -2/21
d) -2/3
Answer: a) 2/21
Explanation: To multiply these fractions, we simply multiply the numerators together and the denominators together. (-1/3) x (-2/7) = 2/21.
3. Which of the following is equivalent to 2/3?
a) 4/5
b) 6/8
c) -2/-3
d) 1/2
Answer: b) 6/8
Explanation: To find an equivalent fraction, we need to multiply both the numerator and denominator by the same number. In this case, we can multiply both by 2 to get 4/6. Simplifying this fraction, we get 2/3, which is equivalent to 6/8.
4. What is the reciprocal of -5/6?
a) -5/6
b) -6/5
c) 5/6
d) 6/5
Answer: b) -6/5
Explanation: The reciprocal of a fraction is found by flipping it upside down. So, the reciprocal of -5/6 is -6/5.
5. What is the difference between -7/9 and -1/3?
a) -2/9
b) 2/9
c) -2/3
d) 2/3
Answer: c) -2/3
Explanation: To subtract these fractions, we need to first find a common denominator, which in this case is 9. -1/3 can be rewritten as -3/9, so the difference becomes (-7/9) - (-3/9) = -4/9. However, we simplify this fraction to get the final answer of -2/3.
6. The sum of three rational numbers is -3/4. If two of the numbers are 2/3 and 1/6, what is the third number?
a) -5/4
b) -7/6
c) -5/6
d) -1/2
Answer: b) -7/6
Explanation: Let x be the third number. Then, we have:
2/3 + 1/6 + x = -3/4
Combining the like terms, we get:
x = -3/4 - 2/3 - 1/6
x = -7/6
Therefore, the third number is -7/6.
7. Which of the following is the additive inverse of -4/5?
a) -4/5
b) 4/5
c) -5/4
d) 5/4
Answer: b) 4/5
Explanation: The additive inverse of a number is the number that when added to it, gives zero. The additive inverse of -4/5 is 4/5, because:
-4/5 + 4/5 = 0
8. Simplify: (4/9) ÷ (2/3)
a) 3/6
b) 8/27
c) 2/3
d) 6/9
Answer: c) 2/3
Explanation: To divide by a fraction, we multiply by its reciprocal. Therefore, we have:
(4/9) ÷ (2/3) = (4/9) x (3/2)
Simplifying the fractions, we get:
(4/9) ÷ (2/3) = (2/3)
9. The sum of two rational numbers is -1/2. If one number is -3/4, what is the other number?
a) 1/4
b) -1/4
c) -1/8
d) -5/8
Answer: d) -5/8
Explanation: Let x be the other number. Then, we have:
-3/4 + x = -1/2
Subtracting -3/4 from both sides, we get:
x = -1/2 + 3/4
Simplifying, we get:
x = -5/8
Therefore, the other number is -5/8.
10. What is the product of -2/3 and 9/4?
a) -3/2
b) -3/4
c) 3/2
d) 3/4
Answer: b) -3/4
Explanation: To multiply two fractions, we simply multiply their numerators and denominators separately. So,
-2/3 x 9/4 = (-2 x 9) / (3 x 4) = -18/12 = -3/2 (after simplification)
Therefore, the product of -2/3 and 9/4 is -3/2.
11. A cake recipe calls for 2/3 cup of flour. If you need to make 3 cakes, how much flour do you need in total?
a) 1/3 cups
b) 2/3 cups
c) 1 cups
d) 2 cups
Answer: d) 2 cups
Explanation: To make 3 cakes, you need to multiply the amount of flour required for one cake by 3: (2/3) x 3 = 2 cups.
12. Simplify: (3/4 - 1/2) / (5/8 + 1/4)
a) 1/7
b) 7/6
c) 6/7
d) 7/4
Answer: c) 6/7
Explanation: (3/4 - 1/2) / (5/8 + 1/4) = (3/4 - 2/4) / (5/8 + 2/8) = 1/4 / 7/8 = 1/4 x 8/7 = 8/28 = 4/14 = 2/7. Therefore, the simplified fraction is 2/7, which is equivalent to 6/21, or 6/7 when simplified.
13. What is the value of the expression [(7/8) ÷ (1/4)] - [(1/2) + (3/4)]?
a) 1/4
b) 1/2
c) 3/4
d) 7/8
Answer: b) 1/2
Explanation: [(7/8) ÷ (1/4)] - [(1/2) + (3/4)] = (7/8) x (4/1) - (5/4) = 7/2 - 5/4 = 14/4 - 5/4 = 9/4 = 2 1/4. Therefore, the correct option is (b) 1/2.
14. What is the value of the expression [(5/6) + (2/3)] x [(3/4) - (1/2)]?
a) 1/6
b) 1/4
c) 1/3
d) 1/2
Answer: b) 1/4
Explanation: [(5/6) + (2/3)] x [(3/4) - (1/2)] = (11/6) x (1/4) = 11/24. Therefore, the correct option is (b) 1/4.
15. What is the value of the expression [2/3 + (1/2 - 1/4)] ÷ [5/6 - (1/3 + 1/4)]?
a) 7/16
b) 1/2
c) 1/3
d) 3/4
Answer: a) 7/16
Explanation: [2/3 + (1/2 - 1/4)] ÷ [5/6 - (1/3 + 1/4)] = [2/3 + 1/4] ÷ [5/6 - 7/12] = [8/12 + 3/12] ÷ [10/12 - 7/12] = 11/12 ÷ 3/12 = 11/3 x 1/11 = 1/3 = 0.3333 (approx). Therefore, the correct option is (a) 7/16.
Important questions - Rational numbers - Concepts
What is a rational number?
Answer: A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, where q ≠ 0.
State three rational numbers between -1 and 0.
Answer: -3/4, -2/5, and -1/2 are three rational numbers between -1 and 0.
State three rational numbers between 3 and 4.
Answer: 7/3, 11/4, and 13/4 are three rational numbers between 3 and 4.
Is the sum of two rational numbers always a rational number?
Answer: Yes, the sum of two rational numbers is always a rational number.
Is the product of two rational numbers always a rational number?
Answer: Yes, the product of two rational numbers is always a rational number.
What is the additive inverse of a rational number?
Answer: The additive inverse of a rational number a/b is -a/b.
What is the multiplicative inverse of a rational number?
Answer: The multiplicative inverse of a rational number a/b is b/a, provided a and b are not both zero.
What is a terminating decimal?
Answer: A terminating decimal is a decimal that has a finite number of digits after the decimal point.
What is a non-terminating repeating decimal?
Answer: A non-terminating repeating decimal is a decimal that has a repeating pattern of digits after the decimal point.
Give an example of a non-terminating repeating decimal.
Answer: 0.666... is an example of a non-terminating repeating decimal.
State the property of rational numbers under addition.
Answer: The property of rational numbers under addition is associative, commutative, and has an identity element 0.
State the property of rational numbers under multiplication.
Answer: The property of rational numbers under multiplication is associative, commutative, and has an identity element 1.
What is the difference between a proper fraction and an improper fraction?
Answer: A proper fraction is a fraction where the numerator is less than the denominator. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.
What is the decimal expansion of the rational number 5/8?
Answer: The decimal expansion of 5/8 is 0.625.
What is the rational number whose decimal expansion is 0.333...?
Answer: The rational number whose decimal expansion is 0.333... is 1/3.
Rational numbers - Sample questions
1. Simplify the expression (3/4 + 5/6 - 1/3) / (1/2 - 1/4 + 1/6).
a) 2/3
b) 3/4
c) 1/2
d) 4/5
Answer: a) 2/3
Explanation: (3/4 + 5/6 - 1/3) / (1/2 - 1/4 + 1/6) = (9/12 + 10/12 - 4/12) / (3/6 - 2/6 + 1/6) = 15/12 / 2/6 = 15/12 x 6/2 = 45/24 = 15/8. Therefore, the correct option is (a) 2/3.
2. What is the difference between a rational number and an irrational number?
a) Rational numbers can be expressed as fractions, while irrational numbers cannot.
b) Rational numbers are whole numbers, while irrational numbers are not.
c) Rational numbers can be negative, while irrational numbers cannot.
d) Rational numbers are always decimals, while irrational numbers are always fractions.
Answer: a) Rational numbers can be expressed as fractions, while irrational numbers cannot.
Explanation: A rational number is any number that can be expressed as a quotient or fraction p/q of two integers, where q ≠ 0. An irrational number is any number that cannot be expressed as a quotient or fraction of two integers. Therefore, the correct option is (a) Rational numbers can be expressed as fractions, while irrational numbers cannot.
3. What is the value of the expression 2/3 - 1/2 + 1/6?
a) 1/3
b) 1/6
c) 1/2
d) 2/3
Answer: b) 1/6
Explanation: 2/3 - 1/2 + 1/6 = 8/12 - 6/12 + 2/12 = 4/12 = 1/3 - 1/2 + 1/6 = 2/6 - 3/6 + 1/6 = 0. Therefore, the correct option is (b) 1/6.
4. State the property of rational numbers under division.
a) Associative property
b) Commutative property
c) Distributive property
d) None of the above
Answer: d) None of the above
Explanation: Rational numbers do not have a division property like the associative, commutative, or distributive properties under addition and multiplication. Therefore, the correct option is (d) None of the above.
5. What is the value of the expression (3/5 - 1/2) / (1/10 + 1/5)?
a) 1/10
b) 1/5
c) 1/2
d) 2
Answer: b) 1/5
Explanation: (3/5 - 1/2) / (1/10 + 1/5) = [(6/10) - (5/10)] / [(1/10) + (2/10)] = 1/10 / 3/10 = 1/10 x 10/3 = 1/3. Therefore, the correct option is (b) 1/5.
6. What is the sum of the rational numbers -1/4, 1/6, and 5/12?
a) 1/2
b) 1/3
c) 1/4
d) 1/6
Answer: a) 1/2
Explanation: -1/4 + 1/6 + 5/12 = -3/12 + 2/12 + 5/12 = 4/12 = 1/3. Therefore, the correct option is (a) 1/2.
7. What is the product of the rational numbers -3/5 and 7/4?
a) -21/20
b) -21/16
c) 21/20
d) 21/16
Answer: a) -21/20
Explanation: (-3/5) x (7/4) = -21/20. Therefore, the correct option is (a) -21/20.
8. Simplify the rational number 16/20.
a) 4/5
b) 3/4
c) 5/4
d) 5/3
Answer: a) 4/5
Explanation: 16/20 can be simplified by dividing both numerator and denominator by their greatest common factor, which is 4. So, 16/20 = (16/4) / (20/4) = 4/5. Therefore, the correct option is (a) 4/5.
9. What is the multiplicative inverse of the rational number -2/7?
a) -7/2
b) -7/5
c) -2/7
d) 7/2
Answer: b) -7/5
Explanation: The multiplicative inverse of a rational number a/b is b/a, provided a and b are not both zero. So, the multiplicative inverse of -2/7 is -7/2.
10. What is the decimal expansion of the rational number 7/25?
a) 0.27
b) 0.28
c) 0.29
d) 0.30
Answer: b) 0.28
Explanation: To convert the fraction 7/25 into a decimal, we perform long division as follows:
0.28
--------
25 | 7.00
6.25
-----
0.750
0.750
-----
0
Therefore, the decimal expansion of 7/25 is 0.28.
Rational numbers - Revision notes
These are some of the key points to remember while revising Rational Numbers for CBSE Class 8.
- Rational numbers are numbers that can be written in the form of p/q, where p and q are integers and q ≠ 0.
- Rational numbers can be positive, negative, or zero.
- Fractions with the same denominator can be added or subtracted by adding or subtracting the numerators while keeping the denominator the same.
- Fractions with different denominators must be converted to a common denominator before they can be added or subtracted.
- Multiplying two rational numbers is done by multiplying the numerators and denominators separately.
- Dividing two rational numbers is done by multiplying the first fraction by the reciprocal of the second fraction.
- A terminating decimal is a decimal that has a finite number of digits after the decimal point, while a non-terminating decimal has an infinite number of digits after the decimal point.
- A repeating decimal is a decimal in which a digit or group of digits repeats infinitely.
- Every rational number can be represented as a terminating decimal, a non-terminating but repeating decimal, or a non-repeating non-terminating decimal.
- The set of rational numbers is closed under addition, subtraction, multiplication, and division, which means that the result of any operation on two rational numbers is always a rational number.
- Rational numbers have the associative, commutative, and distributive properties for addition and multiplication.
- The multiplicative inverse of a non-zero rational number a/b is b/a.
- The additive inverse of a rational number a/b is -a/b.
- The reciprocal of a non-zero rational number a/b is b/a.
- Rational numbers can be plotted on a number line.
- Rational numbers can be compared using the concept of the greater than and less than symbols.
Rational numbers - Sample paper
You will get an instant merit certificate if your score in this sample paper is >70%
Rational numbers - Extra questions
1. Find the multiplicative inverse of the rational number 2/3.
a) 3/2
b) 2/3
c) 1/2
d) 3/4
Answer: a) 3/2
Explanation: The multiplicative inverse of a rational number is the reciprocal of that number. So, the reciprocal of 2/3 is 3/2. Therefore, the correct option is (a) 3/2.
2. Which of the following rational numbers has a multiplicative inverse?
a) 0
b) 1
c) -2
d) 2/3
Answer: d) 2/3
Explanation: A rational number has a multiplicative inverse if it is not equal to zero. So, options (a) and (b) do not have a multiplicative inverse, while option (c) is a negative number and its multiplicative inverse is also negative, which is not a rational number. Therefore, the only rational number with a multiplicative inverse is option (d) 2/3.
3. What is the multiplicative inverse of the rational number -5/6?
a) -6/5
b) 6/5
c) -5/6
d) 5/6
Answer: a) -6/5
Explanation: The multiplicative inverse of a rational number is the reciprocal of that number. So, the reciprocal of -5/6 is -6/5. Therefore, the correct option is (a) -6/5.
4. Find the value of (5/7) x (7/5).
a) 1
b) 2
c) 5/7
d) 7/5
Answer: a) 1
Explanation: When we multiply the reciprocals of two rational numbers, we get the value of 1. So, (5/7) x (7/5) = (5/7) x (5/7) = 25/49. Therefore, the correct option is (a) 1.
5. Find the multiplicative inverse of the rational number -3/4.
a) -4/3
b) 3/4
c) -3/4
d) 4/3
Answer: a) -4/3
Explanation: The multiplicative inverse of a rational number is the reciprocal of that number. So, the reciprocal of -3/4 is -4/3. Therefore, the correct option is (a) -4/3.
6. What is the multiplicative inverse of 3/4?
a) 1 1/3
b) 4/3
c) 3/4
d) 4/7
Answer: b) 4/3
Explanation: The multiplicative inverse of 3/4 is the reciprocal of 3/4, which is 4/3. This is because when you multiply a number by its reciprocal (multiplicative inverse), you get the multiplicative identity, which is 1. So, (3/4) x (4/3) = 12/12 = 1. Therefore, the correct option is (b) 4/3.
7. What is the multiplicative inverse of -5/8?
a) 8/5
b) -8/5
c) -5/8
d) 5/8
Answer: b) -8/5
Explanation: The multiplicative inverse of -5/8 is the reciprocal of -5/8, which is -8/5. This is because when you multiply a number by its reciprocal (multiplicative inverse), you get the multiplicative identity, which is 1. So, (-5/8) x (-8/5) = 40/40 = 1. Therefore, the correct option is (b) -8/5.
8. Simplify the rational number 16/20.
a) 4/5
b) 3/4
c) 5/4
d) 5/3
Answer: a) 4/5
Explanation: 16/20 can be simplified by dividing both numerator and denominator by their greatest common factor, which is 4. So, 16/20 = (16/4) / (20/4) = 4/5. Therefore, the correct option is (a) 4/5.
9. Find the multiplicative inverse of the rational number 2/3.
a) 3/2
b) 2/3
c) 3/4
d) 4/3
Answer: a) 3/2
Explanation: The multiplicative inverse of 2/3 is the reciprocal of 2/3, which is 3/2. This is because when you multiply a number by its reciprocal (multiplicative inverse), you get the multiplicative identity, which is 1. So, (2/3) x (3/2) = 6/6 = 1. Therefore, the correct option is (a) 3/2.
10. What is the multiplicative inverse of -7?
a) 7
b) -1/7
c) -7
d) 1/7
Answer: b) -1/7
11. If a is the additive inverse of b, what is the additive inverse of a?
a) b
b) -b
c) a
d) -a
Answer: d) -a
Explanation: The additive inverse of a number is the number that, when added to the original number, results in zero. So, if a is the additive inverse of b, then a + b = 0. To find the additive inverse of a, we need to find a number x such that a + x = 0. We can rewrite this as x = -a. Therefore, the additive inverse of a is -a, which is option (d).
12. Which of the following pairs of rational numbers are additive inverses of each other?
a) 1/3 and 2/3
b) 2/5 and -2/5
c) -5/8 and -8/5
d) 3/7 and -7/3
Answer: d) 3/7 and -7/3
Explanation: Two rational numbers are additive inverses of each other if their sum is zero. In option (d), 3/7 + (-7/3) = (9/21) + (-49/21) = -40/21, which is equal to zero. Therefore, 3/7 and -7/3 are additive inverses of each other, and the correct option is (d).
13. Which of the following is not the additive inverse of -7/12?
a) 7/12
b) -7/12
c) -12/7
d) 12/7
Answer: a) 7/12
Explanation: The additive inverse of a rational number is the number that, when added to the original number, results in zero. So, if x is the additive inverse of -7/12, then x + (-7/12) = 0. We can rearrange this equation to get x = 7/12. Therefore, option (a) is not the additive inverse of -7/12.
14. What is the additive inverse of the rational number -3/4?
a) 3/4
b) -3/4
c) -4/3
d) 4/3
Answer: a) 3/4
Explanation: The additive inverse of a rational number is the number that, when added to the original number, results in zero. So, if x is the additive inverse of -3/4, then x + (-3/4) = 0. We can rearrange this equation to get x = 3/4. Therefore, the additive inverse of -3/4 is 3/4, which is option (a).
15. What should be added to 3/4 to get 2/3?
a) -1/12
b) -1/6
c) -1/3
d) 1/12
Answer: c) -1/3
Explanation: To find what should be added to 3/4 to get 2/3, we need to solve the equation: 3/4 + x = 2/3. Solving for x, we get x = -1/3. Therefore, the correct option is (c) -1/3.
16. What should be added to -7/8 to get -1/4?
a) 1/8
b) 3/8
c) 5/8
d) 7/8
Answer: d) 7/8
Explanation: To find what should be added to -7/8 to get -1/4, we need to solve the equation: -7/8 + x = -1/4. Solving for x, we get x = 7/8. Therefore, the correct option is (d) 7/8.
17. What should be added to -3/5 to get 1/10?
a) 1/5
b) 3/5
c) 7/10
d) 9/10
Answer: d) 9/10
Explanation: To find what should be added to -3/5 to get 1/10, we need to solve the equation: -3/5 + x = 1/10. Solving for x, we get x = 9/10. Therefore, the correct option is (d) 9/10.
18. What should be added to -5/6 to get -1/3?
a) 1/3
b) 2/3
c) 3/4
d) 5/6
Answer: b) 2/3
Explanation: To find what should be added to -5/6 to get -1/3, we need to solve the equation: -5/6 + x = -1/3. Solving for x, we get x = 2/3. Therefore, the correct option is (b) 2/3.
19. What should be added to -4/7 to get 3/5?
a) 13/35
b) 17/35
c) 19/35
d) 23/35
Answer: c) 19/35
Explanation: To find what should be added to -4/7 to get 3/5, we need to solve the equation: -4/7 + x = 3/5.
20. Simplify the expression (5/6 - 2/3) / (1/2 + 1/3).
a) 5/12
b) 1/2
c) 7/12
d) 2/3
Answer: b) 1/2
Explanation: First, we simplify the expression inside the brackets. 5/6 - 2/3 = 5/6 - 4/6 = 1/6, and 1/2 + 1/3 = 3/6 + 2/6 = 5/6. So, (5/6 - 2/3) / (1/2 + 1/3) = (1/6) / (5/6) = 1/5. Finally, we raise this result to the power of 4/3, which gives (1/5)^(4/3) = (1/5)^(4/3) * (5/5)^(4/3) = 1/5^(4/3) = 1/∛(5^4) = 1/125. The reciprocal of 1/125 is 125, so the final answer is 125, which is option (b).
21. Simplify the rational number 50/66.
a) 5/6
b) 25/33
c) 100/132
d) 75/99
Answer: b) 25/33
Explanation: We can simplify the rational number 50/66 by dividing both the numerator and denominator by their greatest common factor, which is 2. So, 50/66 = (50/2) / (66/2) = 25/33. Therefore, the correct option is (b) 25/33.
22. Simplify the expression 4/5 - 5/7 + 1/2.
a) 7/70
b) 3/10
c) 9/70
d) 1/10
Answer: b) 3/10
Explanation: We need to first find the common denominator, which is 70. Then, we can rewrite the expression as (4/5) * (14/14) - (5/7) * (10/10) + (1/2) * (35/35) = 56/70 - 50/70 + 35/70 = 41/70. Therefore, the correct option is (b) 3/10.
23. Simplify the expression 2/3 + 4/5 - 1/4.
a) 37/60
b) 47/60
c) 7/12
d) 1/2
Answer: a) 37/60
Explanation: We need to first find the common denominator, which is 60. Then, we can rewrite the expression as (2/3)