# Mathematics Questions and Answers – Irrational and Rational Numbers

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Irrational and Rational Numbers”.

1. If x is a number whose simplest form is $$\frac {p}{q}$$, where p and q are integers and q≠0, then x is a terminating decimal only when q is of the form _________
a) 3m×5n
b) 2m×6n
c) 2m×5n
d) 7m×5n

Explanation: Let’s, take a number where q is of the form 2m×5n, say 250×510and p can be any integer
$$\frac {p}{2^{50}\times 5^{10}} = \frac {p \times 5^{40}}{10^{50}}$$
The number $$\frac {p \times 5^{40}}{10^{50}}$$ will terminate after 50 decimal places.
Hence, if q is of the form 2m×5n, it will terminate after some decimal places.

2. If x is a number whose simplest form is $$\frac {p}{q}$$, where p and q are integers and q ≠ 0, then x is a non-terminating repeating decimal only when q is not of the form ________
a) 2m×2n
b) 5m×5n
c) 2m×5n
d) 3m×4n

Explanation: Let’s, take a number where q is of the form 2m×5n, say 250×510and p can be any integer
$$\frac {p}{2^{50}\times 5^{10}} = \frac {p \times 5^{40}}{10^{50}}$$
The number $$\frac {p \times 5^{40}}{10^{50}}$$ will terminate after 50 decimal places.
Hence, if q is of the form 2m×5n, it will terminate after some decimal places.

3. Which of the following rational is non-terminating repeating decimal?
a) 0.25
b) $$\frac {4}{5}$$
c) $$\frac {4}{55}$$
d) $$\frac {2}{5}$$

Explanation: The value of $$\frac {4}{55}$$ is 0.07272727272…., which is non-terminating repeating decimal.
The other numbers terminate after few places of decimal.

4. The terminating rational number from the following numbers is _________
a) $$\frac {4}{9}$$
b) $$\frac {4}{3}$$
c) 0.146
d) $$\frac {4}{5}$$

Explanation: The value of $$\frac {4}{5}$$ is 0.8, which is terminating decimal.

5. The simplest form of the rational number 0.196 is ________
a) $$\frac {1}{6}$$
b) $$\frac {3}{6}$$
c) $$\frac {13}{66}$$
d) $$\frac {2}{5}$$

Explanation:
10x = 1.969696…..(1)
1000x = 196.9696…(2)
Subtracting (1) from (2)
We get,
990x=195
x = $$\frac {195}{990} = \frac {13}{66}$$

6. The numbers of the form $$\frac {p}{q}$$ are integers, and q≠0 are called irrational number.
a) True
b) False

Explanation:
Irrational numbers cannot be written in the form of $$\frac {p}{q}$$.
For example, ∛4 cannot be written in a fraction form as it has non-terminating and non-repeating decimals.

7. After how many places of decimal, will the decimal expansion of the rational number $$\frac {57}{2^4 5^6}$$ terminate?
a) 4
b) 6
c) 7
d) 8

Explanation:
We have,
$$\frac {57}{2^4 5^6} = \frac {57 \times 2^2}{2^6 5^6} = \frac {228}{10^6}$$ = 0.000228
The number $$\frac {57}{2^4 5^6}$$ will terminate after 6 decimal places.

8. From the following numbers, which number is not a rational number?
a) π
b) $$\frac {22}{7}$$
c) $$\frac {3}{4}$$
d) 0.666666…..

Explanation: A rational number has terminating or non-terminating but repeating decimals.
In case of π, it has a non-terminating as well as non-repeating decimal.
The other three numbers have terminating or non-terminating but repeating decimal, therefore, they are rational numbers.
Hence, it is an irrational number.

9. An irrational number has ________
a) Non-terminating decimal
b) Non-repeating decimal
c) Non-terminating and non-repeating decimal
d) Terminating decimal

Explanation: An irrational number has both non-terminating as well as non-repeating decimals.
For example, the number 1.353353335… has non-terminating as well as non-repeating decimals.

10. Which of the following numbers is not an irrational number?
a) π
b) $$\frac {22}{7}$$
c) 1.5353353335….
d) 2.7878878887….

Explanation:
An irrational number is expressible in the decimal form as non-terminating and non-repeating decimals.
From the given options,
π, 1.5353353335…, 2.7878878887… are non-terminating and non-repeating decimal.
Whereas, $$\frac {22}{7}$$ is non-terminating but repeating decimal.

11. The product of $$\frac {33}{2}$$ and $$\frac {5}{4}$$ is an irrational number.
a) True
b) False

Explanation:
$$\frac {33}{2} \times \frac {5}{4} = \frac {165}{8}$$
$$\frac {165}{8}$$ is a rational number

12. The product of a rational and an irrational number is rational number.
a) True
b) False

Explanation: Take a rational and an irrational number, say 2 and 3√3
Product of 2 × 3√3 = 6√3.
6√3 is an irrational number
Hence, the product of a rational and an irrational number is a irrational number.

13. The product of two irrational numbers is an irrational number.
a) True
b) False

Explanation: Consider an irrational number, say √10
√10 × √10=10
10 is a rational number. Hence, the product of two irrational numbers is not always irrational.

14. The sum of two rational numbers is a rational number.
a) False
b) True

Explanation: Consider two rational numbers, say $$\frac {8}{9}, \frac {3}{5}$$
Sum of these number = $$\frac {8}{9} + \frac {3}{5} = \frac {67}{45}$$, which is rational number.
Hence, the sum of two rational numbers is a rational number.

15. The sum of two irrational numbers is a rational number.
a) False
b) True

Explanation: Consider two irrational numbers, say, √2 and √5
Sum of these number = √2 + √3 = 3.14626… which is an irrational number.
Hence, the sum of two irrational numbers is an irrational number.

Sanfoundry Global Education & Learning Series – Mathematics – Class 10.

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