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
View Answer

Answer: b
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.
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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
View Answer

Answer: c
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) \(\frac {2}{5}\)
View Answer

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

Answer: d
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) \(\frac {2}{5}\)
View Answer

Answer: c
Explanation:
10x = 1.969696…..(1)
1000x = 196.9696…(2)
Subtracting (1) from (2)
We get,
990x=195
x = \(\frac {195}{990} = \frac {13}{66}\)
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6. The numbers of the form \(\frac {p}{q}\) are integers, and q≠0 are called irrational number.
a) True
b) False
View Answer

Answer: b
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
View Answer

Answer: b
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.
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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…..
View Answer

Answer: a
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
View Answer

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

Answer: b
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
View Answer

Answer: b
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
View Answer

Answer: b
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
View Answer

Answer: b
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
View Answer

Answer: b
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
View Answer

Answer: a
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.

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Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter