RBSE Solutions Class 10 Maths Chapter 2 Real Numbers Exercise 2.4

Question 1. Without actually performing the long division method, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
(i) \( \frac{15}{1600} \)
(ii) \( \frac{13}{3125} \)
(iii) \( \frac{23}{2^3 \times 5^2} \)
(iv) \( \frac{17}{6} \)
(v) \( \frac{129}{2^2 \times 5^7 \times 7^5} \)
(vi) \( \frac{35}{50} \)
(vii) \( \frac{7}{80} \)
Answer:
(i) \( \frac{15}{1600} \): To determine the decimal expansion type, we first look at the prime factors of the denominator. We can write \( 1600 = 16 \times 100 = 2^4 \times (2^2 \times 5^2) = 2^6 \times 5^2 \). Since the denominator \( 1600 \) only has prime factors of 2 and 5, its decimal expansion is terminating. This means the decimal form will have a finite number of digits.
(ii) \( \frac{13}{3125} \): The denominator \( 3125 \) can be expressed as \( 5 \times 5 \times 5 \times 5 \times 5 = 5^5 \). Since the denominator \( 3125 \) only has the prime factor 5 (which fits the form \( 2^m \times 5^n \) where \( m=0 \)), its decimal expansion is terminating. A decimal that stops is easy to work with.
(iii) \( \frac{23}{2^3 \times 5^2} \): The denominator for this rational number is already given in the form \( 2^m \times 5^n \), where \( m=3 \) and \( n=2 \). Because the prime factors in the denominator are only 2 and 5, this rational number will have a terminating decimal expansion. Numbers like this do not repeat endlessly.
(iv) \( \frac{17}{6} \): For the denominator \( 6 \), its prime factors are \( 2 \times 3 \). Since the denominator includes a prime factor of 3 (which is not 2 or 5), its decimal expansion is non-terminating and repeating. This means the decimal form will have a pattern that keeps repeating.
(v) \( \frac{129}{2^2 \times 5^7 \times 7^5} \): The denominator contains the prime factor 7, along with 2 and 5. Because of this prime factor 7, which is not 2 or 5, its decimal expansion is non-terminating and repeating. A non-2, non-5 prime factor always causes a repeating decimal.
(vi) \( \frac{35}{50} \): First, we should simplify the fraction. \( \frac{35}{50} = \frac{7 \times 5}{10 \times 5} = \frac{7}{10} \). The denominator \( 10 \) has prime factors \( 2 \times 5 \). Since the simplified denominator only has prime factors 2 and 5, its decimal expansion is terminating. This simplifies to a clear 0.7.
(vii) \( \frac{7}{80} \): The denominator \( 80 \) can be written as \( 8 \times 10 = 2^3 \times (2 \times 5) = 2^4 \times 5^1 \). Since the denominator only has prime factors of 2 and 5, its decimal expansion is terminating. This value can be expressed precisely as a decimal.
In simple words: To check if a fraction's decimal form stops or repeats, look at the prime factors of its bottom number. If only 2s and 5s are found, the decimal stops. If any other prime number (like 3 or 7) is found, the decimal will go on forever with a repeating pattern.

🎯 Exam Tip: Always simplify the fraction to its simplest form before analyzing the prime factors of the denominator to correctly identify if it's terminating or non-terminating repeating.

Question 2. Classify the following numbers as having a terminating decimal, non-terminating repeating decimal, or non-terminating non-repeating decimal, and state if they are rational or irrational.
(i) \( 0.120120012000120000... \)
(ii) \( 43.123456789 \)
(iii) \( 27.\overline{142857} \)
Answer:
(i) \( 0.120120012000120000... \): This number has a decimal expansion that does not end (it is non-terminating) and does not show any repeating pattern of digits (it is non-repeating). Because of this, it cannot be written as a simple fraction \( \frac{p}{q} \). Therefore, this number is irrational. Irrational numbers have decimals that go on forever without ever repeating.
(ii) \( 43.123456789 \): This number has a decimal expansion that stops after a certain number of digits (it is terminating). Any terminating decimal can always be expressed in the form \( \frac{p}{q} \). For example, \( 43.123456789 = \frac{43123456789}{1000000000} \). Since it can be written as a fraction of two integers, this number is rational. The denominator in its fractional form will only have prime factors of 2 and 5.
(iii) \( 27.\overline{142857} \): This number has a decimal expansion that does not end (it is non-terminating) but clearly shows a repeating block of digits, which is \( 142857 \). Any number with a non-terminating repeating decimal can be converted into the form \( \frac{p}{q} \). Therefore, this number is rational. Repeating decimals are a key characteristic of rational numbers.
In simple words: Numbers whose decimals stop or repeat a pattern are called rational numbers, as they can be written as fractions. Numbers whose decimals go on forever without any repeating pattern are called irrational numbers.

🎯 Exam Tip: The main way to tell if a number is rational or irrational from its decimal form is to check if it terminates (stops) or repeats. If it does neither, it's irrational.