U uestions Con ith 7] repeating. Sol. A rational number is either terminating or non terminating repeating. An irrational number is non-terminating and non- [CBSE Marking Scheme, 2016] 3 U Q. 4. Calculate\frac{3}{8} in the decimal form. 8 3 Sol. \frac{3}{8} = \frac{3}{2^3} 3\times5^3
The Correct Answer and Explanation is:
To calculate the decimal form of the fraction 38\frac{3}{8}, we proceed as follows:
38=323\frac{3}{8} = \frac{3}{2^3}
Since the denominator is composed only of the prime factor 2, this indicates that the fraction is a terminating decimal. Let’s perform the division:
38=0.375\frac{3}{8} = 0.375
Explanation
A rational number is defined as a number that can be expressed as the ratio of two integers where the denominator is not zero. Rational numbers include both terminating and repeating decimals. A decimal terminates if, when the fraction is fully simplified, its denominator has no prime factors other than 2 or 5. If any other prime factor remains in the denominator, the decimal form will be repeating and non-terminating.
In the case of 38\frac{3}{8}, the denominator is 8, which equals 232^3. It contains only the prime factor 2, making it a rational number with a terminating decimal.
To understand this deeper, consider converting fractions to decimal form by dividing the numerator by the denominator. When we divide 3 by 8, we get:
3÷8=0.3753 \div 8 = 0.375
This means that 38=0.375\frac{3}{8} = 0.375 exactly. There is no repeating part, and the decimal ends after three digits.
In contrast, irrational numbers cannot be written as fractions of integers. They have decimal expansions that are both non-terminating and non-repeating. Examples include π\pi and 2\sqrt{2}. These numbers defy exact expression in decimal or fractional form.
To summarize, 38\frac{3}{8} is a rational number with a terminating decimal, and its exact decimal representation is:
