The decimal equivalent of 9/4 is a repeating decimal.
True
False
The Correct Answer and Explanation is :
False. The decimal equivalent of ( \frac{9}{4} ) is not a repeating decimal; it is a terminating decimal.
To understand why this is the case, let’s first perform the division of ( 9 ) by ( 4 ). When you divide ( 9 ) by ( 4 ), you find:
[
9 \div 4 = 2.25
]
This division results in ( 2.25 ), which is a terminating decimal. A terminating decimal is one that has a finite number of digits after the decimal point. In this case, ( 2.25 ) has two digits after the decimal point: ( 2. ) and ( 5 ).
On the other hand, a repeating decimal is one that has an infinite number of digits after the decimal point that repeat in a pattern. For example, the fraction ( \frac{1}{3} ) equals ( 0.333… ), where the digit ( 3 ) repeats infinitely.
The distinction between terminating and repeating decimals can be better understood by looking at the properties of fractions. A fraction in its simplest form will result in a terminating decimal if the denominator (when fully simplified) has only the prime factors of ( 2 ) and/or ( 5 ). In contrast, if the denominator contains any prime factors other than ( 2 ) and ( 5 ), the decimal will be repeating.
In the case of ( \frac{9}{4} ), the denominator ( 4 ) can be factored into ( 2^2 ). Since ( 4 ) contains only the prime factor ( 2 ), it confirms that ( \frac{9}{4} ) will produce a terminating decimal.
Thus, the statement that ( \frac{9}{4} ) is a repeating decimal is false; it is a terminating decimal, specifically ( 2.25 ).