Can the fraction 84/250 be written as a finite decimal? Explain why or why not without dividing.
The correct answer and explanation is :
The fraction ( \frac{84}{250} ) can indeed be written as a finite decimal. To understand why, we need to examine the denominator’s prime factorization and apply a rule regarding fractions that have finite decimal representations.
Step 1: Prime Factorization of the Denominator
To check if a fraction can be written as a finite decimal, we start by factoring the denominator into its prime factors. Let’s break down 250:
[
250 = 2 \times 5^3
]
The key here is that a fraction will have a finite decimal representation if and only if the denominator, when fully simplified, consists only of the prime factors 2 and/or 5. This is because the decimal system is based on powers of 10 (which is ( 2 \times 5 )), so if the denominator has only these factors, the fraction can be written as a terminating (finite) decimal.
Step 2: Prime Factorization of the Numerator
Next, we can look at the numerator, 84, to see if it simplifies with the denominator:
[
84 = 2^2 \times 3 \times 7
]
Step 3: Simplifying the Fraction
Now we simplify ( \frac{84}{250} ) by canceling out any common factors in the numerator and the denominator. We observe that both 84 and 250 share a factor of 2:
[
\frac{84}{250} = \frac{84 \div 2}{250 \div 2} = \frac{42}{125}
]
Now we check the denominator of 125:
[
125 = 5^3
]
Since 125 is composed entirely of the prime factor 5, the fraction ( \frac{42}{125} ) has a denominator with only the prime factor 5, which means it can be written as a finite decimal.
Step 4: Conclusion
Since the denominator of the simplified fraction ( \frac{42}{125} ) consists only of the prime factor 5, the fraction can indeed be written as a finite decimal.
Final Answer:
[
\frac{84}{250} = 0.168
]
The fraction ( \frac{84}{250} ) can be written as the finite decimal 0.168 because its denominator, when simplified, only contains the prime factor 5, satisfying the condition for finite decimals.