Question

If the denominator of a fraction has factors other than 2 and 5, the decimal expression
A. Repeats
B. Is that of a whole number
C. Has equal numerator and denominator
D. Terminates

Answer 1

Hint: We will take an example of a fraction whose denominator has factors as 2 and 5, also factors 2 or 5. Then, we will take an example of a fraction whose denominator has factors other than 2 and 5. Now, we will divide and get the decimal form of the results for each case and observe its decimal expansion. This will give us the answer.

Complete step-by-step answer:
We have been given that, if the denominator of a fraction has factors other than 2 and 5 then, we have to determine the nature of its decimal expansion.
Let us take some examples as follows:
$$(i){\displaystyle \frac{1}{20}}={\displaystyle \frac{1}{2\times 2\times 5}}=0.05$$
Here, we have the factors of the denominator as 2 and 5. The result obtained in decimal form is 0.05. So, we can see that the decimal terminates in this case.
$$(ii){\displaystyle \frac{1}{6}}={\displaystyle \frac{1}{3\times 2}}$$
Here, there are prime factors in the denominator other than 2 or 5. We have factor 3 also.
Now, the result of dividing it is $${\displaystyle \frac{1}{6}}=0.166666...$$
We can clearly see that the decimal is non-terminating and is repeating.
$$(iii){\displaystyle \frac{1}{21}}={\displaystyle \frac{1}{3\times 7}}$$
Here, the prime factors of the denominator are $$3\times 7$$ .
Dividing the fraction, we get $${\displaystyle \frac{1}{21}}=0.04761904761......$$
Here also, we can see that the decimal is non-terminating and is repeating.
Hence, we can conclude that if the denominator of a fraction has factors other than 2 and 5, then, the decimal expansion repeats.
Therefore, the correct option is A.

Note: Remember that, if the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, then the decimal expression terminates and if there is any prime factor in the denominator other than 2 or 5, then the decimal expression repeats. Options b and c are not relevant here and can be eliminated. The result of the fraction need not be a whole number always and if the denominator and numerator are the same, the result would be 1.