Question

If the denominator of a fraction has only two factors of 2 and factors of 5, the decimal expression…………..
A. Has equal numerator and denominator.
B. Becomes a whole number.
C. Does not terminate.
D. Terminates.

Answer 1

Hint: According to the question, we have to find the decimal expression If the denominator of a fraction has only two factors of 2 and factors of 5. So, first of all we have to understand the prime factorization of the denominator which is mentioned below:
Prime factorization method: first start by dividing the number by first prime number 2 and continue dividing by 2 until you get a decimal or remainder. Then divide by 3,5,7, etc, until the only numbers left are prime numbers. Then write the numbers as a product of prime numbers. Then check which option is correct for that prime factorized method which we have done. Then write the repeated prime factors in the form of power and check that form by the option.
Let’s we can understand above method by a simple example which is mentioned below:
Example: prime factorization of 300.
Start by dividing the number by first prime number 2 and continue dividing by 2 then divide by 3,5,7, etc, until the only numbers left are prime numbers.
Factors of 300 $ = 2 \times 2 \times 3 \times 5 \times 5$
$ \Rightarrow {2^2} \times {3^1} \times {5^2}$

Complete step-by-step solution:
Step 1: As given in the question that the denominator of a fraction has only two factors of 2 and factors of 5. So, first of all we have to let the fraction in which the denominator has only two factors of 2 and factors of 5.
So, let the fraction is $\dfrac{1}{{125}}$
Step 2: Now, we have to use the prime factorization method which is mentioned in the solution hint for the denominator of fraction which we have let.
Factors of 125 $ = 5 \times 5 \times 5$
Now, as we know that the power 0 of any number is equals to zero so, we have to write the above factors in the power of 2 and 5 as mentioned below:
$
   \Rightarrow 1 \times {5^3} \\
   \Rightarrow {2^0} \times {5^3} \\
 $
Step 3: So, we have to get the prime factorization of the fraction which we have to let,
$\dfrac{1}{{125}}$$ = \dfrac{1}{{{2^0} \times {5^3}}}$
Now, we can check by the option that the powers of 2 and 5 have a definite number that’s why the fraction value comes to an end after a few repetitions after the decimal point. Hence, it is a terminating decimal.
Hence, the decimal expression is terminated.

Hence, option (D) is correct.

Note: It is necessary to understand about the prime factorized method to find the prime factors of the fraction which we have to let.
It is necessary to let a fraction which has denominator a factors of 2 and 5 both.