Question

Convert the following into fraction of simplest form: 0.085

Answer 1

Hint: We know that any number, not in the fraction form, can be represented as a fraction with 1 as the denominator. And then, we can multiply both the numerator and denominator to free them from the decimal point, and then cancel the common factors to get the simplest form.

Complete step by step solution:
We know that the decimal point shifts to the right on multiplication by the multiples of 10, and shifts to the left on division by multiples of 10.
So, if we multiply by 10, the decimal will shift right one place. And if we multiply by 100, the decimal point will shift right by two places. Similarly, if we multiply the number by 1000, the decimal point will shift right by 3 places.
We know very well that the number 0.085 is equivalent to the fraction $\dfrac{0.085}{1}$.
Hence, we can write
$0.085=\dfrac{0.085}{1}...\left( i \right)$
Now, let us multiply 1000 into the right hand side of equation (i) in both the numerator and denominator. So, we get
$0.085=\dfrac{0.085\times 1000}{1\times 1000}$
Thus, on simplification, we can write
$0.085=\dfrac{85}{1000}$
So, we have converted the number 0.085 into a fraction, but this fraction is not in the simplest form.
So, let us write the prime factorization of numerator 85 and denominator 1000.
$85=5\times 17$
$1000=2\times 2\times 2\times 5\times 5\times 5$
From these two prime factorizations, we can clearly see that the only common factor we have is 5.
Hence, by cancelling both the numerator and denominator by 5, we get
$0.085=\dfrac{17}{200}$
So, there are no common factors for 17 and 200, except 1. Hence, the fraction is in its simplest form.

Thus, we can say that 0.085 can be expressed as a fraction in its simplest form as $\dfrac{17}{200}$.

Note:
We know that a fraction is in its simplest form, when the numerator and denominator have no common factors, other than 1. Also it is worth mentioning that any number that has no significant digit after the decimal point, like in 85.0, can be expressed without the decimal point, 85 in this case.