Question

Which of the following rational numbers in \[\dfrac{p}{q}\] form \[2.\bar 1\bar 7\].
A) \[\dfrac{{215}}{{909}}\]
B) \[\dfrac{{215}}{{999}}\]
C) \[\dfrac{{215}}{{99}}\]
D) \[\dfrac{{215}}{{19}}\]

Answer 1

Hint: Rational number is a number that can be expressed as the quotient or fraction p/q of two integers. The decimal expansion of a rational number either terminates after a finite number of digits, or begins to repeat the same finite sequence of digits over and over.

Complete step-by-step answer:
We can check each option or can directly cancel out those options that won’t satisfy our requirements.
Method 1:
Check them one by one.
a. \[\dfrac{{215}}{{909}} = 0.2365\]
b. \[\dfrac{{215}}{{999}} = 0.2152\]
c. \[\dfrac{{215}}{{99}} = 2.1717\]
d. \[\dfrac{{215}}{{19}} = 11.31\]
Thus, it is clear that option c is the correct answer.
Method 2:
Our requirement is that the denominator must be 100 or equal to hundred.
Because the decimal digits are two. That is there are two digits after the decimal point and are recurring.
So here we eliminate option a and b. Because their denominators are greater than 100.
Let’s check for c and d.
c. \[\dfrac{{215}}{{99}} = 2.1717\]
d. \[\dfrac{{215}}{{19}} = 11.31\]
Thus, it is clear that option c is the correct answer

Additional Information:
Bar on digits indicates that they are recurring.
Recurring means they are repeating there on.
Decimal is nothing but division of a number by 10 or powers of 10.

Note: If there is variation in options students can get the correct answer easily.
If the options appear to be the same then check them one by one.
If you are very clear with your concept then you can go for the elimination method.