express the given number as a rational number.

Question

express the given number as a rational number.

3.816161616161616161616161616...

Answer 1

Let's write this as 3.8 + 0.016 + 0.00016 + 0.0000016 + . . .

Ignoring the "3.8" for the moment, we see we have an infinite geometric series where the first term (a₁) is 0.016 and the multiplier to go from one term to the next is 1/100 (0.01).

Since the absolute value of the multiplier is < 1, this series converges and the sum is as follows:

S = a₁/(1 - r)

S = 0.016/(1 - 0.01)

S = 0.016/0.99

We can multiply the top and bottom of this fraction by 1000/1000 to get 16/990 = 8/495.

Now, we need to add in the 3.8 we left out. 3.8 = 38/10 = 19/5 so we add 19/5 and 8/495 by getting a common denominator of 495:

19/5•99/99 = 1881/495 and 1881/495 + 8/495 = 1889/495

So the fractional equivalent to 3.81616... is 1889/495