what's .37 (all repeating) as a fraction in simplest form

Answer 1

To convert \(0.373737...\) (which we can denote as \(x\)) into a fraction, we can use the following steps:

1. Let \(x = 0.373737...\).
2. Since the repeating part has 2 digits, multiply both sides by 100: \[ 100x = 37.373737... \]
3. Now, subtract the original \(x\) from this equation: \[ 100x - x = 37.373737... - 0.373737... \] This simplifies to: \[ 99x = 37 \]
4. Now, solve for \(x\): \[ x = \frac{37}{99} \]

Now, let's check if \(\frac{37}{99}\) is in simplest form. The greatest common divisor (GCD) of 37 and 99 is 1 since 37 is a prime number and does not divide 99.

Therefore, the fraction \(\frac{37}{99}\) is already in simplest form.

Thus, the decimal \(0.373737...\) as a fraction in simplest form is: \[ \frac{37}{99} \]