Question

Convert the following repeating decimal to a fraction in simplest form. 0.73 (73 is repeated)

Answer 1

To convert the repeating decimal \( 0.737373...\) to a fraction, we can use a common algebraic method.

Let:
\[
x = 0.737373...\ \text{(This is our repeating decimal)}
\]

Since the two digits "73" repeat, we can multiply \( x \) by 100 (which shifts the decimal point two places to the right):
\[
100x = 73.737373...
\]

Next, we can set up an equation using these two expressions for \( x \):
\[
100x = 73.737373...\
\]
\[
x = 0.737373...
\]

Now, we subtract the second equation from the first:
\[
100x - x = 73.737373... - 0.737373...
\]
This results in:
\[
99x = 73
\]

To solve for \( x \), we divide both sides by 99:
\[
x = \frac{73}{99}
\]

Now, we need to check if \(\frac{73}{99}\) can be simplified. The greatest common divisor (GCD) of 73 and 99 is 1, since 73 is a prime number and does not divide 99.

Thus, the fraction \(\frac{73}{99}\) is already in its simplest form.

Therefore, the repeating decimal \( 0.737373...\) converted to a fraction in simplest form is:
\[
\boxed{\frac{73}{99}}
\]