Question

What is the fraction form of the following decimal expansion? 4.372372372372372732...

Answer 1

To convert the decimal \( 4.372372372372372732... \) into a fraction, we can break it down into a whole part and a repeating part.

1. **Separate the whole part and the repeating decimal:**
- The whole number part is \( 4 \).
- The decimal part is \( 0.372372372372... \).

2. **Let \( x = 0.372372372372... \):**
- This can be expressed as follows:
\[
x = 0.372372372372...
\]

3. **Multiply \( x \) by \( 1000 \) to shift the decimal point three places to the right:**
\[
1000x = 372.372372372372...
\]

4. **Now, subtract \( x \) from \( 1000x \):**
\[
1000x - x = 372.372372372372... - 0.372372372372...
\]
\[
999x = 372
\]

5. **Solve for \( x \):**
\[
x = \frac{372}{999}
\]

6. **Simplify the fraction \( \frac{372}{999} \):**
- The greatest common divisor (GCD) of \( 372 \) and \( 999 \) is \( 9 \).
- Dividing both the numerator and the denominator by \( 9 \):
\[
\frac{372 \div 9}{999 \div 9} = \frac{41.3333...}{111}
\]
This simplifying shows that \( 372/999 \) reduces to \( 41/111 \).

7. **Now, combine the whole number part with the fractional part:**
- Therefore, the full number is:
\[
4 + \frac{41}{111} = \frac{4 \times 111 + 41}{111} = \frac{444 + 41}{111} = \frac{485}{111}
\]

Thus, the decimal \( 4.372372372372372732... \) can be expressed in fractional form as:

\[
\boxed{\frac{485}{111}}
\]