To convert the decimal \( 4.372372372372372732... \) into a fraction, we can break it down into a whole part and a repeating part.
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Separate the whole part and the repeating decimal:
- The whole number part is \( 4 \).
- The decimal part is \( 0.372372372372... \).
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Let \( x = 0.372372372372... \):
- This can be expressed as follows: \[ x = 0.372372372372... \]
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Multiply \( x \) by \( 1000 \) to shift the decimal point three places to the right: \[ 1000x = 372.372372372372... \]
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Now, subtract \( x \) from \( 1000x \): \[ 1000x - x = 372.372372372372... - 0.372372372372... \] \[ 999x = 372 \]
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Solve for \( x \): \[ x = \frac{372}{999} \]
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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 \( 41/111 \).
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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:
\[ \boxed{\frac{485}{111}} \]