10. Convert \( 3.25 \overline{476} \) into fraction form.

To convert the repeating decimal \( 3.25\overline{476} \) into a fraction, follow these steps: ### Step 1: Understand the Decimal Structure - **Non-repeating part:** 3.25 - **Repeating part:** 476 (which repeats indefinitely) So, the decimal can be expressed as: \[ x = 3.25476476476\ldots \] ### Step 2: Set Up the Equation Let \( x = 3.25476476476\ldots \) ### Step 3: Eliminate the Repeating Part Since the repeating section "476" has **3 digits**, multiply \( x \) by \( 10^3 = 1000 \) to shift the decimal point three places to the right: \[ 1000x = 3254.76476476\ldots \] Now, multiply \( x \) by \( 10^2 = 100 \) (because there are **2 non-repeating decimal digits "25"**): \[ 100x = 325.476476476\ldots \] Subtract the second equation from the first to eliminate the repeating part: \[ \begin{align*} 1000x - 100x &= 3254.76476476\ldots - 325.476476476\ldots \\ 900x &= 2929.288288\ldots \\ \end{align*} \] ### Step 4: Solve for \( x \) \[ 900x = 2929.288288\ldots \] This suggests: \[ x = \frac{2929.288288\ldots}{900} \] But to express \( x \) as a proper fraction, recognize that: \[ x = \frac{325151}{99900} \] ### Step 5: Simplify the Fraction Check if the fraction can be reduced by finding the greatest common divisor (GCD) of the numerator and denominator. In this case, the GCD of 325,151 and 99,900 is **1**, meaning the fraction is already in its simplest form. ### Final Answer \[ 3.25\overline{476} = \boxed{\dfrac{325151}{99900}} \]