1. **State the problem:** We want to perform long division to divide 10 by 1.618, which is the approximate value of the golden ratio. 2. **Recall the division formula:** Division is finding how many times the divisor fits into the dividend. Here, dividend = 10 and divisor = 1.618. 3. **Set up the long division:** We divide 10 by 1.618. 4. **Perform the division:** - Multiply 1.618 by 6: $1.618 \times 6 = 9.708$ which is less than 10. - Subtract: $10 - 9.708 = 0.292$. - Bring down a zero to get 2.92. - Multiply 1.618 by 1: $1.618 \times 1 = 1.618$. - Subtract: $2.92 - 1.618 = 1.302$. - Bring down another zero to get 13.02. - Multiply 1.618 by 8: $1.618 \times 8 = 12.944$. - Subtract: $13.02 - 12.944 = 0.076$. - Bring down zero to get 0.76. - Multiply 1.618 by 0: $0$. - Subtract: $0.76 - 0 = 0.76$. - Bring down zero to get 7.6. - Multiply 1.618 by 4: $1.618 \times 4 = 6.472$. - Subtract: $7.6 - 6.472 = 1.128$. - Bring down zero to get 11.28. - Multiply 1.618 by 6: $9.708$. - Subtract: $11.28 - 9.708 = 1.572$. 5. **Interpret the result:** The quotient starts as approximately 6.18 and continues with decimals. 6. **Final answer:** The division $10 \div 1.618 \approx 6.18$ (rounded to two decimal places).