Determine the fraction of the amount remaining each month.

Here are the solutions for question 10: a) What fraction remains after 6 months? Step 1: Determine the fraction of the amount remaining each month. Each month, half of the remaining amount is given away, so the fraction remaining is $\frac{1}{2}$. Step 2: Calculate the fraction remaining after 6 months. After 6 months, the fraction remaining will be $\left(\frac{1}{2}\right)^6$. $$ \left(\frac{1}{2}\right)^6 = \frac{1^6}{2^6} = \frac{1}{64} $$ The fraction remaining after 6 months is $\boxed{\frac{1}{64}}$. b) What fraction remains after 12 months? Step 1: Calculate the fraction remaining after 12 months. After 12 months, the fraction remaining will be $\left(\frac{1}{2}\right)^{12}$. $$ \left(\frac{1}{2}\right)^{12} = \frac{1^{12}}{2^{12}} = \frac{1}{4096} $$ The fraction remaining after 12 months is $\boxed{\frac{1}{4096}}$. c) Write each fraction as a power of 2 with a negative exponent. Step 1: Express the fraction from part a) as a power of 2 with a negative exponent. $$ \frac{1}{64} = \frac{1}{2^6} = 2^{-6} $$ Step 2: Express the fraction from part b) as a power of 2 with a negative exponent. $$ \frac{1}{4096} = \frac{1}{2^{12}} = 2^{-12} $$ The fractions as powers of 2 with negative exponents are $\boxed{2^{-6}}$ and $\boxed{2^{-12}}$. d) What amount is remaining at the end of the year? Step 1: Identify the initial amount and the fraction remaining at the end of the year. The initial amount is $\$1\,000\,000$. The end of the year means after 12 months. From part b), the fraction remaining is $\frac{1}{4096}$. Step 2: Calculate the remaining amount. $$ \text{Remaining amount} = \$1\,000\,000 \times \frac{1}{4096} $$ $$ = \frac{\$1\,000\,000}{4096} $$ $$ = \$244.140625 $$ Step 3: Round the amount to two decimal places for currency. $$ \approx \$244.14 $$ The amount remaining at the end of the year is $\boxed{\$244.14}$. That's 2 down. 3 left today — send the next one.