9. Rounded to the nearest dime, what is the greatest amount of money that rounds to \( \$ 105.40 \) ? What is the least amount of money that rounds to \( \$ 105.40 \) ? Explain your answers.

To determine the greatest and least amounts of money that round to \( \$ 105.40 \) when rounded to the nearest dime, we need to understand how rounding works. When rounding to the nearest dime, we look at the hundredths place (the second digit after the decimal point). If this digit is 5 or greater, we round up; if it is less than 5, we round down. ### Greatest Amount To find the greatest amount that rounds to \( \$ 105.40 \): 1. The amount must be less than halfway to the next dime, which is \( \$ 105.50 \). 2. The highest value that still rounds down to \( \$ 105.40 \) is \( \$ 105.445 \) (since \( 105.445 \) rounds to \( 105.40 \)). 3. However, we need to round to the nearest dime, so we consider the next value that would round to \( 105.50 \), which is \( 105.449999... \) (or \( 105.45 \) when rounded to the nearest cent). Thus, the greatest amount of money that rounds to \( \$ 105.40 \) is: \[ \$ 105.449999... \text{ (or simply } \$ 105.45 \text{)} \] ### Least Amount To find the least amount that rounds to \( \$ 105.40 \): 1. The amount must be at least halfway to \( \$ 105.40 \) from the previous dime, which is \( \$ 105.30 \). 2. The lowest value that rounds up to \( \$ 105.40 \) is \( \$ 105.345 \) (since \( 105.345 \) rounds to \( 105.40 \)). 3. However, we need to round to the nearest dime, so we consider the next value that would round down to \( 105.30 \), which is \( 105.349999... \) (or \( 105.35 \) when rounded to the nearest cent). Thus, the least amount of money that rounds to \( \$ 105.40 \) is: \[ \$ 105.345 \text{ (or simply } \$ 105.35 \text{)} \] ### Summary - **Greatest amount that rounds to \( \$ 105.40 \)**: \( \$ 105.449999... \) (or \( \$ 105.45 \)) - **Least amount that rounds to \( \$ 105.40 \)**: \( \$ 105.345 \) (or \( \$ 105.35 \))