Question: In a jar containing $x$ dimes and $y$ pennies, the total amount is $9.16$. There are $81$ coins in the jar. Which equation below best represents the situation?
$0.10x + 0.01y = 9.16$
$x + y = 81$
$0.01x + 0.10y = 9.16$
$x + y = 81$
$0.10x + 0.01y = 81$
$x + y = 9.16$
$x + 25y = 9.16$
$x + y = 81$
1. **State the problem:** We have $x$ dimes and $y$ pennies in a jar.
2. **Given:** Total amount of money is $9.16$ dollars and total number of coins is $81$.
3. **Formulate equations:**
- Each dime is worth $0.10$ dollars, so total value from dimes is $0.10x$.
- Each penny is worth $0.01$ dollars, so total value from pennies is $0.01y$.
- Total value equation: $$0.10x + 0.01y = 9.16$$
- Total coins equation: $$x + y = 81$$
4. **Check other options:**
- $0.01x + 0.10y = 9.16$ swaps values of dimes and pennies, which is incorrect.
- $0.10x + 0.01y = 81$ incorrectly uses total coins as money amount.
- $x + 25y = 9.16$ incorrectly multiplies pennies by 25 and mixes units.
5. **Conclusion:** The correct system of equations representing the situation is:
$$\begin{cases} 0.10x + 0.01y = 9.16 \\ x + y = 81 \end{cases}$$