Let:

• $x$ represent the number of nickels.
• $y$ represent the number of dimes.

We can set up two equations based on the given conditions.

First Equation:

The total value of the coins in the bank is $2.65. Since nickels are worth $0.05 each and dimes are worth $0.10 each, we can write the first equation as: $0.05x + 0.10y = 2.65$ Multiply through by 100 to eliminate decimals: $5x + 10y = 265$ Simplifying: $x + 2y = 53 \quad \text{(Equation 1)}$

Second Equation:

If the nickels were dimes and the dimes were nickels, the total value of the coins would be $3.50. So, the second equation becomes: $0.10x + 0.05y = 3.50$ Again, multiply through by 100 to eliminate decimals: $10x + 5y = 350$ Simplifying: $2x + y = 70 \quad \text{(Equation 2)}$

Solving the System of Equations:

Now, we have the system:

1. $x + 2y = 53$
2. $2x + y = 70$

To solve this system, we can use substitution or elimination. Let's solve for $y$ in the second equation: $y = 70 - 2x$ Substitute this expression for $y$ into the first equation: $x + 2(70 - 2x) = 53$ Simplify: $x + 140 - 4x = 53$ $-3x + 140 = 53$ Subtract 140 from both sides: $-3x = -87$ Divide by -3: $x = 29$

Thus, the number of nickels in the bank is $\boxed{29}$.

Finding $y$:

Substitute $x = 29$ into $y = 70 - 2x$: $y = 70 - 2(29) = 70 - 58 = 12$

So, there are 29 nickels and 12 dimes.

Would you like more details or have any questions?

Related Questions:

1. How can you check if the solution is correct by plugging the values back?
2. What other systems of equations methods can be used to solve such problems?
3. What would the total be if the dimes were pennies instead of nickels?
4. How does changing the coin values affect the system of equations?
5. Can you use matrix methods to solve this system of equations?

Tip:

When solving word problems, carefully translate each condition into a mathematical equation before solving the system.