Let’s break down this problem step-by-step.

---

Step 1: Set up variables

Let:

• $n$ = number of nickels
• $d$ = number of dimes

Each nickel is worth 5 cents, and each dime is worth 10 cents.

---

Step 2: Create equations for the values

The total value of the coins is $2.75, which is 275 cents.

5n + 10d = 275 \tag{1}

If the nickels were dimes and the dimes were nickels, the total value would be $3.55, or 355 cents.

10n + 5d = 355 \tag{2}

---

Step 3: Solve the system of equations

We have two equations:

1. $5n + 10d = 275$
2. $10n + 5d = 355$

Step 3.1: Simplify both equations

Divide both sides of Equation (1) by 5:

n + 2d = 55 \tag{3}

Divide both sides of Equation (2) by 5:

2n + d = 71 \tag{4}

Step 3.2: Solve Equation (3) for $n$

From Equation (3):

n = 55 - 2d \tag{5}

Step 3.3: Substitute Equation (5) into Equation (4)

Substitute $n = 55 - 2d$ into Equation (4):

$2(55 - 2d) + d = 71$

Simplify:

$110 - 4d + d = 71$

$110 - 3d = 71$

$-3d = 71 - 110$

$-3d = -39$

$d = 13$

Step 3.4: Find $n$

Now that we know $d = 13$, substitute it into Equation (5):

$n = 55 - 2(13) = 55 - 26 = 29$

---

Step 4: Solution

The number of nickels in the bank is:

$\boxed{29}$

---

Step 5: Verification

• Original configuration: $29$ nickels and $13$ dimes.

  • Value: $5(29) + 10(13) = 145 + 130 = 275$ cents (Correct).

• Swapped configuration: $13$ nickels and $29$ dimes.

  • Value: $5(13) + 10(29) = 65 + 290 = 355$ cents (Correct).

Thus, the solution is correct: 29 nickels.

---

Would you like a deeper breakdown of any step?

Here are five follow-up questions for further practice:

1. How many dimes are in the bank?
2. What would the total value be if there were 50 nickels and 5 dimes?
3. How would the solution change if the total value was $3.00 instead of $2.75?
4. How can systems of equations help in other coin problems?
5. What is the role of substitution in solving systems of equations?

Tip: When solving word problems with coins, focus on creating clear equations based on the coin values and quantities!