A coin bank contains only nickels and dimes. The total value of the coins in the bank is $2.75. If the nickels were dimes and the dimes were nickels, the total value of the coins would be $3.55. Find the number of nickels in the bank.
 nickels

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!

A coin bank contains only nickels and dimes. The total value of the coins in the bank is $2.75. If the nickels were dimes and the dimes were nickels, the total value of the coins would be $3.55. Find the number of nickels in the bank.

Learn how to solve a word problem involving nickels and dimes using systems of equations. This problem explores algebraic methods such as substitution to determine the number of nickels in a coin bank with a total value of $2.75. Suitable for Grades 8-10.

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