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Sasha has some pennies, nickels, and dimes in her pocket. The number of coins is 18. The expression [tex]$0.01p + 0.05n + 0.10d$[/tex] represents the value of the coins, which is [tex][tex]$\$[/tex]1.14$[/tex]. She has twice as many dimes as pennies. How many of each coin does Sasha have?

A. 5 pennies, 3 nickels, and 10 dimes
B. 2 pennies, 12 nickels, and 4 dimes
C. 3 pennies, 9 nickels, and 6 dimes
D. 4 pennies, 6 nickels, and 8 dimes



Answer :

GinnyAnswer
To solve the problem, we will create a system of equations based on the information given and solve it step-by-step.

### Step 1: Define the Variables
Let's define the variables:
- [tex]\( p \)[/tex] for the number of pennies
- [tex]\( n \)[/tex] for the number of nickels
- [tex]\( d \)[/tex] for the number of dimes

### Step 2: Set Up the Equations
We are given three key pieces of information which we can translate into equations:

1. The total number of coins is 18:
[tex]\[ p + n + d = 18 \][/tex]

2. The total value of the coins is \$1.14:
[tex]\[ 0.01p + 0.05n + 0.10d = 1.14 \][/tex]

3. Sasha has twice as many dimes as pennies:
[tex]\[ d = 2p \][/tex]

### Step 3: Substitute and Simplify
Using the third equation [tex]\( d = 2p \)[/tex], we can substitute [tex]\( d \)[/tex] into the first and second equations.

1. Substitute [tex]\( d = 2p \)[/tex] into [tex]\( p + n + d = 18 \)[/tex]:
[tex]\[ p + n + 2p = 18 \][/tex]
Simplify:
[tex]\[ 3p + n = 18 \quad \text{(Equation 4)} \][/tex]

2. Substitute [tex]\( d = 2p \)[/tex] into [tex]\( 0.01p + 0.05n + 0.10d = 1.14 \)[/tex]:
[tex]\[ 0.01p + 0.05n + 0.10(2p) = 1.14 \][/tex]
Simplify:
[tex]\[ 0.01p + 0.05n + 0.20p = 1.14 \][/tex]
Combine like terms:
[tex]\[ 0.21p + 0.05n = 1.14 \quad \text{(Equation 5)} \][/tex]

### Step 4: Solve the System of Equations
Now we solve the system of equations consisting of Equation 4 and Equation 5:

- Equation 4: [tex]\( 3p + n = 18 \)[/tex]
- Equation 5: [tex]\( 0.21p + 0.05n = 1.14 \)[/tex]

First, solve Equation 4 for [tex]\( n \)[/tex]:
[tex]\[ n = 18 - 3p \][/tex]

Next, substitute [tex]\( n = 18 - 3p \)[/tex] into Equation 5:
[tex]\[ 0.21p + 0.05(18 - 3p) = 1.14 \][/tex]

Distribute the 0.05:
[tex]\[ 0.21p + 0.90 - 0.15p = 1.14 \][/tex]

Combine like terms:
[tex]\[ 0.06p + 0.90 = 1.14 \][/tex]

Subtract 0.90 from both sides:
[tex]\[ 0.06p = 0.24 \][/tex]

Divide both sides by 0.06:
[tex]\[ p = \frac{0.24}{0.06} = 4 \][/tex]

### Step 5: Calculate [tex]\( n \)[/tex] and [tex]\( d \)[/tex]
Using [tex]\( p = 4 \)[/tex]:

From Equation 4:
[tex]\[ n = 18 - 3(4) = 18 - 12 = 6 \][/tex]

From the given relationship [tex]\( d = 2p \)[/tex]:
[tex]\[ d = 2(4) = 8 \][/tex]

### Conclusion
Sasha has:

- 4 pennies
- 6 nickels
- 8 dimes

So, the correct answer choice is:
- 4 pennies, 6 nickels, and 8 dimes.

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