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Sagot :
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.
### 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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