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Magazine Chapter 2: Problem 9
Solve \(A=P+\) Prt for \(A\), given that \(P=\$ 1000\), \(r=12 \%\), and \(t=5\) years.
Short Answer
Expert verified
The final amount \(A\) is $1600.
Step by step solution
01
Identify the Formula
The exercise involves using the formula for simple interest: \[ A = P + Prt \] This can also be expressed as: \[ A = P (1 + rt) \] where: - \(A\) is the final amount- \(P\) is the principal amount- \(r\) is the interest rate (expressed as a decimal)- \(t\) is the time in years.
02
Convert Percentage Rate to Decimal
The interest rate \(r\) is given as 12%, which needs to be converted to a decimal for calculations.To do this, divide the percentage by 100:\[ r = \frac{12}{100} = 0.12 \]
03
Substitute Known Values
Substitute the known values into the formula.Given: \(P = 1000\), \(r = 0.12\), \(t = 5\).The formula becomes: \[ A = 1000 + 1000 \times 0.12 \times 5 \]
04
Calculate the Interest
Calculate the interest by multiplying the values of \(P\), \(r\), and \(t\):\[ Prt = 1000 \times 0.12 \times 5 \]\[ Prt = 1000 \times 0.6 \]\[ Prt = 600 \]
05
Compute the Final Amount
Add the calculated interest back to the principal:\[ A = 1000 + 600 \]\[ A = 1600 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mathematical Formulas
Formulas provide a structure to solve mathematical problems. In the context of simple interest calculations, the formula you need is straightforward:
- Complete formula: \( A = P + Prt \)
- Simplified formula: \( A = P(1 + rt) \)
Percentage Conversion
Understanding percentages and how to convert them to decimals is crucial. This is especially important in financial calculations like interest. A percentage is simply a fraction of 100. For example, converting 12% into a decimal requires dividing by 100. This process helps to simplify calculations.
- Original percentage: 12%
- Conversion formula: \( r = \frac{12}{100} \)
- Result as a decimal: \( r = 0.12 \)
Interest Calculation
Calculating interest is a step-by-step process that combines different values like the principal amount, interest rate, and time. Here's how you break it down:First, substitute the known values into the formula: \( P = 1000 \), \( r = 0.12 \), \( t = 5 \).The calculation steps are:
- Calculate \( Prt \): \( 1000 \times 0.12 \times 5 \)
- Intermediate steps yield \( Prt = 1000 \times 0.6 \)
- Final result of \( Prt = 600 \)
Algebraic Substitution
Algebraic substitution is the step where you place known values into a formula to find an unknown quantity. In the case of simple interest, you've already outlined the formula. The next step is using this formula:Substitution requires plugging the values of \( P \), \( r \), and \( t \) into the equation:
- Start by positioning values \( P = 1000 \), \( r = 0.12 \), \( t = 5 \) into \( A = P + Prt \)
- The equation then reads \( A = 1000 + 1000 \times 0.12 \times 5 \)
- Solve to find \( A = 1600 \)
