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Difficulty: HardSimple Interest

A developer issues a short-term promissory note to cover construction costs. The note carries an annual simple interest rate of 8%8\%. After 99 months, the developer repays a total of $42,400\$42,400, which includes both the original principal borrowed and the interest earned. What was the original principal amount borrowed, in dollars?

Answer: 40000

Answer

40000
The original principal is found by utilizing the total amount formula for simple interest, A=P(1+rt)A = P(1 + rt), and solving for PP. By substituting A=42,400A = 42,400, r=0.08r = 0.08, and t=0.75t = 0.75 years (from 99 months), the equation simplifies to 1.06P=42,4001.06P = 42,400, which yields a principal of $40,000.

Step-by-Step Solution

1
Identify the known variables from the word problem.
Total Amount (AA) = 42,40042,400, Rate (rr) = 8%=0.088\% = 0.08, Time (tt) = 99 months.
Extracting the specific numbers and their corresponding variables is required before applying any formulas.
2
Convert the time duration from months into years.
t=912=0.75t = \frac{9}{12} = 0.75 years.
Simple interest formulas require the time unit to match the rate unit, which is annual (8%8\% per year).
3
Apply the total amount simple interest formula and substitute the values.
42,400=P(1+0.08×0.75)42,400 = P(1 + 0.08 \times 0.75)
The total amount paid back (42,40042,400) includes both the principal (PP) and the interest (PrtPrt), factored out as P(1+rt)P(1 + rt).
4
Simplify the equation.
42,400=P(1+0.06)⇒42,400=1.06P42,400 = P(1 + 0.06) \Rightarrow 42,400 = 1.06P
Multiplying the rate by the time yields the total interest multiplier (6%6\% or 0.060.06), adding it to 11 gives the total amount multiplier.
5
Solve for the principal (PP).
P=42,4001.06=40,000P = \frac{42,400}{1.06} = 40,000
Isolating PP gives the original amount borrowed. The division can be calculated mentally by noting that 424424 is 4×1064 \times 106.

Key Concept

Solving for Principal in a Simple Interest Total Amount Scenario
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Topics

Simple Interest

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