Math Problem Statement
-
Jimmy invested P55,490 where after 36 months it reached to P68,211.38 compounded semiannually. Find the interest rate?
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How long will it take for Rina's borrowed amount of $45,000 reach to $50,692.17 with an interest rate of 6% compounded quarterly?
Solution
Let's solve the problems step-by-step:
1. Find the interest rate when compounded semiannually
The formula for compound interest is:
Where:
- (future value)
- (initial principal)
- (compounding semiannually per year)
Substituting values into the formula:
Simplify:
Take the 6th root (or raise to the power of ):
Solve for :
The interest rate is approximately 6.86% per year.
2. Find the time required when compounded quarterly
The formula remains:
Where:
- r = 0.06 \, \text{(6% annual interest rate)}
Substitute the known values:
Simplify:
Take the natural logarithm on both sides:
Apply the logarithmic rule :
Solve:
It will take 2 years for the amount to grow from $45,000 to $50,692.17.
Do you want further details or have any questions?
Here are 5 related questions to expand your understanding:
- What happens to the time if the interest rate increases in the second problem?
- How does the compounding frequency affect the interest rate in the first problem?
- How is semiannual compounding different from quarterly compounding in terms of formula adjustments?
- How would the answers change if was given in months instead of years?
- Can you derive a similar formula for simple interest instead of compound interest?
Tip: Always double-check units (years, months, etc.) to ensure the formula is applied correctly!
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Math Problem Analysis
Mathematical Concepts
Compound Interest
Exponential Growth
Logarithms
Formulas
A = P (1 + r/n)^(nt)
ln(A/P) = nt * ln(1 + r/n)
Theorems
Logarithmic properties
Compound interest formula
Suitable Grade Level
Grades 10-12
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