If you invest $10,000 at 6.125% interest compounded daily, how long will it take for you to accumulate $15,000? (Give the number of periods and the number of years, rounded to the nearest hundredth.)
How long will it take for you to accumulate $100,000?
Answer (1)
Calculate the daily interest rate: d ai l y _ r a t e = ann u a l _ r a t e /365 = 0.06125/365 .
Use the compound interest formula to find the number of days (n) to reach 15 , 000 : days = \frac{{\ln(15000/10000)}}{{\ln(1 + daily_rate)}} \approx 2416.44$, which is approximately 6.62 years. - Use the compound interest formula to find the number of days (n) to reach 100 , 000 : days = \frac{{\ln(100000/10000)}}{{\ln(1 + daily_rate)}} \approx 13722.68$, which is approximately 37.60 years.
The time to accumulate $15,000 is approximately 2416.44 days or 6.62 years, and the time to accumulate 100 , 000 i s a pp ro x ima t e l y 13722.68 d a ysor 37.60 ye a rs . \boxed{2416.44 \text{ days}, 6.62 \text{ years}; 13722.68 \text{ days}, 37.60 \text{ years}}$
Explanation
Understanding the Problem Let's break down this compound interest problem. We're starting with an initial investment and want to know how long it takes to reach two different target amounts, considering the interest is compounded daily.
The Compound Interest Formula We'll use the compound interest formula to solve this. The formula is: A = P ( 1 + r / n ) n t Where:
A is the future value of the investment/loan, including interest
P is the principal investment amount (the initial deposit or loan amount)
r is the annual interest rate (as a decimal)
n is the number of times that interest is compounded per year
t is the number of years the money is invested or borrowed for
Setting up the Formula In our case:
P = $10,000
r = 6.125% = 0.06125
n = 365 (compounded daily) We want to find 't' for two different values of A: $15,000 and 100 , 000. S in ce w e a reco m p o u n d in g d ai l y , i t ′ se a s i er t oc a l c u l a t e t h e n u mb ero fd a ys f i rs t an d t h e n co n v er t i tt oye a rs . T h e f or m u l a c anb ere a rr an g e d t oso l v e f or t h e n u mb ero f co m p o u n d in g p er i o d s ( n ∗ t ) , w hi c hin o u rc a se i s t h e n u mb ero fd a ys : n × t = l n ( 1 + r / n ) l n ( A / P ) S o , t h e n u mb ero fd a ys i s : d a ys = l n ( 1 + d ai l y _ r a t e ) l n ( A / P ) Wh ere daily_rate = annual_rate / 365$
Calculating Time to Reach $15,000 First, let's calculate the number of days and years to reach 15 , 000 : d a ys = l n ( 1 + 0.06125/365 ) l n ( 15000/10000 ) d a ys = l n ( 1 + 0.0001678 ) l n ( 1.5 ) d a ys ≈ 0.00016778 0.405465 d a ys ≈ 2416.44 ye a rs = 365 2416.44 ≈ 6.62 $
So, it will take approximately 2416.44 days or 6.62 years to reach $15,000.
Calculating Time to Reach $100,000 Now, let's calculate the number of days and years to reach 100 , 000 : d a ys = l n ( 1 + 0.06125/365 ) l n ( 100000/10000 ) d a ys = l n ( 1 + 0.0001678 ) l n ( 10 ) d a ys ≈ 0.00016778 2.302585 d a ys ≈ 13722.68 ye a rs = 365 13722.68 ≈ 37.60 $
So, it will take approximately 13722.68 days or 37.60 years to reach $100,000.
Final Answer Therefore, to accumulate $15,000, it will take approximately 2416.44 days or 6.62 years. To accumulate $100,000, it will take approximately 13722.68 days or 37.60 years.
Examples
Compound interest is a powerful concept that applies to many real-life situations. For example, when you're saving for retirement, understanding compound interest helps you estimate how much your investments will grow over time. Let's say you invest a certain amount each month in a retirement account. The interest earned on those investments also earns interest, leading to exponential growth over the years. This is also applicable when calculating loan payments, understanding how quickly debt can accumulate if you're not careful. By understanding the principles of compound interest, you can make informed decisions about your finances and plan for your future.
