Alright, let's walk through the solution step-by-step to determine in how many years the deposited amount will become 10 times the initial amount, given that it becomes 7 times in 15 years at simple interest.
### Step 1: Understanding Simple Interest
The formula for the future amount [tex]\(A\)[/tex] using simple interest is:
[tex]\[ A = P (1 + rt) \][/tex]
where:
- [tex]\(A\)[/tex] is the final amount.
- [tex]\(P\)[/tex] is the principal amount (initial deposit).
- [tex]\(r\)[/tex] is the annual interest rate.
- [tex]\(t\)[/tex] is the time in years.
### Step 2: Finding the Interest Rate
Given that the money becomes 7 times in 15 years, we can set up the equation using the known values:
[tex]\[ 7P = P(1 + r \cdot 15) \][/tex]
Dividing both sides by [tex]\(P\)[/tex]:
[tex]\[ 7 = 1 + 15r \][/tex]
Rearranging to solve for [tex]\(r\)[/tex]:
[tex]\[ 7 - 1 = 15r \][/tex]
[tex]\[ 6 = 15r \][/tex]
[tex]\[ r = \frac{6}{15} \][/tex]
[tex]\[ r = 0.4 \][/tex]
So, the annual interest rate [tex]\(r\)[/tex] is 0.4 (or 40%).
### Step 3: Finding the Time for the Amount to Become 10 Times
Now, we need to determine the number of years [tex]\(t\)[/tex] for the money to become 10 times the initial amount. We set up the equation as follows:
[tex]\[ 10P = P(1 + r \cdot t) \][/tex]
Dividing both sides by [tex]\(P\)[/tex]:
[tex]\[ 10 = 1 + 0.4 \cdot t \][/tex]
Rearranging to solve for [tex]\(t\)[/tex]:
[tex]\[ 10 - 1 = 0.4 \cdot t \][/tex]
[tex]\[ 9 = 0.4 \cdot t \][/tex]
[tex]\[ t = \frac{9}{0.4} \][/tex]
[tex]\[ t = 22.5 \][/tex]
### Results
Therefore, it will take 22.5 years for the initial deposit to become 10 times its original amount at the given annual interest rate of 40%.
