karim deposited a sum $ 9000 in a bank after 2 years he withdrew $ 4000 and the end of 5 years he received $ 8300. find the rate of interest on the sum he recieved
at simple interest:
let the rate be r
at the end of 2 years : 9000 + 9000(2)(r)
he withdrew 4000
amount left = 5000 + 9000(2r) = 5000 + 18000r
Three more years :
5000 + 18000r + [5000 + 18000r](3r)
= 5000 + 18000r + 15000r + 54000r^2
= 8300
.....
r =
To find the rate of interest, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the amount received at the end of the period
P is the principal (initial deposit)
r is the interest rate (in decimal form)
n is the number of times the interest is compounded per year
t is the number of years
We can use the information given to solve for the interest rate.
1. Let's first find the amount received after 2 years, when Karim withdrew $4000. We can use the formula to find A.
A = P(1 + r/n)^(nt)
A = $9000(1 + r/n)^(2n)
2. After 5 years, Karim received $8300. We can use the formula again to find A.
A = P(1 + r/n)^(nt)
A = $9000(1 + r/n)^(5n)
3. We can set up a ratio using the amounts received in steps 1 and 2:
($9000 - $4000) / $9000 = A (from step 1) / $8300 (from step 2)
4. Simplify the ratio:
(5000 / 9000) = [(9000(1 + r/n)^(2n)) / 8300]
5. Now solve for r/n, after cross-multiplying and simplifying:
(5000 * 8300 * 9000) / (9000 * 9000) = (1 + r/n)^(2n)
6. Simplify further:
41500000 = (1 + r/n)^(2n)
7. Since the term "r/n" appears in the exponent, we can solve this equation using logarithms. Take the natural logarithm (ln) of both sides of the equation:
ln(41500000) = ln((1 + r/n)^(2n))
8. Apply the power rule of logarithms:
ln(41500000) = 2n * ln(1 + r/n)
9. Isolate "n" by dividing both sides by 2ln(1 + r/n):
ln(41500000) / (2ln(1 + r/n)) = n
10. Now we have found the value of "n". Let's substitute it back into the equation in Step 4 to solve for r/n.
(5000 * 8300 * 9000) / (9000 * 9000) = (1 + r/n)^(2 * n)
11. Simplify:
41500000 = (1 + r/n)^(2n)
12. Finally, solve for r/n by taking the 2n-th root of each side:
(1 + r/n) = (41500000)^(1 / 2n)
13. Now we know (1 + r/n), we can solve for r. Subtract 1 from both sides:
r/n = (41500000)^(1 / 2n) - 1
14. Multiply both sides by n:
r = n * [(41500000)^(1 / 2n) - 1]
These calculations will give you the rate of interest (r) on the sum Karim received. Substitute the value of "n" obtained in Step 9 to get the final result.