Fund A accumulate at 8% effective and fund B accumulates at 10% effective. at the end of 20 years, the total of the two funds is $3000. at the end of 10 years the amount in fund A is half the in Fund B. what is the total of the two funds at the end of 5 years
To solve this problem, we need to use the concept of compound interest. Compound interest is the interest that is calculated on both the initial principal and the accumulated interest from previous periods.
Let's start by finding the amount in each fund after 10 years. We know that the amount in Fund A is half of the amount in Fund B. Let's assume the amount in Fund B after 10 years is x dollars. Therefore, the amount in Fund A after 10 years would be x/2 dollars.
Now, we can calculate the amounts in each fund after 20 years. Since compound interest is calculated based on the initial principal and the accumulated interest, we can use the following formula:
A = P(1 + r/n)^(nt)
Where:
A is the final amount
P is the principal amount (initial investment)
r is the annual interest rate (in decimal form)
n is the number of times that interest is compounded per year
t is the number of years
For Fund A after 20 years:
A_A = (x/2)(1 + 0.08/1)^(1*20)
For Fund B after 20 years:
A_B = x(1 + 0.10/1)^(1*20)
Now, we know that the total of the two funds after 20 years is $3000. Therefore, we can set up the following equation:
A_A + A_B = 3000
Substituting the calculated values, we have:
(x/2)(1 + 0.08/1)^(1*20) + x(1 + 0.10/1)^(1*20) = 3000
Now, the total of the two funds at the end of 5 years can be calculated using the same formula. We just need to replace the value of t with 5 in the equation:
A_A' = (x/2)(1 + 0.08/1)^(1*5)
A_B' = x(1 + 0.10/1)^(1*5)
Finally, we can find the total amount in both funds by adding A_A' and A_B':
Total = A_A' + A_B'