An investment of $3,000 is made at an annual simple interest of 5%. How much additional money must be invested at an annual simple interest rate of 8%, so that the total annual interest earned is 7.5 of the original amount that you invested?
My answer: $279,375
.05(3000) + .08x = .075(3000+x)
150 + .08x = 225 + .075x
.005x = 75
x = 15,000
check:
total invested = 3000+15000 = 18000
.075(18000) = 1350
interest at partial investments
= .05(3000) + .08(15000)
= 150 + 1200
= 1350
How did you possible get $279,375 ???
Did that sound logical to you ?
To solve this problem, we need to break it down into steps. Let's go through each step:
Step 1: Calculate the interest earned on the initial investment.
To find the interest earned on the initial investment of $3,000 at an annual interest rate of 5%, we can use the simple interest formula: I = P * r * t, where I is the interest, P is the principal (initial investment), r is the interest rate, and t is the time in years.
Given that the principal is $3,000 and the interest rate is 5%, we can calculate the interest as follows: I1 = 3000 * 0.05 * 1 (assuming the investment is for one year).
So, the interest earned on the initial investment is $150.
Step 2: Determine the total annual interest needed.
The problem states that the total annual interest earned should be 7.5 times the original amount invested. Since the original amount invested is $3,000, we can calculate the desired total annual interest as follows: Total Annual Interest = 7.5 * 3000 = $22,500.
Step 3: Calculate the additional amount needed to be invested.
Let's assume the additional amount to be invested is x dollars. This additional investment will earn interest at an annual rate of 8%. Now, we need to find the value of x that will result in a total annual interest of $22,500.
Using the simple interest formula again, the interest earned on the additional investment will be I2 = x * 0.08 * 1 (assuming the investment is for one year).
To meet the requirement of a total annual interest of $22,500, the sum of the interest earned on the initial investment and the interest earned on the additional investment must equal $22,500: I1 + I2 = 22,500.
Substituting the known values, we have: 150 + x * 0.08 = 22,500.
Simplifying the equation, we get: 0.08x = 22,500 - 150.
Combining like terms gives us: 0.08x = 22,350.
Finally, we can solve for x by dividing both sides of the equation by 0.08: x = 22,350 / 0.08.
Evaluating this expression gives us: x = $279,375.
Therefore, the additional amount that needs to be invested at an annual interest rate of 8% is $279,375, to ensure the total annual interest earned is 7.5 times the original investment amount.