An investment of $4600 is made at an annual simple interest rate of 6.8%. How much additional money must be invested at an annual simple interest rate of 9% so that the total interest earned is 8% of the total investment?
If x is invested at 9%, then
.068(4600) + .09(x) = .08(4600+x)
To solve this problem, let's break it down step by step.
Step 1: Calculate the interest earned on the original investment.
The interest earned on the original investment of $4600 at a simple interest rate of 6.8% can be calculated using the formula:
Interest = (Principal * Rate * Time)
In this case, the principal is $4600, the rate is 6.8%, and the time is not given. Since the time is not mentioned, let's assume it to be 1 year for simplicity.
Interest = (4600 * 0.068 * 1)
Step 2: Calculate the total investment.
The total investment consists of the original investment ($4600) plus the additional money to be invested.
Total Investment = 4600 + Additional Money
Step 3: Calculate the total interest earned.
The question states that the total interest earned should be 8% of the total investment. So, we can express this as an equation:
Total Interest = 0.08 * Total Investment
Step 4: Set up another equation based on the additional money to be invested.
Let's assume the additional money to be invested is denoted by X.
The interest earned on the additional money at a simple interest rate of 9% can be calculated as:
Interest on Additional Money = (X * 0.09 * 1)
Step 5: Set up the final equation to solve for X.
Based on the information given, the total interest earned should be equal to the sum of the interest earned on the original investment and the interest earned on the additional money. So, we can express this as an equation:
Total Interest = Interest on Original Investment + Interest on Additional Money
Now, let's substitute the values we calculated into the equation and solve for X:
0.08 * Total Investment = (4600 * 0.068 * 1) + (X * 0.09 * 1)
Step 6: Solve for X.
To find the value of X, we can rearrange the equation and isolate X:
X * 0.09 = 0.08 * Total Investment - (4600 * 0.068)
X = (0.08 * Total Investment - (4600 * 0.068)) / 0.09
Step 7: Substitute the value of Total Investment in terms of X.
Remember, Total Investment = 4600 + Additional Money. So, substitute that in the equation:
X = (0.08 * (4600 + X) - (4600 * 0.068)) / 0.09
Now you can solve for X by simplifying and solving the equation for X.