An investor earns $1100 per year from bonds yielding returns of 4% and 5% each year. If the amounts at 4% and 5% were interchanged, she would earn $50 more per year. Find the total sum invested.
Let x be the amount invested at 4% and y be the amount invested at 5%. Then we have:
0.04x + 0.05y = 1100
0.05x + 0.04y = 1150
Multiplying the first equation by 0.05 and the second equation by 0.04, we get:
0.002x + 0.0025y = 55
0.002x + 0.0016y = 46
Subtracting the second equation from the first, we get:
0.0009y = 9
y = 10000
Substituting y = 10000 into the first equation, we get:
0.04x + 500 = 1100
0.04x = 600
x = 15000
Therefore, the total sum invested is:
x + y = 15000 + 10000 = 25000
Let's assume the amount invested at 4% is x, and the amount invested at 5% is y.
According to the given information, the annual return from the 4% investment is 1100.
We know that the formula for calculating interest is I = P * R * T, where I is the interest, P is the principal, R is the interest rate, and T is the time.
So, the interest earned from the 4% investment can be calculated as 0.04 * x.
Similarly, the interest earned from the 5% investment can be calculated as 0.05 * y.
According to the given information, if the amounts at 4% and 5% were interchanged, the investor would earn $50 more per year.
So, the new interest earned from the 4% investment (which was previously the 5% investment) can be calculated as 0.04 * y.
And the new interest earned from the 5% investment (which was previously the 4% investment) can be calculated as 0.05 * x.
We can now set up the equation:
1100 + 50 = (0.04 * y) + (0.05 * x)
Simplifying the equation:
1150 = 0.04y + 0.05x
Now, let's reverse the amounts invested and calculate the new annual return.
The new interest earned from the 4% investment (which was previously the 5% investment) can be calculated as 0.04 * x.
And the new interest earned from the 5% investment (which was previously the 4% investment) can be calculated as 0.05 * y.
We can now set up the equation:
1100 = (0.04 * x) + (0.05 * y)
Now, we have a system of equations:
1150 = 0.04y + 0.05x (Equation 1)
1100 = 0.04x + 0.05y (Equation 2)
To solve this system of equations, we can use the method of substitution. Let's solve Equation 2 for x:
0.04x = 1100 - 0.05y
x = (1100 - 0.05y) / 0.04
Substituting this value of x in Equation 1:
1150 = 0.04y + 0.05((1100 - 0.05y) / 0.04)
1150 = 0.04y + 55 - y / (0.04)
Multiplying both sides of the equation by 0.04 to eliminate the fraction:
1150 * 0.04 = 0.04y * 0.04 + 55 * 0.04 - y
46 = 0.0016y + 2.2 - y
Moving the terms involving y to one side of the equation:
0.0016y - y = 46 - 2.2
-0.9984y = 43.8
Dividing both sides of the equation by -0.9984:
y = 43.8 / -0.9984
y ≈ -43.9
Since we're dealing with investments, we can disregard the negative value of y as it doesn't make sense in this context.
Therefore, the sum invested at 5% is y ≈ $43.9.
Now, let's substitute this value back into Equation 2 to find the value of x:
1100 = 0.04x + 0.05 * 43.9
1100 = 0.04x + 2.195
0.04x = 1100 - 2.195
0.04x = 1097.805
x = 1097.805 / 0.04
x ≈ 27445.125
So, the sum invested at 4% is x ≈ $27445.125.
To find the total sum invested, we add the amounts invested at 4% and 5%:
Total sum invested = x + y ≈ 27445.125 + 43.9
Total sum invested ≈ $27489.025
Therefore, the total sum invested is approximately $27489.025.