An executive invests $26,000, some at 8% and some at 6% annual interest. If he receives an annual return of $1,800, how much is invested at each rate?
0.08(X) + 0.06(26000-X) = 1800
Distribute
0.08(X) + 1560 - 0.06(X) = 1800
Re-arrange
0.08(X) - 0.06(X) = 240
Subtract Like-terms
0.02(X) = 240
Divide
X = 12000
Confirm Results
12000(0.08) = 960
14000(0.06) = 840
960 + 840 = 1800
Let's assume the amount invested at 8% is x dollars.
Therefore, the amount invested at 6% would be (26,000 - x) dollars.
According to the problem, the interest earned on the investment at 8% is (x * 8%) or 0.08x dollars.
Similarly, the interest earned on the investment at 6% is ((26,000 - x) * 6%) or 0.06(26,000 - x) dollars.
The total return from the investments is given as $1,800. So, we can set up the following equation:
0.08x + 0.06(26,000 - x) = 1,800
Let's solve this equation step-by-step:
Step 1: Distribute the 0.06 through the parentheses:
0.08x + 0.06 * 26,000 - 0.06x = 1,800
Step 2: Simplify:
0.08x + 1,560 - 0.06x = 1,800
Step 3: Combine like terms:
0.02x + 1,560 = 1,800
Step 4: Subtract 1,560 from both sides:
0.02x = 240
Step 5: Divide both sides by 0.02:
x = 12,000
So, $12,000 is invested at 8% and ($26,000 - $12,000) or $14,000 is invested at 6%.
To solve this problem, we can set up a system of equations. Let's represent the amount invested at 8% as "x" and the amount invested at 6% as "y".
We know that the total amount invested is $26,000, so we have the equation:
x + y = 26,000
We also know that the annual return on the investments is $1,800. The annual return on the amount invested at 8% is 8% of x, which is 0.08x. Similarly, the annual return on the amount invested at 6% is 6% of y, which is 0.06y. Therefore, we have another equation:
0.08x + 0.06y = 1,800
Now we have a system of two equations with two variables:
x + y = 26,000
0.08x + 0.06y = 1,800
We can solve this system of equations using substitution, elimination, or matrix methods. In this case, let's use the substitution method.
First, solve the first equation for x:
x = 26,000 - y
Substitute this expression for x in the second equation:
0.08(26,000 - y) + 0.06y = 1,800
Now we can simplify and solve for y:
2,080 - 0.08y + 0.06y = 1,800
0.02y = 1,800 - 2,080
0.02y = -280
y = -280 / 0.02
y = 14,000
Now substitute this value for y back into the first equation to solve for x:
x + 14,000 = 26,000
x = 26,000 - 14,000
x = 12,000
Therefore, the executive invested $12,000 at 8% and $14,000 at 6%.