two investments totaling $35,000 produce an annual income of $1395. One investment yields 6% per year, while the other yields 3% per year. How much is invested at each rate?
.06x + .03(35000-x) = 1395
x=11,500
Let's assume the amount invested at 6% per year is x dollars.
Therefore, the amount invested at 3% per year would be (35,000 - x) dollars, as the total investment is $35,000.
The income from the 6% investment would be 0.06x dollars per year, while the income from the 3% investment would be 0.03(35,000 - x) dollars per year.
According to the given information, the total annual income is $1395.
So, we can set up the following equation:
0.06x + 0.03(35,000 - x) = 1395
Let's solve this equation step by step:
0.06x + 0.03(35,000 - x) = 1395
0.06x + 1050 - 0.03x = 1395
(Combine like terms)
0.03x + 1050 = 1395
(Subtract 1050 from both sides)
0.03x = 345
(Divide by 0.03)
x = 345 / 0.03
(Simplify)
x = 11,500
Therefore, $11,500 is invested at 6% per year, while the remaining amount, $35,000 - $11,500 = $23,500, is invested at 3% per year.
To solve this problem, we can use a system of two equations with two variables.
Let's assume that the amount invested at 6% per year is represented by "x" and the amount invested at 3% per year is represented by "y".
According to the given information, the two investments total $35,000. So we have our first equation:
x + y = 35,000 ---(Equation 1)
We are also given that the annual income from these investments is $1,395. The income from the 6% investment can be calculated as 6% of x, which is 0.06x. Similarly, the income from the 3% investment can be calculated as 3% of y, which is 0.03y. So we have our second equation:
0.06x + 0.03y = 1,395 ---(Equation 2)
Now we have a system of equations:
x + y = 35,000
0.06x + 0.03y = 1,395
We can solve this system using various methods such as substitution or elimination. Let's use the substitution method.
From Equation 1, we can isolate one variable in terms of the other. Let's solve for x:
x = 35,000 - y
Now substitute this expression for x in Equation 2:
0.06(35,000 - y) + 0.03y = 1,395
Expand and simplify the equation:
2,100 - 0.06y + 0.03y = 1,395
Combine like terms:
0.03y = 1,395 - 2,100
0.03y = -705
Divide both sides by 0.03:
y = -705 / 0.03
y ≈ 23,500
Now substitute this value of y back into Equation 1 to solve for x:
x + 23,500 = 35,000
x = 35,000 - 23,500
x ≈ 11,500
Therefore, approximately $11,500 is invested at 6% per year and approximately $23,500 is invested at 3% per year.