A man has two investments. The first pays him 3% interest per year and the other, 4%. The total income from these two investments is $170. If the interest rates were interchanged, the yearly income would be $180. How much is the investment?

Question

A man has two investments. The first pays him 3% interest per year and the other, 4%. The total income from these two investments is $170. If the interest rates were interchanged, the yearly income would be $180. How much is the investment?

Answer 1

.03x + .04y = 170

.04x + .03y = 180

now crank it out.

Answer 2

To solve this problem, we can set up a system of equations.

Let's assume the amount of money invested at 3% is x, and the amount invested at 4% is y.

According to the given information, the first investment pays 3% per year, so the income from this investment would be 0.03x.

Similarly, the second investment pays 4% per year, so the income from this investment would be 0.04y.

From the given information, we know that the total income from these two investments is $170, so we can write the first equation:

0.03x + 0.04y = 170 .......(Equation 1)

Now, let's consider the second scenario where the interest rates are interchanged. In this case, the first investment would pay 4% per year, and the second investment would pay 3% per year.

The income from the first investment would be 0.04x, and the income from the second investment would be 0.03y.

From the given information, we know that the total income from these investments in the second scenario is $180, so we can write the second equation:

0.04x + 0.03y = 180 .......(Equation 2)

Now, we have a system of equations:

0.03x + 0.04y = 170 .......(Equation 1)
0.04x + 0.03y = 180 .......(Equation 2)

To solve this system, we can use the method of elimination or substitution.

Let's use the method of elimination to eliminate one of the variables.

Multiplying Equation 1 by 100 and Equation 2 by 100, we get:

3x + 4y = 17000 .......(Equation 3)
4x + 3y = 18000 .......(Equation 4)

Now, let's multiply Equation 3 by 3 and Equation 4 by 4 to eliminate x:

9x + 12y = 51000 .......(Equation 5)
16x + 12y = 72000 .......(Equation 6)

Subtracting Equation 5 from Equation 6, we get:

16x - 9x = 72000 - 51000
7x = 21000

Dividing both sides by 7, we find:

x = 3000

Now, substitute the value of x into Equation 1:

0.03(3000) + 0.04y = 170

90 + 0.04y = 170

0.04y = 170 - 90

0.04y = 80

Divide both sides by 0.04, we get:

y = 2000

So, the amount invested at 3% is $3000, and the amount invested at 4% is $2000.