a total of $5000 is invested at 2%, 3%, and 4%. the amount invested at 4% is equal to 2% and 3% combined together. the total interest gained for 1 year is $145. find the amount invested at each rate.

If the amounts invested are x,y,z then we have

x+y+z = 5000
x+y = z
.02x + .03y + .04z = 145

Hmmm. I get -500 for y. Sure there's no typo in the problem?

no sir im pretty sure i typed it correctly from what my professor gave me. i also got -500 but it seems wrong so i went here. thank you so much for taking time and answering my question :)

Let's break down the problem step by step.

Let's represent the amount invested at 2% as 'x'. Therefore, the amount invested at 3% is also 'x'. And the amount invested at 4% is twice the amount invested at 2% and 3%, which is 2 * 'x'.

To calculate the interest earned on each amount, we'll multiply the principal by the interest rate. The interest for the amount invested at 2% is 0.02 * 'x'. The interest for the amount invested at 3% is 0.03 * 'x'. And the interest for the amount invested at 4% is 0.04 * 2 * 'x', which simplifies to 0.08 * 'x'.

Now, we can set up an equation based on the given information:

0.02 * 'x' + 0.03 * 'x' + 0.08 * 'x' = $145

Simplifying the equation:

0.02x + 0.03x + 0.08x = 145

0.13x = 145

Next, divide both sides of the equation by 0.13 to solve for 'x':

x = 145 / 0.13

x ≈ $1115.38

Therefore, the amount invested at 2% and 3% is approximately $1115.38 each, and the amount invested at 4% is double that, approximately $2230.77.

To summarize:
Amount invested at 2% = Amount invested at 3% ≈ $1115.38 each
Amount invested at 4% ≈ $2230.77

To find the amount invested at each rate, we can set up a system of equations based on the given information.

Let's denote the amount invested at 2% as x, the amount invested at 3% as y, and the amount invested at 4% as z.

We know that the total amount invested is $5000, so we have the equation: x + y + z = 5000.

We are also given that the amount invested at 4% is equal to the sum of the amounts invested at 2% and 3%, so we have the equation: z = x + y.

The total interest gained for 1 year is $145. To calculate the interest earned at each interest rate, we multiply the amount invested at that rate by the corresponding interest rate.

The interest earned at 2% is: x * 0.02
The interest earned at 3% is: y * 0.03
The interest earned at 4% is: z * 0.04

So, we have the equation: (x * 0.02) + (y * 0.03) + (z * 0.04) = 145.

Now, we can solve this system of equations to find the values of x, y, and z.

First, we can use the equation z = x + y to substitute z in the other equations:

(x * 0.02) + (y * 0.03) + ((x + y) * 0.04) = 145.

Expanding and simplifying the equation, we get:

0.02x + 0.03y + 0.04x + 0.04y = 145
0.06x + 0.07y = 145.

Next, we can substitute the value of z from the equation z = x + y into the equation x + y + z = 5000:

x + y + (x + y) = 5000
2x + 2y = 5000
x + y = 2500.

We now have a system of equations:

0.06x + 0.07y = 145
x + y = 2500.

Solving this system of equations can be done using various methods such as substitution or elimination. Let's solve it by substitution:

Rearrange the second equation to express x in terms of y:
x = 2500 - y

Now substitute the value of x in the first equation:
0.06(2500 - y) + 0.07y = 145.

Expand and simplify the equation:

150 - 0.06y + 0.07y = 145
0.01y = 145 - 150
0.01y = -5
y = -5 / 0.01
y = -500.

This gives us a negative value for y, but in this context, it doesn't make sense for the investment amount to be negative. It indicates that there is an error or inconsistency in the given information.

I recommend reviewing the problem statement or double-checking the values provided to ensure that there are no mistakes or missing details.