You invested money into two funds. Last year,the first fund paid a dividend of 8% and the second a dividend of 5%, and you received a total of $1330. This year, the first fund paid a 12% dividend and the second only 2% and you received a total of $1500. How much did you invest in each fund?
.08 f + .05 s = 1330
.12 f + .02 s = 1500
solve the system using elimination or substitution
.O8 f + .05 s = $1,300. .12 f + .02 s = $1,500. This should be right.
To solve this problem, we can set up a system of equations based on the given information.
Let's say you invested x dollars in the first fund and y dollars in the second fund.
According to the information, last year you received a total of $1330 in dividends. The first fund paid a dividend of 8% and the second fund paid a dividend of 5%. So, we can write the equation for last year's dividends as:
0.08x + 0.05y = 1330 ........(Equation 1)
Similarly, this year you received a total of $1500 in dividends. The first fund paid a dividend of 12% and the second fund paid a dividend of 2%. So, we can write the equation for this year's dividends as:
0.12x + 0.02y = 1500 ........(Equation 2)
Now we have a system of two equations with two variables. We can solve this system of equations to find the values of x and y, which will give us the amounts invested in each fund.
There are several methods to solve a system of equations, such as substitution or elimination. Let's solve this system using the elimination method.
To eliminate the decimal points in the equations, we can multiply both sides of Equation 1 by 100 and Equation 2 by 1000. This will simplify the calculations. The equations will become:
8x + 5y = 133000 ........(Equation 3)
120x + 20y = 1500000 ........(Equation 4)
Now we can use the elimination method. By multiplying Equation 3 by 24 and subtracting it from Equation 4, we can eliminate the variable x.
24(8x + 5y) = 24(133000)
192x + 120y = 3192000
-(8x + 5y) = -133000
-8x - 5y = -133000
Subtracting these two equations, we get:
192x + 120y - (-8x - 5y) = 3192000 - (-133000)
200x + 125y = 3325000 ........(Equation 5)
Now we have a new equation (Equation 5) in terms of y. We can solve this equation by rearranging it and substituting it back into Equation 3 to find the value of x.
Let's rearrange Equation 5:
200x + 125y = 3325000
200x = 3325000 - 125y
x = (3325000 - 125y) / 200
Now we can substitute this expression for x in Equation 3:
8x + 5y = 133000
8((3325000 - 125y) / 200) + 5y = 133000
Now we can simplify and solve for y:
(26600000 - 1000y) / 200 + 5y = 133000
(26600000 - 1000y + 1000y) / 200 = 133000
26600000 / 200 = 133000
133000 = 133000
We can see that the equation is true for any value of y. This means that y can take any value.
To determine the value of x, let's go back to Equation 3 and substitute the value of y:
8x + 5y = 133000
8x + 5(0) = 133000
8x = 133000
x = 16625
So, the amount invested in the first fund (x) is $16,625, and the amount invested in the second fund (y) can be any value.