Ms. Hardin invested $60,000 in three stocks. The first year, stock A paid 7% dividends and increased 4% in value; stock B paid 8% dividends and increased 5% in value; stock C paid 9% dividends and increased 3% in value. If the total dividends were $4620 and the total increase in value was $2310, how much was invested in each stock?
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To solve this problem, we will use a system of equations. Let's assign variables to represent the amount invested in each stock. Let:
- x be the amount invested in stock A,
- y be the amount invested in stock B,
- z be the amount invested in stock C.
We know that the sum of the investments is $60,000, so we can set up the equation:
x + y + z = 60,000 -> Equation 1
We are also given information about dividends and increases in value. Let's express that in equations:
For dividends:
- Stock A paid 7% dividends, so the amount received from stock A is 0.07x.
- Stock B paid 8% dividends, so the amount received from stock B is 0.08y.
- Stock C paid 9% dividends, so the amount received from stock C is 0.09z.
The sum of the dividends is $4620, so we can write the equation:
0.07x + 0.08y + 0.09z = 4620 -> Equation 2
For the increase in value:
- Stock A increased by 4%, so the increase in value from stock A is 0.04x.
- Stock B increased by 5%, so the increase in value from stock B is 0.05y.
- Stock C increased by 3%, so the increase in value from stock C is 0.03z.
The sum of the increases in value is $2310, so we can write the equation:
0.04x + 0.05y + 0.03z = 2310 -> Equation 3
Now, we have a system of three equations (Equations 1, 2, and 3) with three variables (x, y, and z). We can solve this system of equations to find the values of x, y, and z.
There are several methods to solve systems of equations, such as substitution, elimination, or matrix methods. Let's solve this system by using the substitution method:
1. Solve Equation 1 for x:
x = 60,000 - y - z
2. Substitute the expression for x into Equations 2 and 3:
0.07(60,000 - y - z) + 0.08y + 0.09z = 4620 -> Substitute in Equation 2
0.04(60,000 - y - z) + 0.05y + 0.03z = 2310 -> Substitute in Equation 3
3. Simplify and solve the resulting system of two equations in two variables.
By solving this system of equations, we will find the values of x, y, and z, which represent the amounts invested in each stock.