invests 7000 for 1 year part at 5% part at 11% and the remainder at 13% total amount of income from these investments was $774. The amount of money invested at 13% was $1200 more than the amounts invested at 5% and 11% combined find the amount invested at each rate
if x at 5%, y at 11% and z at 13%,
x+y+z = 7000
.05x + .11y + .13z = 774.00
z = 1200+x+y
(x,y,z) = (1300,1600,4100)
To solve this problem, let's break down the given information step by step.
Let's denote:
- the amount invested at 5% as "x",
- the amount invested at 11% as "y", and
- the amount invested at 13% as "z".
Given information:
1. The total amount invested is $7000. So, we can write the equation: x + y + z = 7000.
2. The income from investments was $774. This means that the sum of the incomes from each investment should be $774. We can calculate the income from each investment using the formula: income = principal * rate * time (1 year, in this case).
- The income from the investment at 5% is: x * 0.05 * 1 = 0.05x.
- The income from the investment at 11% is: y * 0.11 * 1 = 0.11y.
- The income from the investment at 13% is: z * 0.13 * 1 = 0.13z.
So, we have the equation: 0.05x + 0.11y + 0.13z = 774.
3. The amount invested at 13% is $1200 more than the amounts invested at 5% and 11% combined. Mathematically, it can be represented as: z = x + y + 1200.
Now we have a system of equations:
x + y + z = 7000 (Equation 1)
0.05x + 0.11y + 0.13z = 774 (Equation 2)
z = x + y + 1200 (Equation 3)
To solve this system, we can use substitution or elimination method. Let's solve it using substitution:
1. Substitute z in Equation 2 with the value from Equation 3:
0.05x + 0.11y + 0.13(x + y + 1200) = 774
Simplify the equation:
0.05x + 0.11y + 0.13x + 0.13y + 156 = 774
2. Combine like terms:
0.18x + 0.24y = 618 (Equation 4)
3. Now, we have two equations:
x + y + z = 7000 (Equation 1)
0.18x + 0.24y = 618 (Equation 4)
From Equation 3, we can deduce z in terms of x and y:
z = x + y + 1200
Rewrite it: x + y = z - 1200
Substitute this value into Equation 1:
z - 1200 + z = 7000
2z - 1200 = 7000
2z = 8200
z = 4100
Now we know z = 4100. We can substitute it into Equation 4:
0.18x + 0.24y = 618
0.18x + 0.24y = 618
Solve this equation to find the values of x and y.
After finding the values of x, y, and z, we can determine the amount invested at each rate.