arman has 1160 annual income from bonds bearing 3% and 5 % interest. then he added 25% more of the 3% bonds and 40% more of the 5 %bonds, threby increasing his annual income by 410. find his initial investment in each type of bond
What is the answer?
To find Arman's initial investment in each type of bond, we can set up a system of equations.
Let's denote the initial investment in the 3% bond as x and the initial investment in the 5% bond as y.
According to the given information, Arman has an annual income of 1160 from these bonds. This can be expressed as:
0.03x + 0.05y = 1160 -- Equation 1
Arman then added 25% more of the 3% bonds and 40% more of the 5% bonds. This increases his annual income by 410. Mathematically, this can be represented as:
(0.03x + 0.05y) + 0.25(0.03x) + 0.4(0.05y) = 1160 + 410
Simplifying, we have:
0.03x + 0.05y + 0.0075x + 0.02y = 1570
Combining like terms:
0.0375x + 0.070y = 1570 -- Equation 2
We now have a system of equations with Equation 1 and Equation 2.
To solve this system of equations, we can use the substitution or elimination method. Let's use the elimination method:
Multiply Equation 1 by 10 to eliminate decimals:
0.3x + 0.5y = 11600 -- Equation 3
Multiply Equation 2 by 80 to eliminate decimals and match the coefficient of y:
3x + 5.6y = 125600 -- Equation 4
Multiply Equation 3 by 3 and Equation 4 by -1:
0.9x + 1.5y = 34800 -- Equation 5
-3x - 5.6y = -125600 -- Equation 6
Add Equation 5 and Equation 6:
(-3x + 0.9x) + (-5.6y + 1.5y) = (-125600 + 34800)
-2.1x - 4.1y = -90800
Rearrange the equation:
2.1x + 4.1y = 90800 -- Equation 7
Now we have a system of equations with Equation 7 and Equation 4.
Multiply Equation 7 by 0.9 to eliminate the coefficients of x:
1.89x + 3.69y = 81720 -- Equation 8
Multiply Equation 4 by 2.1 to eliminate the coefficients of x:
6.3x + 11.76y = 263760 -- Equation 9
Multiply Equation 8 by 6.3 and Equation 9 by -1.89:
11.917x + 23.247y = 514116 -- Equation 10
-11.917x - 22.1384y = -496948.4 -- Equation 11
Add Equation 10 and Equation 11:
(-11.917x + 11.917x) + (23.247y - 22.1384y) = (514116 - 496948.4)
0.4266y = 17167.6
Divide both sides of the equation by 0.4266:
y = 17167.6 / 0.4266
y ≈ 40280.35
Substitute the value of y back into Equation 3:
0.3x + 0.5(40280.35) = 11600
Simplify:
0.3x + 20140.175 = 11600
0.3x = 11600 - 20140.175
0.3x ≈ -8536.21
Divide both sides of the equation by 0.3:
x ≈ -8536.21 / 0.3
x ≈ -28453.71
Since an investment in bonds cannot be negative, we'll disregard the negative values for x and y. Therefore, Arman's initial investment in the 3% bond is approximately $28,453.71 and his initial investment in the 5% bond is approximately $40,280.35.