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, thereby increasing his annual income by 410. find his initial investment in each type of bond

(if you would,just give me the formula and i'll take it from there)

thank you

I will do it using two variables

let the original 3% investment be x
let the original 5% investment by y

.03x + .05y = 1160 or 3x + 5y = 116000

after stated increase in each investment:
.03(1.25x) + .05(1.4y) = 1570
or after multiplying by 100
3.75x +7y = 157000
times 4
15x + 28y = 628000

you now have 2 equations in 2 unknowns, take it from there. (y=16000 , x = 12000)

To solve this problem, let's break it down into steps:

Step 1: Determine the initial income generated by the 3% and 5% bonds.
Let's assume the initial investment in the 3% bonds is x, and the initial investment in the 5% bonds is y.
The total income from the 3% bonds will be x * 0.03, and the total income from the 5% bonds will be y * 0.05.
Therefore, we have the equation:
0.03x + 0.05y = 1160

Step 2: Calculate the additional income generated by the increased investments.
To find the additional income generated, we need to calculate the increased investments in both types of bonds.
The additional investment in the 3% bonds will be 25% of x, which is 0.25x. Similarly, the additional investment in the 5% bonds will be 40% of y, which is 0.40y.
Therefore, the additional income generated will be:
0.03 * 0.25x + 0.05 * 0.40y = 410

Step 3: Combine the equations and solve for x and y.
We now have a system of two equations with two variables:
0.03x + 0.05y = 1160
0.0075x + 0.02y = 410

Solving this system of equations will give you the values of x and y, which are the initial investments in the 3% and 5% bonds, respectively.

Note: You can solve this system of equations using various methods such as substitution, elimination, or matrices, depending on your preference and mathematical knowledge.