How to find the total investment from interest received

Question

Dave Horn invested half of his money at $5$%, one-third of his money at $4$%, and the rest of his money at $3.5$%. If his total annual investment income was $\$530$, how much had he invested?

I found this, but I was still confused as to why he didn't change the 530, if he was putting 6 into the equation:

6m = total money (choose something it's easy to get half of and 1/3 of)

3m = money at 5% 2m = money at 4% (6m - 5m) = 1m = money at 3.5%

So 0.05(3m) + 0.04(2m) + 0.035 (1m) = 530

Multiply out each, then combine, then divide 530 by the number in front of m after you combine. Then plug that in to get each amount.

Answer 1

Hint: let the whole amount invested be $x$. How much interest did he receive from he part at $5\%$? Add up the amount received from each piece, which must total $530$.

Answer 2

Let $I$ represent the amount invested.

$$0.05\cdot \frac 12I + 0.04\cdot \frac 13 I + 0.035 \cdot \frac 16 I = 530 $$

Solve for I.

Answer 3

First you have to calculate the Proportion of the investment, (x), which is invested at 3,5%.

It is $1-\frac{1}{2}-\frac{1}{3}=a$

Thus the equation is: paid interest investment - investment=530

$\frac{x}{2}\cdot 1.05 +\frac{x}{3}\cdot 1.04+a\cdot x\cdot 1.035-x=530$