A person invested $7000 for 1 year, part at 5%, part at 10%, and the remainder at 15%. The total annual income from these investments was $815. The amount of money invested at 15% was $200 more than the amount invested at 5% and 10% combined. Find the amount invested at each rate.
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To solve this problem, let's set up some equations to represent the given information.
Let X be the amount invested at 5%.
The amount invested at 10% is also X since it is the same as the amount invested at 5%.
The amount invested at 15% is X + 200 since it's $200 more than the amount invested at 5% and 10% combined.
Now, let's calculate the income from each investment.
The income from the investment at 5% is X * (5/100).
The income from the investment at 10% is X * (10/100).
The income from the investment at 15% is (X + 200) * (15/100).
According to the given information, the total income from all investments is $815.
So, we can set up the equation: (X * (5/100)) + (X * (10/100)) + ((X + 200) * (15/100)) = 815.
Now, let's solve the equation step by step:
First, let's simplify the equation by multiplying the fractions:
0.05X + 0.10X + (0.15X + 30) = 815.
Next, let's combine like terms:
0.30X + 30 = 815.
Now, let's isolate the X term:
0.30X = 815 - 30.
Simplifying the right side:
0.30X = 785.
Now, let's solve for X by dividing both sides of the equation by 0.30:
X = 785 / 0.30.
Calculating the value:
X ≈ 2616.67
So, the amount invested at 5% and 10% is approximately $2616.67 each.
The amount invested at 15% is $200 more than the amount invested at 5% and 10%, so it is:
2616.67 + 200 = 2816.67.
Therefore, the amount invested at each rate is approximately:
$2616.67 at 5%,
$2616.67 at 10%, and
$2816.67 at 15%.