(13) Dan invested a total of 12, 000. He invested some of that money at 3%
and some at 5%. After one year, he earned a total of 460 in interest.
How much did Dan invest at each rate of interest?
If x is at 3%, then the rest (12000-x) is at 5%. So, add up the interest:
.03x + .05(12000-x) = 460
To solve this problem, we can use a system of two equations with two variables. Let's assume Dan invested x amount of money at 3% interest and y amount of money at 5% interest.
Based on the given information, we know that the total amount invested is $12,000:
x + y = 12,000 Equation 1
We also know that after one year, the total interest earned is $460:
0.03x + 0.05y = 460 Equation 2
Now we can solve this system of equations to find the values of x and y.
To solve Equation 1, we can express y in terms of x:
y = 12,000 - x
Substituting this value into Equation 2:
0.03x + 0.05(12,000 - x) = 460
Now we can solve for x:
0.03x + 600 - 0.05x = 460
-0.02x = -140
x = -140 / -0.02
x = 7,000
Now that we have found the value of x, we can substitute it back into Equation 1 to find y:
7,000 + y = 12,000
y = 12,000 - 7,000
y = 5,000
Therefore, Dan invested $7,000 at 3% interest and $5,000 at 5% interest.