Micheal loaned two of his roommates a total of $4,000. He charged his first roommate 3% interest and his second roommate 4% interest. After six months he collected a total of $332 in interest. How much did he loan each friend?
Something is wrong with your numbers. Even if he loaned all the $4000 at the higher annual rate of 4%, he would only collect $80 in interest in 6 months
This question was posted by a classmate and we had to answer it. I would let my classmate know that something is wrong or missing. Thanks for your help drwls
To solve this problem, we can set up a system of linear equations. Let's denote the amount that Michael loaned to his first roommate as "x" and the amount he loaned to his second roommate as "y."
Since the total loan amount is $4,000, we have the equation:
x + y = 4000 ---(equation 1)
In this equation, we are considering the loan amounts, not including the interest.
Next, we need to find the total interest collected after six months. Michael charged 3% interest to the first roommate and 4% interest to the second roommate. The equation for the total interest collected can be written as:
0.03x + 0.04y = 332 ---(equation 2)
Now, we have a system of two equations (equations 1 and 2) that we can solve simultaneously to find the values of x and y.
We can use various methods to solve this system of equations, such as substitution or elimination. Let's use the elimination method by multiplying equation 1 by 0.03 and equation 2 by 100 to eliminate the decimals.
(0.03)*(x + y) = 0.03*4000
0.03x + 0.03y = 120 ---(equation 3)
(0.04x + 0.04y) = 100*(332)
4x + 4y = 33200 ---(equation 4)
Now we have two equations without decimals:
0.03x + 0.03y = 120 ---(equation 3)
4x + 4y = 33200 ---(equation 4)
We can solve this system of equations by eliminating one variable.
Let's multiply equation 3 by 4 to make the coefficients of "y" in equation 3 and equation 4 equal:
4*(0.03x + 0.03y) = 4*120
0.12x + 0.12y = 480 ---(equation 5)
Now we have the following two equations:
0.12x + 0.12y = 480 ---(equation 5)
4x + 4y = 33200 ---(equation 4)
Subtract equation 5 from equation 4:
(4x + 4y) - (0.12x + 0.12y) = 33200 - 480
3.88x + 3.88y = 32720 ---(equation 6)
Now we have two equations:
0.12x + 0.12y = 480 ---(equation 5)
3.88x + 3.88y = 32720 ---(equation 6)
We'll use these two equations to solve for x and y.
To do so, we can multiply equation 5 by 3.88 and equation 6 by 0.12 so that the x coefficients will be equal:
(3.88)*(0.12x + 0.12y) = (3.88)*480
0.4656x + 0.4656y = 186.24 ---(equation 7)
Now we have the following two equations:
0.4656x + 0.4656y = 186.24 ---(equation 7)
3.88x + 3.88y = 32720 ---(equation 6)
Subtract equation 7 from equation 6:
(3.88x + 3.88y) - (0.4656x + 0.4656y) = 32720 - 186.24
3.4144x + 3.4144y = 32533.76 ---(equation 8)
Now we have two equations:
0.4656x + 0.4656y = 186.24 ---(equation 7)
3.4144x + 3.4144y = 32533.76 ---(equation 8)
We'll use these two equations to solve for x and y.
To solve this system of linear equations, we can subtract equation 7 from equation 8:
(3.4144x + 3.4144y) - (0.4656x + 0.4656y) = 32533.76 - 186.24
2.9488x + 2.9488y = 32347.52
Now we have a new equation:
2.9488x + 2.9488y = 32347.52
Let's divide this equation by 2.9488:
x + y = 10989.23
Now we have the equation:
x + y = 10989.23
From equation 1, we have:
x + y = 4000
The resulting equation is:
10989.23 = 4000
This implies that there is an error in the given information or the calculation process. Please re-check the values provided in the problem to ensure accuracy.