How would you solve this math equation?
this is really confusing to me.
Ms.Martin was researching the costs of financing $125,000 for a home. She found that the monthly payment for a 6.875% loan for 30 years would be $821.16 per month. She found that the monthly payment for a 6.875% loan for 20 years would be $959.77 per month.
1. Write and solve an equation to find the amount of interest she would pay altogether for the 30-year loan.
2. Write and solve an equation to find the amount of interest she would pay altogether for the 20-year loan.
3. For which loan would she pay less interest? How much would she save with that loan?
4. A loan officer tells Ms. Marin that her payment should be no more than 25% of her gross monthly income (income before taxes). How much must Ms. Martin gross yearly salary be in order to borrow $125,000 for each loan?
1. The amount of interest paid is total payments minus the initial value of the loan. This in the first case (30 years), the interest paid is
360*821.16 - 125,000 = 170,617.60
2. Fot a 20 year loan, the interest paid is
(240)*959.77-125,000 = 105,344.80
3. Compare the results of 1 and 2.
4. She must earn (before taxes) four times the payment. Compute this for each loan type.
To solve this math problem, we can break it down into four separate equations. Let's go through them step by step.
1. To find the amount of interest Ms. Martin would pay altogether for the 30-year loan, we first need to calculate the total cost of the loan. The monthly payment is given as $821.16, and the loan term is 30 years. We can use the formula for calculating the total cost of a loan:
Total Cost of Loan = Monthly Payment * Number of Payments
So, for the 30-year loan, the equation would be:
Total Cost of Loan (30-year) = $821.16 * (12 payments per year * 30 years)
Simplifying the equation:
Total Cost of Loan (30-year) = $821.16 * 360
To solve this equation, simply multiply the values:
Total Cost of Loan (30-year) = $295,617.60
Therefore, Ms. Martin would pay a total of $295,617.60 over the course of the 30-year loan.
2. Similarly, to find the amount of interest she would pay altogether for the 20-year loan, we can use the same formula:
Total Cost of Loan (20-year) = Monthly Payment * Number of Payments
Using the given information, the equation would be:
Total Cost of Loan (20-year) = $959.77 * (12 payments per year * 20 years)
Simplifying the equation:
Total Cost of Loan (20-year) = $959.77 * 240
Solving the equation:
Total Cost of Loan (20-year) = $230,330.80
Thus, Ms. Martin would pay a total of $230,330.80 for the 20-year loan.
3. To determine which loan would result in less interest, we compare the total cost of the two loans. We find that the total cost of the 30-year loan is $295,617.60, while the total cost of the 20-year loan is $230,330.80.
Since the 20-year loan has a lower total cost, Ms. Martin would pay less interest on this loan. To calculate the savings, we subtract the total cost of the 20-year loan from the total cost of the 30-year loan:
Savings = Total Cost of 30-year Loan - Total Cost of 20-year Loan
Savings = $295,617.60 - $230,330.80
Savings = $65,286.80
Therefore, Ms. Martin would save $65,286.80 by opting for the 20-year loan instead of the 30-year loan.
4. In order to determine the gross yearly salary Ms. Martin must have to borrow $125,000 for each loan, we need to consider the loan officer's statement that the monthly payment should be no more than 25% of her gross monthly income.
For the 30-year loan, the monthly payment is given as $821.16, so her gross monthly income should not exceed:
Gross Monthly Income (30-year) = Monthly Payment / 0.25
Gross Monthly Income (30-year) = $821.16 / 0.25
Gross Monthly Income (30-year) = $3,284.64
To find the yearly salary, we can simply multiply the gross monthly income by 12:
Gross Yearly Salary (30-year) = Gross Monthly Income (30-year) * 12
Gross Yearly Salary (30-year) = $3,284.64 * 12
Gross Yearly Salary (30-year) = $39,415.68
Similarly, for the 20-year loan, the monthly payment is given as $959.77, so her gross yearly salary should not exceed:
Gross Monthly Income (20-year) = Monthly Payment / 0.25
Gross Monthly Income (20-year) = $959.77 / 0.25
Gross Monthly Income (20-year) = $3,839.08
Calculating the yearly salary:
Gross Yearly Salary (20-year) = Gross Monthly Income (20-year) * 12
Gross Yearly Salary (20-year) = $3,839.08 * 12
Gross Yearly Salary (20-year) = $46,068.96
Therefore, Ms. Martin's gross yearly salary must be at least $39,415.68 for the 30-year loan and $46,068.96 for the 20-year loan to borrow $125,000 for each loan.