My friend let me borrow an assignment to study and for these 2 questions they didn't show their work, but got them correct. I was wondering if you could help me figure out how they got it? Thanks
A person borrowed $1400 at 7.2% compounded monthly. He decided to pay off the loan with a single payment after 22 months. What is the total amount that will be paid back?
- They put $1596.92 (no work shown)
A person wants to buy a rare coin because she believes the value will increase. She plans to buy the coin at 3.1% compounded monthly. She plans to make a single payment of $4700 at the end of 2 years. What is the greatest amount of money that she can borrow?
- They put $4417.80
1400(1+.072/12)^22 = 1596.92
P(1+.031/12)^(12*2) = 4700
P = 4417.80
Thanks Steve
To find out how your friend got the correct answers, let's break down the steps for each question:
Question 1:
A person borrowed $1400 at 7.2% compounded monthly and decided to pay off the loan with a single payment after 22 months. We need to calculate the total amount that will be paid back.
To solve this, your friend most likely used the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the loan (the total amount that will be paid back)
P = the principal amount (the initial loan amount)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the time in years
Given values:
P = $1400
r = 7.2% (or 0.072 in decimal form)
n = 12 (compounded monthly)
t = 22/12 years (since the loan was paid off after 22 months)
Plugging in these values into the formula, your friend most likely calculated it as follows:
A = 1400(1 + 0.072/12)^(12 * (22/12))
= 1400(1 + 0.006)^22
≈ 1596.92
So, according to the compound interest formula, the total amount paid back would be approximately $1596.92.
Question 2:
A person wants to buy a rare coin and plans to make a single payment of $4700 at the end of 2 years. They want to know the greatest amount of money they can borrow, given an interest rate of 3.1% compounded monthly.
To solve this, your friend most likely used the formula for present value:
P = A/(1 + r/n)^(nt)
Where:
P = the principal amount (the maximum borrowing amount)
A = the future value (the planned payment at the end of 2 years)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the time in years
Given values:
A = $4700
r = 3.1% (or 0.031 in decimal form)
n = 12 (compounded monthly)
t = 2
Plugging in these values into the formula, your friend most likely calculated it as follows:
P = 4700/(1 + 0.031/12)^(12 * 2)
= 4700/(1 + 0.00258)^24
≈ 4417.80
So, according to the present value formula, the greatest amount of money that can be borrowed would be approximately $4417.80.
By using the appropriate formulas for compound interest and present value, your friend arrived at the correct answers for both questions.