Based on the summary below, the first statement is true. That is, it is true that if each loan has a term of 3 years, Ramon's monthly payments will be higher. This is because Ramon’s monthly payments of $35.59 is higher than Stephen's monthly payments of $34.58.
Step-by-step explanation:
(a) If each loan has a term of 3 years, Ramon's monthly payments will be higher.
The monthly rate can be calculated using the formula for calculating the present value of an ordinary annuity as follows:
PV = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (1)
Where;
For Ramon:
PV = Present value of loan = $1,200
P = Monthly payments = ?
r = Monthly interest rate = 4.3% / 12 = 0.043 / 12 = 0.00358333333333333
n = number of months = 3 years * 12 months = 36
Substitute the values into equation (1) and solve for P, we have:
$1,200 = P * ((1 - (1 / (1 + 0.00358333333333333))^36) / 0.00358333333333333)
$1,200 = P * 33.7181420201235
P = $1,200 / 33.7181420201235
P = 35.5891495825547
Approximating to 2 decimal places, we have:
P = $35.59
Therefore, Ramon's monthly payments is $35.59.
For Stephen:
PV = Present value of loan = $1,200
P = Monthly payments = ?
r = Monthly interest rate = 2.4% / 12 = 0.024 / 12 = 0.002
n = number of months = 3 years * 12 months = 36
Substitute the values into equation (1) and solve for P, we have:
$1,200 = P * ((1 - (1 / (1 + 0.002))^36) / 0.002)
$1,200 = P * 34.7010963779533
P = $1,200 / 34.7010963779533
P = $34.5810399455389
Approximating to 2 decimal places, we have:
P = $34.58
Therefore, Stephen's monthly payments is $34.58.
(b) If they both make $300 payments a month, Ramon will pay off his loan faster
To determine this we also make use of equation (1) in part (a) as follows:
Where;
For Ramon:
PV = Present value of loan = $1,200
P = Monthly payments = $300
r = Monthly interest rate = 4.3% / 12 = 0.043 / 12 = 0.00358333333333333
n = number of months = ?
Substitute the values into equation (1) and solve for n, we have:
1,200 = 300 * ((1 - (1 / (1 + 0.00358333333333333))^n) / 0.00358333333333333)
1,200 / 300 = (1 - (1 / (1 + 0.00358333333333333))^n) / 0.00358333333333333
4 = (1 - (1 / (1 + 0.00358333333333333))^n) / 0.00358333333333333
4 * 0.00358333333333333 = 1 - (1 / (1 + 0.00358333333333333))^n
0.0143333333333333 = 1 - (1 / (1 + 0.00358333333333333))^n
0.0143333333333333 = 1 - 0.996429461097733^n
Rearrange, we have:
0.996429461097733^n = 1 - 0.0143333333333333
0.996429461097733^n = 0.985666666666667
Loglinearize both sides, we have:
n * log(0.996429461097733) = log(0.985666666666667)
n * (-0.00155344030551834) = -0.00626993019354464
n = -0.00626993019354464 / -0.00155344030551834
n = 4.03615779201283 = 4.04
Therefore, Ramon will pay off his loan within 4.04 months.
For Stephen:
PV = Present value of loan = $1,200
P = Monthly payments = $300
r = Monthly interest rate = 2.4% / 12 = 0.024 / 12 = 0.002
n = number of months = ?
Substitute the values into equation (1) and solve for n, we have:
1,200 = 300 * ((1 - (1 / (1 + 0.002))^n) / 0.002)
1,200 / 300 = (1 - (1 / (1 + 0.002))^n) / 0.002
4 = (1 - (1 / (1 + 0.002))^n) / 0.002
4 * 0.002 = 1 - (1 / (1 + 0.00358333333333333))^n
0.008 = 1 - (1 / (1 + 0.002))^n
0.008 = 1 - 0.998003992015968^n
Rearrange, we have:
0.998003992015968^n = 1 - 0.008
0.998003992015968^n = 0.992
Loglinearize both sides, we have:
n * log(0.998003992015968) = log(0.992)
n * (-0.000867721531226935) = -0.00348832784582135
n = -0.00348832784582135 / -0.000867721531226935 = 4.02010059712238 = 4.02
Therefore, Stephen will pay off his loan within 4.02 months.
(c) Both men would likely get a better interest rate if they used a credit card, rather than a personal loan, to make their purchases.
This is a likely occurence, not a fact, as both Ramon and Stephen have not actually chose to used a credit card. Therefore, it is not possible to know whether it is true or not.
(d) If Ramon applies to Stephen's bank, instead, for his loan, he's guaranteed to get the same 2.4% interest rate that Stephen's been offered.
This is a conditional statement that may be true or not. Therefore, cannot be verified to determine whether it is true or not since Ramon has not actually applied to Stephen's bank.
Summary
The following facts can be obtained from the above:
Ramon's monthly payments = $35.59
Stephen's monthly payments is $34.58
Number of months Ramon will pay off his loan = 4.04 months
Number of months Stephen will pay off his loan = 4.02 months
Conclusion
Based on the summary above, the first statement is true. That is, it is true that if each loan has a term of 3 years, Ramon's monthly payments will be higher. This is because Ramon’s monthly payments of $35.59 is higher than Stephen's monthly payments of $34.58.
