Question

megan borrowed $50,000 at 5% simple interest for 6 years. joseph borrowed $60,000 at 4% simple interest for 8 years. the formula $m = \frac{p + prt}{12t}$ can be used to calculate the monthly payment, $m$, where $p$ is the principle amount borrowed, $r$ is the rate expressed as a decimal, and $t$ is the amount of time for the loan, in years. who will have a greater monthly payment, and by how much?\
\

• megan will pay approximately $8 dollars more per month.\
• megan will pay approximately $78 dollars more per month\
• joseph will pay approximately $8 dollars more per month.\
• joseph will pay approximately $78 dollars more per month.

Explanation:

Step1: Calculate Megan's monthly payment

Given \( P = 50000 \), \( r = 0.05 \), \( t = 6 \).
First, calculate \( P + Prt \):
\( P + Prt = 50000 + 50000\times0.05\times6 \)
\( = 50000 + 50000\times0.3 \)
\( = 50000 + 15000 = 65000 \)
Then, use the formula \( m=\frac{P + Prt}{12t} \):
\( m_{Megan}=\frac{65000}{12\times6}=\frac{65000}{72}\approx902.78 \)

Step2: Calculate Joseph's monthly payment

Given \( P = 60000 \), \( r = 0.04 \), \( t = 8 \).
First, calculate \( P + Prt \):
\( P + Prt = 60000 + 60000\times0.04\times8 \)
\( = 60000 + 60000\times0.32 \)
\( = 60000 + 19200 = 79200 \)
Then, use the formula \( m=\frac{P + Prt}{12t} \):
\( m_{Joseph}=\frac{79200}{12\times8}=\frac{79200}{96} = 825 \) Wait, no, wait, 12*8=96? Wait 12*8 is 96? Wait 12*8=96, yes. But 79200/96: 79200 ÷ 96. Let's calculate 96*825= 96*(800+25)= 76800 + 2400=79200. Wait, but earlier Megan's was ~902.78, Joseph's 825? That can't be. Wait, no, wait the formula: wait the formula is \( m=\frac{P + Prt}{12t} \). Wait, maybe I made a mistake. Wait, no, the formula is for monthly payment, so total amount is \( P + Prt \) (principal + interest), then divide by total number of months, which is \( 12t \). So for Megan: t=6 years, so 12*6=72 months. For Joseph: t=8 years, 12*8=96 months. Wait, but when I calculated Megan's: 50000 + 50000*0.05*6=50000+15000=65000. 65000/72≈902.78. Joseph: 60000 + 60000*0.04*8=60000+19200=79200. 79200/96=825. Wait, but that would mean Megan pays more. But the options say Joseph pays more? Wait, no, I must have messed up the formula. Wait, wait the formula: is it \( m=\frac{P(1 + rt)}{12t} \), which is same as \( \frac{P + Prt}{12t} \). Wait, maybe the formula is wrong? Wait, no, simple interest total amount is \( A = P(1 + rt) \), then monthly payment is \( A/(12t) \), since total months is 12t. So that part is correct. Wait, but let's recalculate Joseph's. Wait 60000*0.04 is 2400 per year, for 8 years: 2400*8=19200. So total amount 60000+19200=79200. Total months: 8*12=96. 79200/96=825. Megan: 50000*0.05=2500 per year, 6 years: 15000. Total amount 65000. Months: 6*12=72. 65000/72≈902.78. So Megan's monthly is ~902.78, Joseph's is 825? But the options say Joseph pays more. Wait, I must have misread the time for Joseph. Wait the problem says "Joseph borrowed $60,000 at 4% simple interest for 8 years"? Wait the original problem: "Joseph borrowed $60,000 at 4% simple interest for 8 years"? Wait the user's image: "Joseph borrowed $60,000 at 4% simple interest for 8 years". Wait, but then my calculation shows Megan pays more. But the options have "Joseph will pay approximately $8 more per month" or "Megan...". Wait, maybe I made a mistake in the formula. Wait, maybe the formula is \( m = \frac{P + Prt}{12t} \), but maybe the formula is \( m = \frac{P(1 + rt)}{12t} \), which is same. Wait, let's check the formula again. The formula given is \( m=\frac{P + Prt}{12t} \). So that's correct. Wait, maybe I miscalculated Joseph's. Wait 60000*0.04*8: 0.04*8=0.32, 60000*0.32=19200. 60000+19200=79200. 79200 divided by (12*8)=96. 79200/96: let's do 79200 ÷ 96. 96*800=76800, 79200-76800=2400. 2400/96=25. So 800+25=825. Correct. Megan: 50000*0.05*6=15000, 50000+15000=65000. 65000/72: 72*900=64800, 65000-64800=200. 200/72≈2.78, so 902.78. So Megan's is ~902.78, Joseph's 825. So Megan pays more. But the options: first option: "Megan will pay approximately $8 dollars more per month." Wait 902.78 - 825 = 77.78≈78? Wait, wait, I think I messed up Joseph's time. Wait the problem says "Joseph borrowed $60,000 at 4% simple interest for 8 years"? Wait no, maybe it's 6 years? Wait the user's image: "Joseph borrowed $60,000 at 4% simple interest for 8 years"? Wait let me check again. The original problem: "Megan borrowed $50,000 at 5% simple interest for 6 years. Joseph borrowed $60,000 at 4% simple interest for 8 years." Wait, no, maybe the time for Joseph is 6 years? Wait no, the user's image shows "for 8 years". Wait, but then my calculation is correct. Wait 902.78 - 825 = 77.78≈78. So Megan pays approximately $78 more per month. But the options: second option is "Megan will pay approximately $78 dollars more per month". Wait, but earlier I thought Joseph pays more, but no, Megan's monthly is higher. So let's recalculate:

Megan:
\( P = 50000 \), \( r = 0.05 \), \( t = 6 \)
\( P + Prt = 50000 + 50000*0.05*6 = 50000 + 15000 = 65000 \)
\( m = 65000 / (12*6) = 65000 / 72 ≈ 902.78 \)

Joseph:
\( P = 60000 \), \( r = 0.04 \), \( t = 8 \)
\( P + Prt = 60000 + 60000*0.04*8 = 60000 + 19200 = 79200 \)
\( m = 79200 / (12*8) = 79200 / 96 = 825 \)

Difference: 902.78 - 825 = 77.78 ≈ 78. So Megan pays approximately $78 more per month. So the correct option is "Megan will pay approximately $78 dollars more per month".

Answer:

Megan will pay approximately $78 dollars more per month