Jennifer owes $13,000 to a friend who is charging her interest of 0.90% p.m. She is required to settle the amount owed with two equal payments, one today and the other in three months. Calculate the size of the payments using three months as the focal date.
I am totally lost - I cannot follow the steps of the answer given "S1+xP1(1+rt1)+xx(1+0.009×3)+x1.027x+x2.027xx=S2=P2(1+rt2)=13,000(1+0.009×3)=13,351=13,351=13,3512.027=$6,586.58"
I have no idea what your equation is supposed to say or do.
Since you are talking about a "focal date", I will assume you have learned to make a time line.
label it "now" or 0, 1,2,3 months
I place the debt above the line , and the payments below the line, so
write 13,000 at 'now' above the line
write x at 'now' and x at 3 months
Using 3 months as the focal date:
x(1.009)^3 + x = 13,000(1.009)^3
x( 1.009^3 + 1) = 13000(1.009^3
x = 13000(1.009)^3 /(1.009^3 + 1) = $6587.35
personally , I would have picked 'now' as the focal date
x + x(1.009)^-3 = 13000
x = 13000/(1.009^-3 + 1) = 6587.35
Don't worry, I'll break down the steps for you to understand how to calculate the size of the payments.
The formula provided is a bit complex, so let's simplify it step by step:
Step 1: Determine the total amount owed with interest after three months.
To calculate the total amount owed after three months, we need to apply the interest rate of 0.90% per month. The formula to calculate the total amount with interest is:
S2 = S1(1 + rt)
where:
S2 is the total amount owed after three months,
S1 is the initial amount owed,
r is the interest rate per period (0.90%),
and t is the number of periods (3 months).
In this case, S1 (the initial amount owed) is $13,000, r (the interest rate per month) is 0.009 (0.90% expressed as a decimal), and t (the number of periods) is 3 months.
S2 = 13,000(1 + 0.009 * 3)
S2 = 13,000(1 + 0.027)
S2 = 13,000 * 1.027
S2 ≈ $13,351.00 (rounded to the nearest dollar)
So, after three months, Jennifer will owe approximately $13,351.00.
Step 2: Determine the size of the equal payments.
Now that we know the total amount owed after three months, we can calculate the size of the equal payments. Jennifer is required to settle the amount owed with two equal payments.
Let's say the size of each payment is x dollars. So, Jennifer will make one payment today and another in three months. The formula to calculate the total amount owed with two equal payments is:
S2 = P1(1 + rt1) + P2(1 + rt2)
where:
S2 is the total amount owed after three months ($13,351.00),
P1 is the first payment made today,
r is the interest rate per period (0.90%),
t1 is the number of periods until the first payment (0 months),
P2 is the second payment made after three months,
and t2 is the number of periods until the second payment (3 months).
We know that the total amount owed after three months is $13,351.00, and the number of periods until the second payment is 3 months. We need to solve for the size of each payment, x.
Let's rewrite the equation using the known values:
$13,351.00 = x(1 + 0.009 * 0) + x(1 + 0.009 * 3)
Simplifying the equation:
$13,351.00 = x + x(1.027)
$13,351.00 = x + 1.027x
Combining the terms with x:
$13,351.00 = 2.027x
To solve for x, we divide both sides of the equation by 2.027:
x = $13,351.00 / 2.027
x ≈ $6,586.58 (rounded to the nearest cent)
So, the size of each payment Jennifer should make is approximately $6,586.58.