1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
DENIUS [597]
3 years ago
6

Vanna has just financed the purchase of a home for $200 000. She agreed to repay the loan by making equal monthly blended paymen

ts of $3000 each at 4%/a, compounded monthly
How long will it take to repay the loan?
How much will be the final payment?
Determine how much interest she will pay for her loan.
How much sooner would the loan be paid if she made a 15% down payment?
How much would Vanna have saved if she had obtained a loan 3%/a, compounded monthly?
Mathematics
1 answer:
Vaselesa [24]3 years ago
4 0
The question describes retiring a dept though annuity. The present value of an annuity, PV, where periodic payments are made for n periods at r% interest rate at t periods per year is given by
PV=P\left[\frac{1-\left(1+\frac{r}{t}\right)^{-nt}}{\frac{r}{t}}\right]
where: PV = $200,000; P = $3,000; r = 4%; t = 12

Part A.
To find, n the number of years it will take to repay the loan, we have
200,000=3000\left[\frac{1-\left(1+\frac{0.04}{12}\right)^{-12n}}{\frac{0.04}{12}}\right] \\  \\ =3000\left[\frac{1-\left(\frac{301}{300}\right)^{-12n}}{\frac{1}{300}}\right]=900,000\left[1-\left(\frac{301}{300}\right)^{-12n}\right] \\  \\ \Rightarrow\left[1-\left(\frac{301}{300}\right)^{-12n}\right]=\frac{2}{9} \\  \\ \Rightarrow\left(\frac{301}{300}\right)^{-12n}=1-\frac{2}{9}=\frac{7}{9} \\  \\ -12n\log{\left(\frac{301}{300}\right)}=\log{\left(\frac{7}{9}\right)}

\Rightarrow-12n=\frac{\log{\left(\frac{7}{9}\right)}}{\log{\left(\frac{301}{300}\right)}}=-75.52 \\  \\ \Rightarrow n=\frac{-75.52}{-12}=6.29

Therefore, it will take 6 years and 4 months to repay the loan.


Part B.
Total amount paid for the loan including interest is given by the formular
A=P\left[\frac{\left(1+\frac{r}{t}\right)^{nt}-1}{\frac{r}{t}}\right]
where: P = $3,000; r = 4%; t = 12; n = 6.29332634

A=3000\left[\frac{\left(1+\frac{0.04}{12}\right)^{6.29332634\times12}-1}{\frac{0.04}{12}}\right] \\  \\ =3000\left[\frac{0.285714286}{0.003333333333}\right]=3000(85.7142858)=\$257,142.86

Total amount paid as at the month before the last payment (i.e. after 6 years 3 months) is given by
A=3000\left[\frac{\left(1+\frac{0.04}{12}\right)^{6.25\times12}-1}{\frac{0.04}{12}}\right] \\ \\ =3000\left[\frac{0.283491702}{0.003333333333}\right]=3000(85.04751061)=\$255,142.53

Therefore, the amount paid in the final payment is $257,142.86 - $255,142.53 = $2,000.33 or approximately $2,000


Part C.
The amount of interest she will pay for her loan is given by the total amount payed for the duration of the loan less the loan amount.
Therefore, interest = $257,142.86 - $200,000 = $57,142.86


Part D.
If she made a 15% down payment, then amount to be repaid is given by
200000 - \frac{15}{100}\times200000=200000-30,000=\$170,000

Thus, the number of periods for the repayment is given by
170,000=3000\left[\frac{1-\left(1+\frac{0.04}{12}\right)^{-12n}}{\frac{0.04}{12}}\right] \\ \\ =3000\left[\frac{1-\left(\frac{301}{300}\right)^{-12n}}{\frac{1}{300}}\right]=900,000\left[1-\left(\frac{301}{300}\right)^{-12n}\right] \\ \\ \Rightarrow\left[1-\left(\frac{301}{300}\right)^{-12n}\right]=\frac{17}{90} \\ \\ \Rightarrow\left(\frac{301}{300}\right)^{-12n}=1-\frac{17}{90}=\frac{73}{90} \\ \\ -12n\log{\left(\frac{301}{300}\right)}=\log{\left(\frac{73}{90}\right)}

\Rightarrow-12n=\frac{\log{\left(\frac{73}{90}\right)}}{\log{\left(\frac{301}{300}\right)}}=-62.91 \\ \\ \Rightarrow n=\frac{-62.91}{-12}=5.24

If she made a 15% down payment, it will take her 5 years 3 months to complete the payment.


Part E.
If she have obtained a loan of 3% / annum compounded monthly. Total payment to be made at 3% interest is given by
A=3000\left[\frac{\left(1+\frac{0.03}{12}\right)^{6.29332634\times12}-1}{\frac{0.04}{12}}\right] \\ \\ =3000\left[\frac{0.207514582}{0.0025}\right]=3000(83.00583299)= \\  \\ \$249,017.50

Therefore, she will save $257,142.86 - $249,017.50 = $8,125.36
You might be interested in
At 6:00 a.m., the outside temperature was –14.5⁰. By noon, the outside temperature had dropped 8.6⁰. What was the outside temper
Lerok [7]
The answer is -23.1°
3 0
3 years ago
Maria uses 2 centimeters of tape for every present she wraps. Write an equation that shows the relationship between the number o
Marizza181 [45]

Answer:

y = 2x

Step-by-step explanation:

Maria uses 2 centimeters of tape for each present she wraps, and we want a equation that shows the relationship between the number of presents x and the tape used y. To do this, we can establish a rule of three with the information given:

1 present wrapped -> 2 centimeters of tape

x presents wrapped -> y centimeters of tape

1/x = 2/y

x = y/2 -> y = 2x

From this relation, we know that to make x presents, Maria will use 2x centimeters of tape.

6 0
3 years ago
The median and mean of these numbers:
aleksandr82 [10.1K]

Answer:

Mean : ≈ 12.7

Median: ≈ 13.5

Step-by-step explanation:

To find the mean you must add all the numbers and divide by the # of numbers in the set.

8+12+13+14+14+15 = 76

76/6 ≈ 12.7

To find the median you have to order the numbers from least to greatest, the middle number should be the median, but in this case two numbers are in the middle.

8, 12, 13, 14, 14, 15

So you have to add 14 and 13, which is 27.

Then divide by 2 which is ≈ 13.5

8 0
2 years ago
A tank currently holds 5 gallons of water. It is being filled at the rate of 1/4 gallon per minute. How long will it take for th
Verdich [7]
So we know the tank already has 5 gallons and we need 9. This means we need 4 more gallons in order to fill up the tank.

If 1/4 gallon is being filled per minute we need to find out how many 1/4 gallons makes up 4 gallons in total.

4 of the 1/4 gallons makes up 1 whole gallon. So since we need 4 whole gallons we would need 16 of the 1/4 gallons.

This means that it will take 16 minutes for the tank to hold 9 gallons of water. 
6 0
3 years ago
Read 2 more answers
Write the equation of the line that is the perpendicular bisector of the segment with endpoints (4, 1) and (2, -5)
Sedbober [7]

Answer:   \bold{y=-\dfrac{1}{3}x-1}

<u>Step-by-step explanation:</u>

(4, 1) & (2, -5)

First, find the slope (m) and then the perpendicular (opposite reciprocal) slope:

m=\dfrac{y_2-y_1}{x_2-x_1}\\\\\\m=\dfrac{-5-1}{2-4} = \dfrac{-6}{-2}=3\quad \rightarrow \quad m_{\perp}=-\dfrac{1}{3}\\

Next, find the midpoint of (4, 1) and (2, -5):

Midpoint=\bigg(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\bigg)\\\\\\.\qquad \qquad=\bigg(\dfrac{4+2}{2},\dfrac{1-5}{2}\bigg)\\\\\\.\qquad \qquad=\bigg(\dfrac{6}{2},\dfrac{-4}{2}\bigg)\\\\\\.\qquad \qquad=(3, -2)

Lastly, input the perpendicular slope and the midpoint into the Point-Slope formula to find the equation of the line:

y - y_1 = m_{\perp}(x - x_1)\\\\y - (-2) = -\dfrac{1}{3}(x - 3)\\\\y + 2=-\dfrac{1}{3}x +1\\\\y =-\dfrac{1}{3}x - 1\\


6 0
2 years ago
Other questions:
  • Is 74 inches greater or less than 2 yards 2 inches?
    7·2 answers
  • <img src="https://tex.z-dn.net/?f=%20%5Csqrt%7B640%7D%20%20%5C%3A%20%20%5C%3A%20simplified" id="TexFormula1" title=" \sqrt{640}
    6·2 answers
  • There are 25 students in Ms.Morris's class. She wants to divide the class into 3,4,or 5 equal teams with all students on a team.
    6·2 answers
  • What is the length of SU? Please show work!
    12·1 answer
  • 45 1/2% in simplest form
    8·1 answer
  • Please Help Me!!!!!!!
    5·2 answers
  • PLEASE help me
    6·1 answer
  • What is the equation of the trend line in the scatter plot when y=6 and x=9
    14·1 answer
  • Diana looks up at an angle of 57 and sees a hot air balloon 150 meters away. To the nearest meter, what is the value of x,
    7·2 answers
  • Determine the composition of
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!