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brooket30057
13.01.2020 •
Mathematics
Dale has purchased a $165,000 home with a 30-year mortgage at 5.05%. he can make a monthly payment of $1100. if he were to make this payment each month, how long will it take him to pay off his mortgage?
a.
243 months
b.
237 months
c.
300 months
d.
217 months
Solved
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Ответ:
The time taken to pay off the Mortgage amount is 216 months .
Step-by-step explanation:
Given as :
The mortgage value of home = p = $165,000
The time period of mortgage = t = 30 years = 30 × 12 = 360 months
The monthly payment amount = $1100
So, The payment amount in 360 months = $1100 × 360 = $396,000
i,e Amount paid after 30 years = A = $396,000
Rate of interest applied = r = 5.05%
Let The time taken to pay off mortgage amount = T years
Now, From compound Interest
Amount = Principal ×![(1+\dfrac{\textrm rate}{100})^{\textrm time}](/tpl/images/0453/0095/b9fe9.png)
Or, A = p ×![(1+\dfrac{\textrm r}{100})^{\textrm T}](/tpl/images/0453/0095/21302.png)
Or, $396,000 = $165,000 ×![(1+\dfrac{\textrm 5.05}{100})^{\textrm T}](/tpl/images/0453/0095/054e0.png)
Or,
= ![(1.0505)^{T}](/tpl/images/0453/0095/b12ce.png)
Or, 2.4 =![(1.0505)^{T}](/tpl/images/0453/0095/b12ce.png)
Taking power
both side
Or,
= ![((1.0505)^{T})^{\frac{1}{T}}](/tpl/images/0453/0095/8c01a.png)
Now, Taking Log both side
So, Log
= Log 1.0505
Or,
× Log2.4 = 0.0213
Or,
× 0.380 = 0.0213
Or, T =![\dfrac{0.380}{0.0213}](/tpl/images/0453/0095/7f06e.png)
Or, T = 17.84 ≈ 18 years
So, The time take= T = 18 × 12 = 216 months
Hence, The time taken to pay off the Mortgage amount is 216 months . Answer
Ответ: