Question 1200766: A loan of $45,000 is made at 5.5% interest, compounded annually. After how many years will the amount due reach $70,000 or more? Write the smallest possible whole number answer.
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: 9 years
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Work Shown:
The compound interest formula we use here is
A = P*(1+r/n)^(n*t)
In this case,
A = 70000 = final amount
P = 45000 = starting amount
r = 0.055 = decimal form of the 5.5% interest rate
n = 1 = since we're compounding once per year
t = unknown number of years.
We use logarithms to isolate the exponent.
Think of it like saying: "The variable is in the trees so we must log it down."
The reason why logs are handy here is that we'll take advantage of the rule log(A^B) = B*log(A) to pull down the exponent.
A = P*(1+r/n)^(n*t)
70000 = 45000*(1+0.055/1)^(1*t)
70000 = 45000*(1.055)^t
70000/45000 = (1.055)^t
1.555556 = (1.055)^t
log( 1.555556 ) = log( (1.055)^t )
log( 1.555556 ) = t*log( 1.055 ) .... use the log rule mentioned
t*log( 1.055 ) = log( 1.555556 )
t = log(1.555556)/log(1.055)
t = 8.252273
t = 9
We round UP to the nearest integer.
It doesn't matter that 8.252273 is closer to 8 than it is to 9.
If we tried t = 8, then,
A = P*(1+r/n)^(n*t)
A = 45000*(1+0.055/1)^(1*8)
A = 69060.8931748269
A = 69060.89
which is about $1000 short of the goal of $70,000.
But t = 9 leads to,
A = P*(1+r/n)^(n*t)
A = 45000*(1+0.055/1)^(1*9)
A = 72859.2422994423
A = 72859.24
we've satisfied the "$70,000 or more" criteria.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
The formulation of the problem in the post is ABSURDIST/unprofessional
and only may to confuse a reader.
In Finance, a loan is a mechanism which assumes regular back payments.
If there are no back payments, the problem at least MUST SAY about it EXPLICITLY.
Normally, such problems are worded in different form. They ask about the growing deposit, not about the loan.
The normal/regular formulation should be like this:
A deposit of $45,000 is made at 5.5% interest, compounded annually.
After how many years will the amount reach $70,000 or more?
Then it makes sense; otherwise it is only good to frighten a reader.
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To see million of solved problems of this type, look into the lessons
- Compounded interest percentage problems
- Problems on discretely compound accounts
in this site, and learn the subject from there.
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