he298c

2021-08-16

A $23,000 college loan with a rate of 4.76% compounded quarterly is taken. After how many years will the taker owe $30,000 if making no payments?

Brighton

Skilled2021-08-17Added 103 answers

The formula for compounf interest:

$A=P{(1+\frac{r}{n})}^{nt}$

Where:

P=principal=$23,000

r=interest rate (in decimal)=0.0476

n=amount of quarters in a year=4

A=the amount you owe=$30,000

t=amount of years=?

Substitute:

$30,000=23,000{(1+\frac{0.0476}{4})}^{4t}$

$30,000=23,000{(1+0.0119)}^{4t}$

$30,000=23,000{\left(1.0119\right)}^{4t}$

$\frac{30}{23}={1.0119}^{4t}$

$\mathrm{log}\left(\frac{30}{23}\right)={\mathrm{log}1.0119}^{4t}$

$\mathrm{log}\left(\frac{30}{23}\right)=4t\mathrm{log}1.0119$

$\frac{\mathrm{log}\left(\frac{30}{23}\right)}{4\mathrm{log}1.0119}=t$

$t\approx 22.5$

Answer: after 22.5 years

Where:

P=principal=$23,000

r=interest rate (in decimal)=0.0476

n=amount of quarters in a year=4

A=the amount you owe=$30,000

t=amount of years=?

Substitute:

Answer: after 22.5 years

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