Question

Bethany needs to borrow $10,000. She can borrow the money at 5.5\% s

Decimals
ANSWERED
asked 2021-09-17
Bethany needs to borrow \(\displaystyle\${10},{000}.\) She can borrow the money at \(\displaystyle{5.5}\%\) simple interest for 4 yr or she can borrow at \(\displaystyle{5}\%\) with interest compounded continuously for 4 yr.
a) How much total interest would Bethany pay at \(\displaystyle{5.5}\%\) simple interest?
b) How much total interest would Bethany pay at \(\displaystyle{5}\%\) interest compounded continuously?
c) Which option results in less total interest?

Expert Answers (1)

2021-09-18
Step 1
a) Simple interest for an amount P, at an interest rate of r (in decimals) per annum and for T years is given by:
Plugging the values:
\(\displaystyle{I}={P}{T}{r}\)
\(\displaystyle{I}={\left({10000}\right)}{\left({4}\right)}{\left({0.055}\right)}\)
\(\displaystyle{I}={2200}\)
Step 2
b) Amount after continuous compounding for T years of principle P at an interest rate of r is given by:
Plugging the values:
So the interest is given by:
\(\displaystyle{A}={P}{e}^{{{r}{T}}}\)
\(\displaystyle{A}={\left({10000}\right)}{e}^{{{\left({0.05}\right)}{\left({4}\right)}}}\)
\(\displaystyle{A}={\left({10000}\right)}{e}^{{{0.2}}}\)
\(\displaystyle{A}={\left({10000}\right)}{\left({1.221402}\right)}\)
\(\displaystyle{A}={12214.03}\)
\(\displaystyle{I}={12214.03}-{10000}={2214.03}\)
Step 3
c) Hence, simple interest results in less total interest.
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