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

Decimals
ANSWERED 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? 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.