[begin{gathered}
{text{a}}{text{.)}} hfill \
P = 850 hfill \
r = 0.09 hfill \
t = 6{text{ years}} hfill \
hfill \
A = Pleft( {1 + rt} right) = 850left( {1 + 0.09 times 6} right) hfill \
A = $ 1309 hfill \
hfill \
I = Prt = 850 times 0.09 times 6 = $ 459 hfill \
end{gathered} ]

[begin{gathered}
{text{b}}{text{.)}} hfill \
P = 5460 hfill \
r = 0.084 hfill \
{text{compounded semi-annually}} hfill \
t = 13{text{ years}} hfill \
i = frac{{0.084}}{2} = 0.042 hfill \
m = 13 times 2 = 26 hfill \
hfill \
A = P{left( {1 + i} right)^m} hfill \
A = 5460{left( {1 + 0.042} right)^{26}} hfill \
A = $ 15913.05 hfill \ \
I = A – P = 15913.05 – 5460 hfill \
I = $10,453.05 hfill \
end{gathered} ]

a) $A =$1309.00$ , $I=$459.00$

b) $A=$15,913.05$ , $I=$10,453.05$

c) $A=$21,005.02$ , $I=$3065.02$

a.) If the difference between amount owned in consecutive year is constant, then it is simple interest. If the ratio between consecutive year is constant, then it is compound interest.

for Loan$#$1

$3942-3796=$146$

$4088-3942=$146$

Therefore, Loan $#1$ is done with simple interest.

Loan $#2$

$dfrac{1036.18}{977.53}approx 1.06$

$dfrac{1098.35}{1036.18}approx 1.06$

Therefore, Loan $#2$ is done with compound interest.

b.) Since loan number 1 is simple interest,

$I=Prt$

$146=(3796-146)times r times 1$

$r=0.04=4.0%$

Since Loan number 2 is compound interest

$A=P(1+i)^m$

$1036.18=977.53(1+i)^1$

$$
i=0.06=6%/a
$$

c) The principal for loan number 1 is

$3796-146=$3650$

The principal loan for loan number 2 is

$$
dfrac{977.53}{1.06^2}=$870
$$

d.) The value after 10 years are

loan 1: $A=P(1+rt)=3650(1+0.04times 10)=$5110$

loan 2: $A=P(1+i)^m=870(1+0.06)^{10}=$1558.04$

a) simple interest, compound interest

b) $4%/a$, $6%/a$

c) $$3,650$ , $$870$

d) $$5110$ , $$1558.04$

[begin{gathered}
{text{The future value of an investment }} hfill \
{text{with compound interest is}} hfill \
hfill \
{text{The given values are}} hfill \
hfill \
A = 25,000 hfill \
i = frac{{0.092}}{4} = 0.023 hfill \
text{compounded quarterly} hfill \
t=8 text{ years} hfill \
m = 8 times 4 = 32 hfill \
hfill \
A = P{left( {1 + i} right)^m} hfill \
P = frac{A}{{{{left( {1 + i} right)}^m}}} hfill \
P = frac{{25000}}{{{{left( {1 + 0.023} right)}^{32}}}} hfill \
P = boxed{{mathbf{$ 12075}}{mathbf{.91}}} hfill \
hfill \
{text{Therefore, Betsy must invest $ 12,075}}{text{.91}} hfill \
hfill \
end{gathered} ]

$$
$12,075.91
$$

[begin{gathered}
{text{In this case,}} hfill \
R = $ 250 hfill \
t = 6.5{text{ years}} hfill \
r = 4.8% /a hfill \
{text{compounded monthly}} hfill \
i = frac{{0.048}}{{12}} = 0.004 hfill \
m = 6.5{kern 1pt} times 12 = 78 hfill \
hfill \
FV = R cdot frac{{{{left( {1 + i} right)}^m} – 1}}{i} hfill \
FV = 250 cdot frac{{{{left( {1 + 0.004} right)}^{78}} – 1}}{{0.004}} hfill \
FV = boxed{{mathbf{$ 22,831}}{mathbf{.55}}} hfill \
hfill \
{text{Therefore, Derek’s investment will be worth $22,831}}{text{.55}} hfill \
{text{at the end of 6}}{text{.5 years}}{text{.}} hfill \
hfill \
end{gathered} ]

$$
$22,831.55
$$

[begin{gathered}
{text{To determine the best option, we shall}} hfill \
{text{calculate the future value of a $1 investment}} hfill \
{text{after 1 year}}{text{.}} hfill \
hfill \
5.88% /a{text{ compounded monthly}} hfill \
A = P{left( {1 + i} right)^m} hfill \
A = 1 cdot {left( {1 + frac{{0.0588}}{{12}}} right)^{12left( 1 right)}} = 1.0604 hfill \
hfill \
6% /a{text{ compounded annually}} hfill \
A = 1 cdot {left( {1 + 0.06} right)^1} = 1.06 hfill \
hfill \
{text{Therefore, she should choose }}boxed{{mathbf{5}}{mathbf{.88% /a; compounded ;monthly}}} hfill \
end{gathered} ]

She should choose $5.88%/a$ compounded monthly

[begin{gathered}
{text{In this case,}} hfill \
R = $ 450{text{ monthly payments}} hfill \
t = 30{text{ years}} hfill \
FV = 450,000 hfill \
hfill \
{text{We shall find the annual interest rate }}r.{text{ }} hfill \
{text{Since payments are done monthly}} hfill \
i = frac{r}{{12}} hfill \
m = 30{kern 1pt} times 12 = 360 hfill \
hfill \
{text{Using the formula for future value of annuity}} hfill \
FV = R cdot frac{{{{left( {1 + i} right)}^m} – 1}}{i} Rightarrow ,frac{{FV}}{R} = frac{{{{left( {1 + i} right)}^m} – 1}}{i} hfill \
frac{{450000}}{{450}} = frac{{{{left( {1 + frac{r}{{12}}} right)}^{360}} – 1}}{{r/12}} hfill \
1000 = frac{{{{left( {1 + frac{r}{{12}}} right)}^{360}} – 1}}{{r/12}} hfill \
hfill \
end{gathered} ]

Therefore, the annual interest is $0.0598=5.98%/a$

$$
5.98%/a
$$

[begin{gathered}
{text{Since Yvette wants to withdraw $ 5000}} hfill \
{text{monthly for 10 years after 17 years,}} hfill \
{text{we shall calculate how much is needed}} hfill \
{text{to support such withdrawals}}{text{.}} hfill \
{text{For the first case, }}r = 7.2% hfill \
hfill \
PV = R cdot frac{{1 – {{left( {1 + i} right)}^m}}}{i} hfill \
hfill \
PV = 5000 cdot frac{{1 – {{left( {1 + frac{{0.072}}{{12}}} right)}^{ – 12left( {10} right)}}}}{{0.072/12}} = 426,832.85 hfill \
hfill \
{text{With this target amount, we shall determine}} hfill \
{text{the quarterly regular payments required that }} hfill \
{text{will result in future value of 436,832}}{text{.85}}{text{.}} hfill \
{text{This time, }}r = 8.4% /a hfill \
hfill \
FV = R cdot frac{{{{left( {1 + i} right)}^m} – 1}}{i} hfill \
hfill \
R = frac{{FV cdot i}}{{{{left( {1 + i} right)}^m} – 1}} = frac{{left( {426,832.85} right)left( {frac{{0.084}}{4}} right)}}{{{{left( {1 + frac{{0.084}}{4}} right)}^{17left( 4 right)}} – 1}} hfill \
hfill \
R = 2882.95 hfill \
hfill \
{text{For the second case, rates are swapped,}} hfill \
{text{that is, }}r = 8.4% /a{text{ during}} hfill \
{text{her withdrawals while }}r = 7.2% /a hfill \
{text{in the 17 years}}{text{.}} hfill \
hfill \
PV = R cdot frac{{1 – {{left( {1 + i} right)}^m}}}{i} hfill \
hfill \
PV = 5000 cdot frac{{1 – {{left( {1 + frac{{0.084}}{{12}}} right)}^{ – 12left( {10} right)}}}}{{0.084/12}} = 405,017.35 hfill \
hfill \
R = frac{{FV cdot i}}{{{{left( {1 + i} right)}^m} – 1}} = frac{{405,017.35left( {frac{{0.072}}{4}} right)}}{{{{left( {1 + frac{{0.072}}{4}} right)}^{17left( 4 right)}} – 1}} hfill \
hfill \
R = 3083.96 hfill \
hfill \
{text{The difference in regular investment is}} hfill \
{text{3083}}{text{.96}} – 2882.95 = boxed{{mathbf{$ 201}}{mathbf{.01}}} hfill \
{text{Thus, if Yvette will choose the second case,}} hfill \
{text{she has to invest $ 201}}{text{.01 more per quarter}}{text{.}} hfill \
hfill \
end{gathered} ]

Yvette has to invest more $$201.01$ more for the second case.