Chapter 6

To write n% in its fractional form is to write the percent as the fraction $\frac{n}{100}$.

$3\mathrm{\%<!--\; \%\; -->}=\frac{3}{100}$

$\frac{3}{100}$

To write n% in its fractional form is to write the percent as the fraction $\frac{n}{100}$.

$94\mathrm{\%<!--\; \%\; -->}=\frac{94}{100}$

$\frac{94}{100}$

To write n% in its fractional form is to write the percent as the fraction $\frac{n}{100}$.

$67.2\mathrm{\%<!--\; \%\; -->}=\frac{67.2}{100}$

$\frac{67.2}{100}$

To write n% in its fractional form is to write the percent as the fraction $\frac{n}{100}$.

$670\mathrm{\%<!--\; \%\; -->}=\frac{670}{100}$

$\frac{670}{100}$

The decimal form of n% is found by calculating the decimal value of n ÷ 100.

9% is 9 divided by 100.

This moves the decimal point two places to the left, resulting in 0.09.

0.09

The decimal form of n% is found by calculating the decimal value of n ÷ 100.

24% is 24 divided by 100.

This moves the decimal point two places to the left, resulting in 0.24.

0.24

The decimal form of n% is found by calculating the decimal value of n ÷ 100.

2.18% is 2.18 divided by 100.

This moves the decimal point two places to the left, resulting in 0.0218.

0.0218

To convert the number x from decimal form to percent, multiply x by 100 and place a percent sign, %, after the number: $(x\times <!--\; \times \; -->100)$.

$0.41=(0.41\times <!--\; \times \; -->100)$

41%

To convert the number x from decimal form to percent, multiply x by 100 and place a percent sign, %, after the number: $(x\times <!--\; \times \; -->100)$.

$0.02=(0.02\times <!--\; \times \; -->100)$

2%

To convert the number x from decimal form to percent, multiply x by 100 and place a percent sign, %, after the number: $(x\times <!--\; \times \; -->100)$.

$9.2481=(9.2481\times <!--\; \times \; -->100)$

924.81%

The formula relating the total (base), the percent in decimal form, and the part (amount) is

$part=percent\times <!--\; \times \; -->total$ or $amount=percent\times <!--\; \times \; -->base$.

26% of 1,300

$part=percent\times <!--\; \times \; -->total$

$part=0.26\left(1,300\right)$

$part=338$

338

The formula relating the total (base), the percent in decimal form, and the part (amount) is

$part=percent\times <!--\; \times \; -->total$ or $amount=percent\times <!--\; \times \; -->base$.

225% of 915

$part=percent\times <!--\; \times \; -->total$

$part=2.25\left(915\right)$

$part=2058.75$

2058.75

The formula relating the total (base), the percent in decimal form, and the part (amount) is

$part=percent\times <!--\; \times \; -->total$.

45 is 18% of the total.

$45=0.18\times <!--\; \times \; -->total$

$\frac{45}{0.18}=\frac{\cancel{.0.18}\times <!--\; \times \; -->total}{\cancel{0.18}}$

The total is 250.

250

The formula relating the total (base), the percent in decimal form, and the part (amount) is

$part=percent\times <!--\; \times \; -->total$.

900 is 15% of the total.

$900=0.15\times <!--\; \times \; -->total$

$\frac{900}{0.15}=\frac{\cancel{.0.15}\times <!--\; \times \; -->total}{\cancel{0.15}}$

The total is 6,000.

6,000

The formula relating the total (base), the percent in decimal form, and the part (amount) is

$part=percent\times <!--\; \times \; -->total$.

25 is what percent of 40?

$25=percent\times <!--\; \times \; -->40$

$\frac{25}{40}=\frac{percent\times <!--\; \times \; -->\cancel{40}}{\cancel{40}}$

$0.625=percent$

To convert the number x from decimal form to percent, multiply x by 100 and place a percent sign, %, after the number: (x × 100)%.

0.625 = 62.5%

62.5%

The formula relating the total (base), the percent in decimal form, and the part (amount) is

$part=percent\times <!--\; \times \; -->total$.

292 is what percent of 730?

$292=percent\times <!--\; \times \; -->730$

$\frac{292}{730}=\frac{percent\times <!--\; \times \; -->\cancel{292}}{\cancel{292}}$

$0.4=percent$

To convert the number x from decimal form to percent, multiply x by 100 and place a percent sign, %, after the number: (x × 100)%.

0.40 = 40%

40%

The formula relating the total (base), the percent in decimal form, and the part (amount) is

$part=percent\times <!--\; \times \; -->total$ or $amount=percent\times <!--\; \times \; -->base$.

30% of $1,765

$part=percent\times <!--\; \times \; -->total$

$part=0.3\left(1,765\right)$

$part=\mathrm{\$<!--\; \$\; -->}529.50$

The amount is $529.50.

$529.50

The formula relating the total (base), the percent in decimal form, and the part (amount) is

$part=percent\times <!--\; \times \; -->total$.

1,679 is what percent of 2,532?

$1,679=percent\times <!--\; \times \; -->2,532$

$\frac{1,679}{2,532}=\frac{percent\times <!--\; \times \; -->\cancel{2,532}}{\cancel{2,532}}$

$0.6631\approx <!--\; \approx \; -->percent$

To convert the number x from decimal form to percent, multiply x by 100 and place a percent sign, %, after the number: (x × 100)%.

0.6631 = 66.31%

66.31%

The formula relating the total (base), the percent in decimal form, and the part (amount) is

$part=percent\times <!--\; \times \; -->total$.

52 is 65% of the total.

$52=0.65\times <!--\; \times \; -->total$

$\frac{52}{0.65}=\frac{\cancel{.0.65}\times <!--\; \times \; -->total}{\cancel{0.65}}$

The total is 80 bulbs.

80 bulbs.

discount = percent discount × original price

$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}=0.32\times <!--\; \times \; -->1,550=\mathrm{\$<!--\; \$\; -->}496.00$

sale price = original price – discount

sale price = 1,550 – 496 = $1,054

The discount is $496 and the sale price is $1,054.

Discount is $496; sale price is $1,054.00

discount = percent discount × original price

$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}=0.10\times <!--\; \times \; -->27.50=\mathrm{\$<!--\; \$\; -->}2.75$

sale price = original price – discount

sale price = 27.50 – 2.75 = $24.75

The discount is $2.75 and the sale price is $24.75.

Discount is $2.75; sale price is $24.75

35%

sale price = original price – discount

162.50 = 250 – discount

discount = 87.50

87.50 is what percent of 250?

87.50 = percent discount × 250

$\frac{87.50}{250}=\frac{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\times <!--\; \times \; -->\cancel{250}}{\cancel{250}}$

$0.35=\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}$

The percent discount is 35%.

12%

sale price = original price – discount

17.16 = 19.50 – discount

discount = 2.34

2.34 is what percent of 19.50?

2.34 = percent discount × 19.50

$\frac{2.34}{19.50}=\frac{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\times <!--\; \times \; -->\cancel{19.50}}{\cancel{19.50}}$

$0.12=\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}$

The percent discount is 12%.

$13.00

If the percent discount was 15%, the sale price is 85% of the original price.

sale price = original price × (1 – percent discount)

$11.05=\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}\times <!--\; \times \; -->\left(1-<!--\; -\; -->0.15\right)$

$11.05=\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}\times <!--\; \times \; -->\left(0.85\right)$

$\frac{11.05}{0.85}=\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}$

The original price was $13.

$220.00

If the percent discount was 9%, the sale price is 91% of the original price.

sale price = original price × (1 – percent discount)

$200.20=\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}\times <!--\; \times \; -->\left(1-<!--\; -\; -->0.09\right)$

$200.20=\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}\times <!--\; \times \; -->\left(0.91\right)$

$\frac{200.20}{0.91}=\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}$

The original price was $220.

discount = percent discount × original price

discount = 0.60 × 550 = $330

sale price = original price – discount

sale price = 550 – 330 = $220

The discount is $330 and the sale price is $220.

Discount is $330; sale price of the bed is $220.

sale price = original price – discount

13.43 = 15.80 – discount

discount = 2.37

2.37 is what percent of 15.80?

2.37 = percent discount × 15.80

$\frac{2.37}{15.80}$ = percent discount

0.15 = percent discount

The percent discount is 15%.

15%

100% – 26% = 74%

If the percent discount was 26%, the sale price is 76% of the original price.

sale price = original price × (1 – percent discount)

$43.66=\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}\times <!--\; \times \; -->\left(1-<!--\; -\; -->0.26\right)$

$43.66=\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}\times <!--\; \times \; -->\left(0.74\right)$

$\frac{43.66}{0.74}=\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}$

The original price was $59.

$59.00

markup = percent markup × cost

markup = 0.22 × $1,800

markup = $396

retail price = cost + markup

retail price = 1,800 + 396 = $2,196

Markup is $396; retail price is $2,196.00.

markup = percent markup × cost

markup = 0.10 × $10.50

markup = $1.05

retail price = cost + markup

retail price = 10.50 + 1.05 = $11.55

Markup is $1.05; retail price is $11.55.

retail price = cost + markup

190 = 120 + markup

70 = markup

The markup is what percent of cost?

70 is what percent of 120?

70 = percent markup × 120

$\frac{70}{120}$ = percent markup

0.5833 ≈ percent markup

58.33% ≈ percent markup

58.33%

retail price = cost + markup

1.14 = 0.38 + markup

0.76 = markup

The markup is what percent of cost?

0.76 is what percent of 0.38?

0.76 = percent markup × 0.38

$\frac{0.76}{0.38}$ = percent markup

2 = percent markup

200% = percent markup

200%

retail price = cost × (1 + percent markup)

$35.00

retail price = cost × (1 + percent markup)

$8.99

markup = percent markup × cost

markup = 2.50 × 360

markup = 900

retail price = cost + markup

retail price = 360 + 900 = $1,260

The retail price is $1,260.

The retail price of the bed is $1,260.00.

retail price = cost + markup

45.36 = markup

The markup is what percent of cost?

45.36 is what percent of 189?

45.36 = percent markup × 189

$\frac{45.36}{189}$ = percent markup

0.24 = percent markup

24% = percent markup

The percent markup is 24 percent.

24%

retail price = cost × (1 + percent markup)

$38.00

sales tax = purchase price × tax rate

sales tax = 1,499 × 0.07

sales tax = $104.93

total price = purchase price + sales tax = 1,499 + 104.93 = $1,603.93

The sales tax is $104.93 and the total price is $1,603.93.

Sales tax is $104.93; total price is $1,603.93.

sales tax = purchase price × tax rate

sales tax = 26.89 × 0.07

total price = purchase price + sales tax = 26.89 + 1.88 = $28.77

The sales tax is $1.88 and the total price is $28.77.

Sales tax is $1.88; total price is $28.77.

total price = purchase price + sales tax

88.30 = 83.90 + sales tax

$4.40 = sales tax

sales tax = purchase price × tax rate

4.40 = 83.90 × tax rate

$\frac{4.40}{83.90}=$ tax rate

0.052443… ≈ tax rate

The tax rate is approximately 5.24%.

Sales tax rate = 5.25%

total price = purchase price + sales tax

509.20 = 477 + sales tax

$32.20 = sales tax

sales tax = purchase price × tax rate

32.20 = 477 × tax rate

$\frac{32.20}{477}=$ tax rate

0.0675 ≈ tax rate

The tax rate is approximately 6.75%.

Sales tax rate = 6.75%

total price = purchase price × (1 + tax rate)

157.81 = purchase price × (1 + 0.0825)

157.81 = purchase price × (1.0825)

$\frac{157.81}{1.0825}$ = purchase price

$145.78

total price = purchase price × (1 + tax rate)

522.01 = purchase price × (1 + 0.0675)

522.01 = purchase price × (1.0675)

$\frac{522.01}{1.0675}$ = purchase price

$489.00

sales tax = purchase price × tax rate

sales tax = 1,149 × 0.0968

total price = purchase price + sales tax = 1,149 + 111.22 = $1,260.22

Daryl’s sales tax is $111.22. The total price is $1,260.22.

Sales tax is $111.22. The total price Daryl pays is $1,260.22.

total price = purchase price × (1 + tax rate)

153 = purchase price × (1 + 0.0625)

153 = purchase price × (1.0625)

$\frac{153}{1.0625}$ = purchase price

$144 = purchase price

total price = purchase price + sales tax

153 = 144 + sales tax

$9 = sales tax

The sales tax is $9.

The sales tax is $9.

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t$

$I=6,700\times <!--\; \times \; -->0.1199\times <!--\; \times \; -->3$

The interest is $2,409.99.

The loan payoff is the sum of the principal and the interest.

$T=P+I=6,700+2,409.99=\mathrm{\$<!--\; \$\; -->}9109.99$

Interest = $2,409.99, the loan payoff is $9,109.99

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t$

$I=25,800\times <!--\; \times \; -->0.069\times <!--\; \times \; -->5$

The interest is $8,901.

The loan payoff is the sum of the principal and the interest.

$T=P+I=25,800+8,901=\mathrm{\$<!--\; \$\; -->}34,701$

Interest = $8,901, the loan payoff is $34,701.00

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t$

$I=9,500\times <!--\; \times \; -->0.0925\times <!--\; \times \; -->1$

The interest is $878.75.

The loan payoff is the sum of the principal and the interest.

$T=P+I=9,500+878.75=\mathrm{\$<!--\; \$\; -->}10,378.75$

The total repayment is $10,378.75.

She will pay $878.75 in interest, total repayment is $10,378.75.

Remember that 6 months is 0.5 years.

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t$

$I=8,400\times <!--\; \times \; -->0.1733\times <!--\; \times \; -->0.5$

The interest is $727.86.

The loan payoff is the sum of the principal and the interest.

$T=P+I=8,400+727.86=\mathrm{\$<!--\; \$\; -->}9,127.86$

The payoff is $9,127.86.

He will pay $727.86 in interest, payoff is $9,127.86.

Remember that 60 days is $\frac{60}{365}$ years.

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t$

$I=3,700\times <!--\; \times \; -->0.1899\times <!--\; \times \; -->\frac{60}{365}$

The interest is $115.51 (rounded to the nearest cent; interest is usually rounded up when paid to the lender).

The loan payoff is the sum of the principal and the interest.

$T=P+I=3,700+115.50=\mathrm{\$<!--\; \$\; -->}3,815.50$

The loan payoff is $3,815.50.

The cost to borrow is $115.51 and their payoff is $3,815.51.

$\mathrm{F}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=FV=P+I$

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t=4,500\times <!--\; \times \; -->0.0188\times <!--\; \times \; -->3=\mathrm{\$<!--\; \$\; -->}253.80$

$FV=P+I=4,500+253.80=\mathrm{\$<!--\; \$\; -->}4,753.80$

Interest = $253.8, $FV$ = $4,753.80

$\mathrm{F}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=FV=P+I$

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t=2,000\times <!--\; \times \; -->0.0203\times <!--\; \times \; -->10=\mathrm{\$<!--\; \$\; -->}406$

$FV=P+I=2,000+406=\mathrm{\$<!--\; \$\; -->}2,406$

Interest = $406.00, $FV$ = $2,406.00

$\mathrm{F}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=FV=P+I$

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t=120,000\times <!--\; \times \; -->0.031\times <!--\; \times \; -->\frac{100}{365}=\mathrm{\$<!--\; \$\; -->}1,019.18$

$FV=P+I=120,000+1,019.18=\mathrm{\$<!--\; \$\; -->}121,019.18$

Interest= $1019.18, $FV$ = $121,019.18

$\mathrm{F}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=FV=P+I$

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t=4,680\times <!--\; \times \; -->0.0155\times <!--\; \times \; -->\frac{42}{12}=\mathrm{\$<!--\; \$\; -->}253.89$

$FV=P+I=4,680+253.89=\mathrm{\$<!--\; \$\; -->}4,933.89$

Interest = $253.89, $FV$ = $4,933.89

$\mathrm{F}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=FV=P+I$

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t=4,900\times <!--\; \times \; -->0.0395\times <!--\; \times \; -->7=\mathrm{\$<!--\; \$\; -->}1,354.85$

$FV=P+I=4,900+1,354.85=\mathrm{\$<!--\; \$\; -->}6,254.85$

Mia will receive $6,254.85.

$846.58

$172.91

Subtract the interest off the loan payment.

$700 – 9.59 = $690.41 went toward paying off the loan.

$1,400 – 690.41 = $709.59 is the remaining balance of the loan.

$709.59

Subtract the interest off the loan payment.

$10,000 – 182.74 = $9,817.26 went toward paying off the loan.

$23,000 – 9,817.26 = $13,182.74 is the remaining balance of the loan.

$13,182.74

Interest on the loan after 75 days:

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t=3,500\times <!--\; \times \; -->0.112\times <!--\; \times \; -->\frac{75}{365}\approx <!--\; \approx \; -->\mathrm{\$<!--\; \$\; -->}80.547$

Interest on the loan after 30 days:

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t=3,500\times <!--\; \times \; -->0.112\times <!--\; \times \; -->\frac{30}{365}\approx <!--\; \approx \; -->\mathrm{\$<!--\; \$\; -->}32.22$ (rounded to the nearest cent)

Subtract the partial interest off the loan payment.

$1,250 – 32.22 = $1,217.78 went toward paying off the loan.

$3,500 – 1217.78 = $2,282.22 is the remaining balance of the loan.

There are 45 more days to pay interest:

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t=2282.22\times <!--\; \times \; -->0.112\times <!--\; \times \; -->\frac{45}{365}\approx <!--\; \approx \; -->\mathrm{\$<!--\; \$\; -->}31.52$ more interest will accrue.

The total payout left of the loan is the balance of the loan plus the additional interest.

$2,282.22+31.52=\mathrm{\$<!--\; \$\; -->}2,313.74$

$2,313.74

The amount of monthly payments, A, for a loan with principal P, monthly simple interest rate r (in decimal form), for t number of months is found using the formula $A=P\times <!--\; \times \; -->\frac{r\times <!--\; \times \; -->{\left(1+r\right)}^t}{{\left(1+r\right)}^t-<!--\; -\; -->1}$. The monthly interest rate is the annual interest rate divided by 12.

5 years = 60 months

$A=23,660\times <!--\; \times \; -->\frac{\frac{0.0476}{12}\times <!--\; \times \; -->{\left(1+\frac{0.0476}{12}\right)}^{60}}{{\left(1+\frac{0.0476}{12}\right)}^{60}-<!--\; -\; -->1}=\mathrm{\$<!--\; \$\; -->}443.90$

$443.90

The present value, PV, of money deposited at an annual, simple interest rate of r (in decimal form) for time t (in years) with a specified future value of FV is calculated with the formula $PV=\frac{FV}{1+rt}$. Present value, in this calculation, is always rounded up. Otherwise, the future value may fall short of the target future value.

$14,285.72 needs to be invested in an account earning 7.5% annual simple interest to have $25,000 at the end of the 10 years.

$14,285.72. $14,285.72 needs to be invested so that, after 10 years at 7.5% interest, the investment will be worth $25,000.

The present value, PV, of money deposited at an annual, simple interest rate of r (in decimal form) for time t (in years) with a specified future value of FV is calculated with the formula $PV=\frac{FV}{1+rt}$. Present value, in this calculation, is always rounded up. Otherwise, the future value may fall short of the target future value.

$97,709.93 needs to be invested in an account earning 6.5% annual simple interest to have $320,000 at the end of the 35 years.

$97,709.93. $97,709.93 needs to be invested so that, after 35 years at 6.5% interest, the investment will be worth $320,000.

The present value, PV, of money deposited at an annual, simple interest rate of r (in decimal form) for time t (in years) with a specified future value of FV is calculated with the formula $PV=\frac{FV}{1+rt}$. Present value, in this calculation, is always rounded up. Otherwise, the future value may fall short of the target future value.

270 months is 22.5 years.

$48,813.56 needs to be invested in an account earning 3.75% annual simple interest to have $90,000 at the end of the 22.5 years.

$48,813.56. $48,813.56 needs to be invested so that, after 270 months at 3.75% interest, the investment will be worth $90,000.

The present value, PV, of money deposited at an annual, simple interest rate of r (in decimal form) for time t (in years) with a specified future value of FV is calculated with the formula $PV=\frac{FV}{1+rt}$. Present value, in this calculation, is always rounded up. Otherwise, the future value may fall short of the target future value.

$10,549.46

The interest $(I)$ is the principal $(P)$ times the annual simple interest rate as a decimal $(r)$ times the time in years $(t)$.

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t$

Step 1: End of the first year

$I=5,000\times <!--\; \times \; -->0.03\times <!--\; \times \; -->1=150$

$T=P+I=5,000+150=\mathrm{\$<!--\; \$\; -->}5,150$

The interest is added into the account and you begin earning interest on $5,150.

Step 2: End of the second year

$I=5,150\times <!--\; \times \; -->0.03\times <!--\; \times \; -->1=154.50$

$T=P+I=5,150+154.50=\mathrm{\$<!--\; \$\; -->}5,304.50$

The interest is added into the account and you begin earning interest on $5,304.50.

Step 3: End of the third year

$T=P+I=5,304.50+159.14=\mathrm{\$<!--\; \$\; -->}5,463.64$

The interest is added into the account and you begin earning interest on $5,463.64.

Step 4: End of the fourth year

$T=P+I=5,463.64+163.91=\mathrm{\$<!--\; \$\; -->}5,627.54$

The interest is added into the account and you begin earning interest on $5,627.54.

The CD is worth $5,627.54 after 4 years.

From the previous exercise, you know she earned $5,627.54 when compounded annually.

Find the amount if earning simple interest:

$I=P\times <!--\; \times \; -->r\times <!--\; \times \; -->t$

$I=5,000\times <!--\; \times \; -->0.03\times <!--\; \times \; -->4=\mathrm{\$<!--\; \$\; -->}600$

$T=P+I=5,000+600=\mathrm{\$<!--\; \$\; -->}5,600$

You earned $27.54 more when compounding interest.

CD would have been worth $5,600. Oksana earned $27.54 more with compound interest.

The future value of an investment, $A$, when the principal, $P$, is invested at an annual interest rate of $r$ (in decimal form), compounded $n$ times per year for $t$ years is found using the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

Hint: Find $nt$ before you plug numbers into your calculator! $nt=\left(12\right)\left(10\right)=120$ It will help you get the right results. Use those parentheses in your calculator, too!

$A=7,600{\left(1+\frac{0.041}{12}\right)}^{120}$

$A\approx <!--\; \approx \; -->\mathrm{\$<!--\; \$\; -->}11,443.81$

$11,443.81

The future value of an investment, $A$, when the principal, $P$, is invested at an annual interest rate of $r$ (in decimal form), compounded $n$ times per year for $t$ years is found using the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

Hint: Find $nt$ before you plug numbers into your calculator! $nt=\left(4\right)\left(25\right)=100$ It will help you get the right results. Use those parentheses in your calculator, too!

$A=13,250{\left(1+\frac{0.0279}{4}\right)}^{100}$

$A\approx <!--\; \approx \; -->\mathrm{\$<!--\; \$\; -->}26,551.23$

$26,551.23

The future value of an investment, $A$, when the principal, $P$, is invested at an annual interest rate of $r$ (in decimal form), compounded $n$ times per year for $t$ years is found using the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

Hint: Find $nt$ before you plug numbers into your calculator! $nt=\left(12\right)\left(13\right)=156$ It will help you get the right results. Use those parentheses in your calculator, too!

$A=3,000{\left(1+\frac{0.051}{12}\right)}^{156}$

$A\approx <!--\; \approx \; -->\mathrm{\$<!--\; \$\; -->}5,813.64$

$5,813.64

The future value of an investment, $A$, when the principal, $P$, is invested at an annual interest rate of $r$ (in decimal form), compounded $n$ times per year for $t$ years is found using the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

Hint: Find $nt$ before you plug numbers into your calculator! $nt=\left(365\right)\left(5\right)=1,825$ It will help you get the right results. Use those parentheses in your calculator, too!

$A=3,000{\left(1+\frac{0.034}{365}\right)}^{\left(1,825\right)}$

$A\approx <!--\; \approx \; -->\mathrm{\$<!--\; \$\; -->}3,555.88$

$3,555.88

The money invested in an account bearing an annual interest rate of $r$ (in decimal form), compounded $n$ times per year for $t$ years, is called the present value, $PV$, of the account (or of the money) and found using the formula $PV=\frac{A}{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}}$ where $A$ is the value of the account at the investment’s end. Always round this value up to the next penny.

Hint: Find $n\times <!--\; \times \; -->t$ before you plug numbers into your calculator! $n\times <!--\; \times \; -->t=\left(12\right)\left(40\right)=480$

$PV=\frac{1,000,000}{{\left(1+\frac{0.575}{12}\right)}^{480}}=\mathrm{\$<!--\; \$\; -->}100,811.07$

$100,811.07

The money invested in an account bearing an annual interest rate of $r$ (in decimal form), compounded $n$ times per year for $t$ years, is called the present value, $PV$, of the account (or of the money) and found using the formula $PV=\frac{A}{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}}$ where $A$ is the value of the account at the investment’s end. Always round this value up to the next penny.

Hint: Find $n\times <!--\; \times \; -->t$ before you plug numbers into your calculator! $n\times <!--\; \times \; -->t=\left(4\right)\left(20\right)=80$

$PV=\frac{175,000}{{\left(1+\frac{0.038}{4}\right)}^{80}}=\mathrm{\$<!--\; \$\; -->}82,135.75$

$82,135.75

The money invested in an account bearing an annual interest rate of $r$ (in decimal form), compounded $n$ times per year for $t$ years, is called the present value, $PV$, of the account (or of the money) and found using the formula $PV=\frac{A}{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}}$ where $A$ is the value of the account at the investment’s end. Always round this value up to the next penny.

It is 35 years until Hajun turns 65.

Hint: Find $n\times <!--\; \times \; -->t$ before you plug numbers into your calculator! $n\times <!--\; \times \; -->t=\left(12\right)\left(35\right)=420$

$PV=\frac{1,200,000}{{\left(1+\frac{0.0723}{12}\right)}^{420}}=\mathrm{\$<!--\; \$\; -->}96,271.23$

Hajun needs to invest $96,271.23.

$96,271.23

Effective annual yield is $Y={\left(1+\frac{r}{n}\right)}^n-<!--\; -\; -->1$ where $Y=$ effective annual yield, $r=$ the interest rate in decimal form, and $n=$ the number of times the interest is compounded in a year. $Y$ is interpreted as the equivalent annual simple interest rate.

$Y={\left(1+\frac{0.07}{4}\right)}^4-<!--\; -\; -->1=\mathrm{0.071859....}$

Effective annual yield: 7.19%

Earning 7% compounded quarterly gives you an effective annual yield of 7.19%, which is the equivalent of the annual simple interest rate.

A rate of 7% compounded quarterly is equivalent to a simple interest rate of 7.19%.

Effective annual yield is $Y={\left(1+\frac{r}{n}\right)}^n-<!--\; -\; -->1$ where $Y=$ effective annual yield, $r=$ the interest rate in decimal form, and $n=$ the number of times the interest is compounded in a year. $Y$ is interpreted as the equivalent annual simple interest rate.

$Y={\left(1+\frac{0.025}{365}\right)}^{365}-<!--\; -\; -->1=\mathrm{0.025314....}$

Effective annual yield: 2.53%

Earning 2.5% compounded daily gives you an effective annual yield of 2.53%, which is the equivalent of the annual simple interest rate.

The effective annual yield is 2.53%.

Find the effective annual yield of each bank.

Community Bank has the highest effective annual yield at 3.14%.

The effective annual yields for the banks are 3.1158% for Smith Bank, 3.11% for Park Bank, 3.1381% for Town Bank, and 3.144% for Community Bank. Community Bank has the best yield.

Add the income amounts: $3,375+300=\mathrm{\$<!--\; \$\; -->}3,675$

Add the expenses: $987+589+312+167+150+400+325+470+375=\mathrm{\$<!--\; \$\; -->}3,775$

The expenses are greater than the income by $100.

Mateo’s budget is below.
His total monthly income is $3,675.00, and his monthly expenses are $3,775. Mateo falls $100.00 short each month.

The income amount is $6,093.75.

Add the expenses: $1,452.89+627.38+179+265+320+150+400+370+175=\mathrm{\$<!--\; \$\; -->}4,389.27$

The income is greater than the expenses by $1,704.48.

Maddy’s budget is below.

Her total monthly income is $6,093.75, and her monthly expenses are $4,389.27. Maddy has $1,704.48 in extra income per month. This is her cushion in the budget.

Her new budget:

New income – expenses = $246.40 with the mortgage and increase in utilities.

She still has an excess in her budget.

She would change from a rent of $1,050 to a mortgage of $1,240.

$1,240-<!--\; -\; -->1,050=\mathrm{\$<!--\; \$\; -->}190$ This is an increase of $190.

Her utilities would also change from $130 to $295.

$295-<!--\; -\; -->130=165$ This is an increase of $165.

This is an overall increase of $190+165=\mathrm{\$<!--\; \$\; -->}355.$

Her budget in the example had an excess of $601.40 as a cushion. Her cushion can cover this $355 increase in expenses.

Heather’s budget is now
The changes have added $355.00 to her budget. As her extra income is $601.40, she can afford the changes.

Income – expenses = $601.40

The 50-30-20 principle suggests spending 50 percent of your budget on necessary expenses (basic living requirements and debt services such as mortgage/rent, utilities, car, insurance, health care, groceries, gasoline, child care, and minimum debt payments), 30 percent of your budget for wants (restaurants, vacations, hobby costs), and 20 percent should be set aside (retirement funds, stocks, an emergency fund should have at least 3 months of income, other investments).

Heather’s monthly income is $4,437.40.

Heather’s total income is $4,437.40.
For the necessities, Heather should budget $2,218.70.
For her wants, she should budget $1,331.22.
For savings and extra debt service, she should budget $887.48.
Her necessities total $3,536.00, which exceeds the suggested budget amount of $2,218.70.
Her wants total $300.00, which is below the suggested budget amount of $1,331.22.
Her excess income is $601.40, which is below the suggested budget amount of $887.48.
Heather should make some changes.

The 50-30-20 principle suggests spending 50 percent of your budget on necessary expenses (basic living requirements and debt services such as mortgage/rent, utilities, car, insurance, health care, groceries, gasoline, child care, and minimum debt payments), 30 percent of your budget for wants (restaurants, vacations, hobby costs), and 20 percent should be set aside (retirement funds, stocks, an emergency fund should have at least 3 months of income, other investments).

The monthly income: $3,263.44.

Spend 50 percent on needs. Fifty percent is $1,631.72.

Spend 30 percent on wants. Thirty percent is $979.03.

Set aside 20 percent in savings, investments, and debt service. Twenty percent is $652.69.

Keeping within or below his budget can build his savings.

Elijah should budget $1,631.72 for needs, $979.03 for wants, and $652.69 for savings and debt service. When choosing where to live, what to eat, and what to drive, he should make choices that keep those costs, combined with debt service costs, gasoline, and utilities, below $1,631.72. This means he will have to make decisions about what his priorities are. Elijah should then figure out his wants, and stay within the limits here, that is, keep those costs below $979.03. Finally, he can begin to build his savings with the remaining $652.69.

The 50-30-20 principle suggests spending 50 percent of your budget on necessary expenses (basic living requirements and debt services such as mortgage/rent, utilities, car, insurance, health care, groceries, gasoline, child care, and minimum debt payments), 30 percent of your budget for wants (restaurants, vacations, hobby costs), and 20 percent should be set aside (retirement funds, stocks, an emergency fund should have at least 3 months of income, other investments).

Monthly income = $43,700 / 12

The monthly income: $3,641.66.

Spend 50 percent on needs. Fifty percent is $1,820.83.

Spend 30 percent on wants. Thirty percent is $1,092.50.

Set aside 20 percent in savings, investments, and debt service. Twenty percent is $728.33.

Her monthly income is $3,641.66.
Needs (50%): $1,820.83, Wants (30%): $1,092.50, Savings (20%): $728.33

$\mathrm{\$<!--\; \$\; -->}1,820.83-<!--\; -\; -->295-<!--\; -\; -->264-<!--\; -\; -->200=\mathrm{\$<!--\; \$\; -->}1,061.83$

$1,061.83

Spend 30 percent on wants. Thirty percent is $1,092.50.

Set aside 20 percent in savings, investments, and debt service. Twenty percent is $728.33.

$1,092.50, $728.33

The amount in the account, A, when the principal, P, is invested at an annual interest rate of r (in decimal form), compounded n times per year for t years is found using the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

It will help you get the right answer from your calculator to find nt separately.

$nt=\left(1\right)\left(5\right)=5$

$A=5,600{\left(1+\frac{0.0123}{1}\right)}^5=\mathrm{\$<!--\; \$\; -->}5,952.97$

$5,952.97

The amount in the account, A, when the principal, P, is invested at an annual interest rate of r (in decimal form), compounded n times per year for t years is found using the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

It will help you get the right answer from your calculator to find nt separately.

$nt=\left(4\right)\left(3\right)=12$

$A=3,500{\left(1+\frac{0.0223}{4}\right)}^{12}=\mathrm{\$<!--\; \$\; -->}3,741.46$

$3,741.46

The amount in the account, A, when the principal, P, is invested at an annual interest rate of r (in decimal form), compounded n times per year for t years is found using the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

It will help you get the right answer from your calculator to find nt separately.

$nt=\left(4\right)\left(3\right)=12$

$A=8,500{\left(1+\frac{0.0183}{4}\right)}^{12}=\mathrm{\$<!--\; \$\; -->}8,978.57$

$8,978.57

The return on investment (ROI) is the percent difference between the initial investment (P) and the final value of the investment (FV), or $\mathrm{R}\mathrm{O}\mathrm{I}=\frac{FV-<!--\; -\; -->P}{P}$, expressed as a percentage. The length of time of the investment is not considered in ROI.

$\mathrm{R}\mathrm{O}\mathrm{I}=\frac{FV-<!--\; -\; -->P}{P}=\frac{15,250-<!--\; -\; -->13,000}{13,000}=\frac{2,250}{13,000}\approx <!--\; \approx \; -->\mathrm{0.17307....}\approx <!--\; \approx \; -->17.31\mathrm{\%<!--\; \%\; -->}$

17.31%

The return on investment (ROI) is the percent difference between the initial investment (P) and the final value of the investment (FV), or $\mathrm{R}\mathrm{O}\mathrm{I}=\frac{FV-<!--\; -\; -->P}{P}$, expressed as a percentage. The length of time of the investment is not considered in ROI.

$\mathrm{R}\mathrm{O}\mathrm{I}=\frac{FV-<!--\; -\; -->P}{P}=\frac{7,358-<!--\; -\; -->6,500}{6,500}=\frac{858}{6,500}=0.132=13.2\mathrm{\%<!--\; \%\; -->}$

13.2%

$42,499.63

The future value of an ordinary annuity is $FV=pmt\times <!--\; \times \; -->\frac{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}{\frac{r}{n}}$, where FV is the future value of the annuity, pmt is the payment, r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. It is important to note that the number of deposits per year and the number of compounding periods per year must be the same.

It will help you get the right answer using your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(4\right)\times <!--\; \times \; -->\left(15\right)=60$

$FV=525\times <!--\; \times \; -->\frac{\left({\left(1+\frac{0.0389}{4}\right)}^{60}-<!--\; -\; -->1\right)}{\left(\frac{0.0389}{4}\right)}=\mathrm{\$<!--\; \$\; -->}42,499.63$

The extra parentheses were added to help you enter the expression properly in your calculator.

The future value of an ordinary annuity is $FV=pmt\times <!--\; \times \; -->\frac{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}{\frac{r}{n}}$, where FV is the future value of the annuity, pmt is the payment, r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. It is important to note that the number of deposits per year and the number of compounding periods per year must be the same.

It will help you get the right answer using your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(12\right)\times <!--\; \times \; -->\left(20\right)=240$

$FV=280\times <!--\; \times \; -->\frac{\left({\left(1+\frac{0.0311}{12}\right)}^{240}-<!--\; -\; -->1\right)}{\left(\frac{0.0311}{12}\right)}=\mathrm{\$<!--\; \$\; -->}93,037.59$

$93,037.59

$1,227.30

To compute the payment to reach a final goal, the payment (pmt) of an ordinary annuity to reach a specific future value (FV) is $pmt=\frac{FV\times <!--\; \times \; -->\frac{r}{n}}{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}$, where r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. Round up!

Once again, it will help you get the right answer using your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(4\right)\left(5\right)=20$

$pmt=\frac{\left(27,800\times <!--\; \times \; -->\frac{0.0516}{4}\right)}{\left({\left(1+\frac{0.0516}{4}\right)}^{20}-<!--\; -\; -->1\right)}=\mathrm{\$<!--\; \$\; -->}1,227.30$

Additional parentheses were added to help you enter the expression properly in your calculator.

Use your knowledge of percents. Remember to convert the percentage to a decimal.

3.1% of 500

$part=percent\times <!--\; \times \; -->total$

$part=\left(0.0475\right)\left(5,000\right)$ $=\mathrm{\$<!--\; \$\; -->}237.50$

Maureen receives $237.50 per year.

On the maturity date, she receives the interest and the initial payment.

$237.50+5,000=\mathrm{\$<!--\; \$\; -->}5,237.50$

$237.50 per year, $5,237.50 on maturity date.

Use the formula for the future value of an investment, A, when the principal, P, is invested at an annual interest rate of r (in decimal form), compounded n times per year for t years: $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

Hint: Find nt before you plug numbers into your calculator!

$nt=\left(1\right)\left(12\right)=12$

It will help you get the right results. Use those parentheses in your calculator, too!

$A=23,000{\left(1+\frac{0.138}{1}\right)}^{12}$

$A=\mathrm{\$<!--\; \$\; -->}108,501.30$

Rixie expects the stock to be worth $108,501.30.

$108,501.30

The percent yield for a stock, Yld%, is $\mathrm{Y}\mathrm{l}\mathrm{d}\mathrm{\%<!--\; \%\; -->}=\frac{\mathrm{A}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{D}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{d}}{\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{P}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}}\times <!--\; \times \; -->100\mathrm{\%<!--\; \%\; -->}$.

$\text{Yld}\mathrm{\%<!--\; \%\; -->}=\frac{\text{1}.\text{60}}{\text{37}.\text{40}}\times <!--\; \times \; -->100\mathrm{\%<!--\; \%\; -->}\approx <!--\; \approx \; -->4.28\mathrm{\%<!--\; \%\; -->}$

4.28%

The percent yield for a stock, Yld%, is $\mathrm{Y}\mathrm{l}\mathrm{d}\mathrm{\%<!--\; \%\; -->}=\frac{\mathrm{A}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{D}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{d}}{\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{P}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{e}}\times <!--\; \times \; -->100\mathrm{\%<!--\; \%\; -->}$.

$\text{Yld}\mathrm{\%<!--\; \%\; -->}=\frac{\text{2}.\text{41}}{\text{73}.\text{22}}\times <!--\; \times \; -->100\mathrm{\%<!--\; \%\; -->}\approx <!--\; \approx \; -->3.29\mathrm{\%<!--\; \%\; -->}$

3.29%

Look for the “Open” label to find the current price of $30.68.

Looking at the table, the current price of a share is $30.68.

Look for the “52 Week Range” label, which tells you the 52-week high of $56.28 and the 52-week low of $30.25.

The high was $56.28, and the low was $30.05.

Look for the “Ex-Dividend Date” label, which tells you when the dividend is expected: August 4, 2022.

August 4, 2022

Look for “Yield” on the display, which tells you the yield is 4.77%.

4.77%

Look for the “EPS” label, which tells you the earnings per share is $4.66.

$4.66

$336.00

Each share pays $1.12.

Because the number of shares is 300, multiply 300 by $1.12.

$\text{300(1}\text{.12) = \$336}$

Yulia earns $336.

Ginny bought 200 shares of stock at $9.76 per share: $200\left(9.76\right)=\mathrm{\$<!--\; \$\; -->}1,952$.

Later, 200 shares sold for $10.02 per share: $200\left(10.02\right)=\mathrm{\$<!--\; \$\; -->}2,004$.

$2,004-<!--\; -\; -->1,952=\mathrm{\$<!--\; \$\; -->}\mathrm{526.67.2}$

$52

The return on investment (ROI) is the percent difference between the initial investment (P) and the final value of the investment (FV), or $\mathrm{R}\mathrm{O}\mathrm{I}=\frac{FV-<!--\; -\; -->P}{P}$, expressed as a percentage. The length of time of the investment is not considered in ROI.

$\mathrm{R}\mathrm{O}\mathrm{I}=\frac{2,004-<!--\; -\; -->1,952}{2,004}=\frac{52}{2,004}=\mathrm{0.0259481....}\approx <!--\; \approx \; -->2.6\mathrm{\%<!--\; \%\; -->}$

2.6%

The future value is $FV=pmt\times <!--\; \times \; -->\frac{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}{\frac{r}{n}}$, where FV is the future value, pmt is the payment, r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. It is important to note that the number of deposits per year and the number of compounding periods per year must be the same.

It will help you get the right answer using your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(1\right)\times <!--\; \times \; -->\left(15\right)=15$

$FV=3,200\times <!--\; \times \; -->\frac{{\left(1+\frac{0.108}{1}\right)}^{15}-<!--\; -\; -->1}{\frac{0.108}{1}}\text{ = \$108,352}\text{.43}$

The extra parentheses were added to help you enter the expression properly in your calculator.

$108,352.43

$1,815.83

To compute the payment to reach a final goal, the payment (pmt) of an ordinary annuity or mutual fund to reach a specific future value (FV) is $pmt=\frac{FV\times <!--\; \times \; -->\frac{r}{n}}{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}$, where r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. Round up!

Once again, it will help you get the right answer using your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(1\right)\left(40\right)=40$

$pmt=\frac{\left(1,000,000\times <!--\; \times \; -->\frac{0.108}{1}\right)}{\left({\left(1+\frac{0.108}{1}\right)}^{40}-<!--\; -\; -->1\right)}\text{ = \$1,815}\text{.83}$

Additional parentheses were added to help you enter the expression properly in your calculator.

23.75%

Each year, Maureen was paid 4.75% of the $5,000 she invested.

$part=percent\times <!--\; \times \; -->total$

$part=\left(0.0475\right)\left(5,000\right)\text{ = \$237}\text{.50}$

Maureen is paid interest each year for five years.

$5\left(237.50\right)=\mathrm{\$<!--\; \$\; -->}1,187.50$

The return on investment (ROI) is the percent difference between the initial investment (P) and the final value of the investment (FV), or $\mathrm{R}\mathrm{O}\mathrm{I}=\frac{FV-<!--\; -\; -->P}{P}$, expressed as a percentage. The length of time of the investment is not considered in ROI.

$\mathrm{R}\mathrm{O}\mathrm{I}=\frac{1,187.50}{5,000}=0.02375=23.75\mathrm{\%<!--\; \%\; -->}$

Maureen’s FV is the $5,000 she invested plus the $1,187.50 she earned for a total of $6,187.50. Her principal was $5,000. The time is 5 years.

The annual return $={\left(\frac{\mathrm{F}\mathrm{V}}{P}\right)}^{\left(\frac{1}{t}\right)}-<!--\; -\; -->1$, where t is the number of years, FV is the new value, and P is the starting principal. Convert the answer to a percent.

annual return $={\left(\frac{\text{6},\text{187}.\text{50}}{5,000}\right)}^{\left(\frac{1}{5}\right)}-<!--\; -\; -->1\text{ = 4}\text{.35\% }$

Maureen’s annual return on investment is 4.35 percent.

4.35%

The return on investment (ROI) is the percent difference between the initial investment (P) and the final value of the investment (FV), or $\mathrm{R}\mathrm{O}\mathrm{I}=\frac{FV-<!--\; -\; -->P}{P}$, expressed as a percentage. The length of time of the investment is not considered in ROI.

$\mathrm{R}\mathrm{O}\mathrm{I}=\frac{108,501.30-<!--\; -\; -->23,000}{23,000}=\frac{85,501.30}{23,000}=\mathrm{3.747449826....}=371.74\mathrm{\%<!--\; \%\; -->}$

Rixie’s ROI is 371.74 percent.6.72.2

371.74%

The annual return $={\left(\frac{\mathrm{F}\mathrm{V}}{P}\right)}^{\left(\frac{1}{t}\right)}-<!--\; -\; -->1$, where t is the number of years, FV is the new value, and P is the starting principal. Convert the answer to a percent.

annual return $={\left(\frac{\text{108},\text{501}.\text{30}}{23,000}\right)}^{\left(\frac{1}{12}\right)}-<!--\; -\; -->1\text{ = 13}\text{.8\% }$

The annual return was 13.8 percent.

13.8%

$3,336.13

To compute the payment to reach a final goal for an annuity or mutual fund, the payment (pmt) is $pmt=\frac{FV\times <!--\; \times \; -->\frac{r}{n}}{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}$, where r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. Round up!

Once again, it will help you get the right answer using your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(1\right)\left(14\right)=14$

$pmt=\frac{\left(90,000\times <!--\; \times \; -->\frac{0.095}{1}\right)}{\left({\left(1+\frac{0.095}{1}\right)}^{14}-<!--\; -\; -->1\right)}\text{ = \$3,336}\text{.13}$

Additional parentheses were added to help you enter the expression properly in your calculator.

Pete and Erin need to set aside $3,336.13 each year.

An employer will match up to 7.5% of any employee’s salary.

$\text{0}\text{.075(72,500) = \$5,460}$

The amount Jamie can deposit that will be matched by the employer is $5,460.

$5,460

The amount Jamie deposits plus the employer’s match: $\text{5,460 + 5,460 = \$10,920}$.

The total is $10,920.

$10,920

Jamie deposits $5,460. The account ends up with $10,920.

The return on investment (ROI) is the percent difference between the initial investment (P) and the final value of the investment (FV), or $\mathrm{R}\mathrm{O}\mathrm{I}=\frac{FV-<!--\; -\; -->P}{P}$, expressed as a percentage. The length of time of the investment is not considered in ROI.

$\mathrm{R}\mathrm{O}\mathrm{I}=\frac{10,920-<!--\; -\; -->5,460}{5,460}=\frac{5,460}{5,460}=1=100\mathrm{\%<!--\; \%\; -->}$

The ROI is 100%.

100% return on the $5,460 deposit

$813,128.60

The future value of an ordinary annuity or mutual fund is $FV=pmt\times <!--\; \times \; -->\frac{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}{\frac{r}{n}}$, where FV is the future value of the annuity, pmt is the payment, r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. It is important to note that the number of deposits per year and the number of compounding periods per year must be the same.

It will help you get the right answer using your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(1\right)\times <!--\; \times \; -->\left(25\right)=25$

The combined monthly payment is $\text{450 + 350 = \$800}$.

It is only compounded annually, so for the payment, multiply the monthly payment by 12.

$\text{12(800) = \$9,600}$

$FV=9,600\times <!--\; \times \; -->\frac{\left({\left(1+\frac{0.09}{1}\right)}^{25}-<!--\; -\; -->1\right)}{\left(\frac{0.09}{1}\right)}\text{ = \$813,128}\text{.60}$

The extra parentheses were added to help you enter the expression properly in your calculator.

Crystal’s account will be worth $813,128.60.

$457,384.83

It is only compounded annually, so for the payment, multiply the monthly payment by 12.

$\text{12(450) = \$5,400}$

$FV=5,400\times <!--\; \times \; -->\frac{\left({\left(1+\frac{0.09}{1}\right)}^{25}-<!--\; -\; -->1\right)}{\left(\frac{0.09}{1}\right)}\text{ = \$457,384}\text{.84}$

Without matching funds, her account will be worth $457,384.84.

The amount of interest, I, to be paid for one period of a loan with remaining principal, P, is $I=P\times <!--\; \times \; -->\frac{r}{n}$, where r is the interest rate in decimal form and n is the number of payments in a year. Because r is the interest rate in a year, t is 1 and does not impact the calculation. Note, interest paid to lenders is always rounded up to the nearest penny.

$I=56,945\times <!--\; \times \; -->\frac{0.075}{12}=\mathrm{\$<!--\; \$\; -->}355.91$

$355.91

The amount of interest, I, to be paid for one period of a loan with remaining principal, P, is $I=P\times <!--\; \times \; -->\frac{r}{n}$, where r is the interest rate in decimal form and n is the number of payments in a year. Because r is the interest rate in a year, t is 1 and does not impact the calculation. Note, interest paid to lenders is always rounded up to the nearest penny.

$I=25,850\times <!--\; \times \; -->\frac{0.029}{12}=\mathrm{\$<!--\; \$\; -->}62.48$

$62.48

To compute the payment to pay down a loan with beginning principal, P, the payment (pmt) is $pmt=\frac{P\times <!--\; \times \; -->\frac{r}{n}\times <!--\; \times \; -->{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}}{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}$, where r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. Note, payment to lenders is always rounded up to the next penny.

Once again, it will help you get the right answer from your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(12\right)\left(10\right)=120$

$pmt=\frac{17,950\times <!--\; \times \; -->\frac{0.075}{12}\times <!--\; \times \; -->{\left(1+\frac{0.075}{12}\right)}^{120}}{{\left(1+\frac{0.075}{12}\right)}^{120}-<!--\; -\; -->1}=\mathrm{\$<!--\; \$\; -->}213.07$

Additional parentheses were added to help you enter the expression properly in your calculator.

The payment is $213.07.

$213.07

To compute the payment to pay down a loan with beginning principal, P, the payment (pmt) is $pmt=\frac{P\times <!--\; \times \; -->\frac{r}{n}\times <!--\; \times \; -->{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}}{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}$, where r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. Note, payment to lenders is always rounded up to the next penny.

Once again, it will help you get the right answer from your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(12\right)\left(20\right)=240$

$pmt=\frac{33,760\times <!--\; \times \; -->\frac{0.043}{12}\times <!--\; \times \; -->{\left(1+\frac{0.043}{12}\right)}^{120}}{{\left(1+\frac{0.043}{12}\right)}^{240}-<!--\; -\; -->1}=\mathrm{\$<!--\; \$\; -->}209.96$

Additional parentheses were added to help you enter the expression properly in your calculator.

The payment is $209.96.

$209.96

Look at the top row. The label to the left says “Loan Amount.” To the right, you see the loan amount, $34,700. This is usually called the principal in the formulas.

Look for the label called “Interest Rate (annual)” in the second row. Beside it, you see the interest rate of 3.79%.

Look for the label in the third row called “Term (Years).” Beside it, you see the number of years for the loan, which is 6 years.

$34,700, 3.79%, 6 years

In the seventh row, second column, you see the header title, “Payment.” Beginning in the ninth row, second column, you see the monthly payment of $539.57.

$539.57

In the seventh row, first column, you see the header title, “Month.” In the last column on the right, you see the header title, “Balance.”

Scroll down in the first column to find Month 20. While in that row, look across to the last column in that row to find the balance is $25,837.42.

$25,837.42

In the seventh row, second column, you see the header title, “Month.” In the fourth column of that row, you see the header title, “Interest.”

Look down the first column for Month 5. While in that row, look across to the fourth column of that row to see that the interest for Month 5 is $104.14.

$104.14

In the seventh row, second column, you see the header title, “Month.” In the fifth column of that row, you see the header title, “Total Interest.”

Look down the first column for Month 24. While in that row, look across to the fifth column of that row to see that the total interest for Month 24 is $2,246.62.

$2,246.62

The cost of finance is the sum of the interest and any fees paid for the loan. Add that to the interest.

$\text{1,400 + 100 + 4,937}\text{.18 = \$6,437}\text{.18}$

$6,437.18

The costs are $34,845.

The awards are $\text{17,500 + 5,600 + 2,000 = 25,100}$.

$34,845-<!--\; -\; -->25,100=\mathrm{\$<!--\; \$\; -->}9,745$

The funding gap is $9,745.

$9,745

The debt so far:

$\text{5,500 + 8,470 + 10,000 + 7,890 + 11,900 = \$43,760}$

Tiana is an independent student, so for the first year, she can borrow $9,500. For the second year, she can borrow $10,500. For additional years, she can borrow $12,500.

The total maximum for undergraduates is $57,500.

How much can she still borrow? $57,500-<!--\; -\; -->43,760=\mathrm{\$<!--\; \$\; -->}13,740$.

Tiana still is $13,740 away from the total allowed maximum, so she can borrow the yearly maximum of $12,500 in her sixth year.

Tiana can take out $12,500, the maximum in year 6.

The costs are $39,200 for the third year.

The awards are $19,500+3,850+5,000=\mathrm{\$<!--\; \$\; -->}28,350$.

$39,200-<!--\; -\; -->28,350=\mathrm{\$<!--\; \$\; -->}10,850$

The funding gap is $10,850.

Assuming that Makenzy is a dependent student, the loan limit for a third-year student is $7,500 (but no more than $5,500 in subsidized loans).

After the maximum amount of loans, find the remaining amount to be funded.

$10,850-<!--\; -\; -->7,500=\mathrm{\$<!--\; \$\; -->}3,350$

Makenzy needs to come up with another $3,350 to fund her education.

Gap = $10,850, Federal loans = $7,500, Remaining $3,350

The amount due for a loan, A, when the principal, P, an interest rate of r (in decimal form), compounded n times per year for t years is found using the formula $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

Hint: Find nt before you plug numbers into your calculator! It will help you get the right results.

$nt=\left(12\right)\left(6\right)=72$ if paid the full six years.

But they are asking for the balance in May when starting in August, so subtract three payments.

$nt=72-<!--\; -\; -->3=69$

Use those parentheses in your calculator, too!

$A=2,000{\left(1+\frac{0.07}{12}\right)}^{69}$

$A=\mathrm{\$<!--\; \$\; -->}2,987.63$

$2,987.63

$82.70

The standard plan is 12 payments a year for 10 years.

To compute the payment to pay down a loan with beginning principal, P, the payment (pmt) is $pmt=\frac{P\times <!--\; \times \; -->\frac{r}{n}\times <!--\; \times \; -->{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}}{{\left(1+\frac{r}{n}\right)}^{n\times <!--\; \times \; -->t}-<!--\; -\; -->1}$, where r is the annual interest rate (in decimal form), n is the number of compounding periods per year, and t is the number of years. Note, payment to lenders is always rounded up to the next penny.

If the number of periods per year is 1, then the formula is simplified to $pmt=\frac{P\times <!--\; \times \; -->r\times <!--\; \times \; -->{\left(1+r\right)}^t}{{\left(1+r\right)}^t-<!--\; -\; -->1}$.

Once again, it will help you get the right answer from your calculator to find $n\times <!--\; \times \; -->t$ first.

$n\times <!--\; \times \; -->t=\left(12\right)\left(10\right)=120$

$pmt=\frac{\left(7,800\times <!--\; \times \; -->\frac{0.0499}{12}\times <!--\; \times \; -->{\left(1+\frac{0.0499}{12}\right)}^{120}\right)}{\left({\left(1+\frac{0.0499}{12}\right)}^{120}-<!--\; -\; -->1\right)}=\mathrm{\$<!--\; \$\; -->}82.70$

Additional parentheses were added to help you enter the expression properly in your calculator.