Chapter 6: Money ManagementContent check — Contemporary Math, Chapter 6: Money Management

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

$3\mathrm{\%}=\frac{3}{100}$

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

$94\mathrm{\%}=\frac{94}{100}$

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

$67.2\mathrm{\%}=\frac{67.2}{100}$

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

$670\mathrm{\%}=\frac{670}{100}$

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.

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.

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.

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

$0.41=(0.41\times 100)$

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

$0.02=(0.02\times 100)$

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

$9.2481=(9.2481\times 100)$

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

$part=percent\times total$ or $amount=percent\times base$.

26% of 1,300

$part=percent\times total$

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

$part=338$

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

$part=percent\times total$ or $amount=percent\times base$.

225% of 915

$part=percent\times total$

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

$part=2058.75$

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

$part=percent\times total$.

45 is 18% of the total.

$45=0.18\times total$

$\frac{45}{0.18}=\frac{.0.18\times total}{0.18}$

The total is 250.

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

$part=percent\times total$.

900 is 15% of the total.

$900=0.15\times total$

$\frac{900}{0.15}=\frac{.0.15\times total}{0.15}$

The total is 6,000.

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

$part=percent\times total$.

25 is what percent of 40?

$25=percent\times 40$

$\frac{25}{40}=\frac{percent\times 40}{40}$

$0.625=percent$

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%

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

$part=percent\times total$.

292 is what percent of 730?

$292=percent\times 730$

$\frac{292}{730}=\frac{percent\times 292}{292}$

$0.4=percent$

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%

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

$part=percent\times total$ or $amount=percent\times base$.

30% of $1,765

$part=percent\times total$

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

$part=\mathrm{\$}529.50$

The amount is $529.50.

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

$part=percent\times total$.

1,679 is what percent of 2,532?

$1,679=percent\times 2,532$

$\frac{1,679}{2,532}=\frac{percent\times 2,532}{2,532}$

$0.6631\approx percent$

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%

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

$part=percent\times total$.

52 is 65% of the total.

$52=0.65\times total$

$\frac{52}{0.65}=\frac{.0.65\times total}{0.65}$

The total is 80 bulbs.

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

100

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

38.7% is 38.7 divided by 100.

This moves the decimal point two places to the left, resulting in 0.387.

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

$part=percent\times total$ or $amount=percent\times base$.

68% of 280

$part=percent\times total$

$part=0.68\left(280\right)$

$part=190.4$

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

$part=percent\times total$.

342 is 41% of the total.

$342=0.41\times total$

$\frac{342}{0.41}=\frac{0.41\times total}{0.41}$

The total is approximately 834.15 (rounded to two decimal places).

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

$part=percent\times total$.

80,000,000 is what percent of 332,403,650?

$80,000,000=percent\times 332,403,650$

$\frac{80,000,000}{332,403,650}=\frac{percent\times 332,403,650}{332,403,650}$

$0.24067\approx percent$

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

$0.24067$≈ 24.07% (an approximation)

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

$part=percent\times total$ or $amount=percent\times base$.

What is 20% of 80?

$part=percent\times total$

$part=0.20\left(80\right)$

$part=16$

Amazon needs 16 more drivers.

discount = percent discount × original price

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

sale price = original price – discount

sale price = 1,550 – 496 = $1,054

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

discount = percent discount × original price

$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}=0.10\times 27.50=\mathrm{\$}2.75$

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.

sale price = original price – discount

162.50 = 250 – discount

discount = 87.50

87.50 is what percent of 250?

87.50 = percent discount × 250

$\frac{87.50}{250}=\frac{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\times 250}{250}$

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

The percent discount is 35%.

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

$\frac{2.34}{19.50}=\frac{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\times 19.50}{19.50}$

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

The percent discount is 12%.

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

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

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

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

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

The original price was $13.

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

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

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

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

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

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.

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%.

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)

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

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

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

The original price was $59.

markup = percent markup × cost

markup = 0.22 × $1,800

markup = $396

retail price = cost + markup

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

markup = percent markup × cost

markup = 0.10 × $10.50

markup = $1.05

retail price = cost + markup

retail price = 10.50 + 1.05 = $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

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

retail price = cost × (1 + percent markup)

retail price = cost × (1 + percent markup)

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.

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.

retail price = cost × (1 + percent markup)

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 = 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.

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%.

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%.

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

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

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.

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.

discount = percent discount × original price

discount = 0.32 × 45 = $14.40

sale price = original price – discount

sale price = 45 – 14.40 = $30.60

The price after the discount is $30.60.

markup = percent markup × cost

markup = 0.60 × $311

markup = $186.60

retail price = cost + markup

retail price = 311 + 186.60 = $497.60

The price after the markup is $497.60.

Yes.

This pair of formulas mirrors each other.

retail price = cost + markup

total price = purchase price + sales tax

This is another pair of formulas:

markup = percent markup × cost

sales tax = purchase price × tax rate

sales tax = purchase price × tax rate

sales tax = 39.95 × 0.068

total price = purchase price + sales tax = 39.95 + 2.72 = $42.67

The total price is $42.67.

$I=P\times r\times t$

$I=6,700\times 0.1199\times 3$

The interest is $2,409.99.

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

$T=P+I=6,700+2,409.99=\mathrm{\$}9109.99$

$I=P\times r\times t$

$I=25,800\times 0.069\times 5$

The interest is $8,901.

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

$T=P+I=25,800+8,901=\mathrm{\$}34,701$

$I=P\times r\times t$

$I=9,500\times 0.0925\times 1$

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.

Remember that 6 months is 0.5 years.

$I=P\times r\times t$

$I=8,400\times 0.1733\times 0.5$

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.

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

$I=P\times r\times t$

$I=3,700\times 0.1899\times \frac{60}{365}$

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.

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

$I=P\times r\times t=4,500\times 0.0188\times 3=\mathrm{\$}253.80$

$FV=P+I=4,500+253.80=\mathrm{\$}4,753.80$

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

$I=P\times r\times t=2,000\times 0.0203\times 10=\mathrm{\$}406$

$FV=P+I=2,000+406=\mathrm{\$}2,406$

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

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

$FV=P+I=120,000+1,019.18=\mathrm{\$}121,019.18$

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

$I=P\times r\times t=4,680\times 0.0155\times \frac{42}{12}=\mathrm{\$}253.89$

$FV=P+I=4,680+253.89=\mathrm{\$}4,933.89$

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

$I=P\times r\times t=4,900\times 0.0395\times 7=\mathrm{\$}1,354.85$

$FV=P+I=4,900+1,354.85=\mathrm{\$}6,254.85$

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.

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.

Interest on the loan after 75 days:

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

Interest on the loan after 30 days:

$I=P\times r\times t=3,500\times 0.112\times \frac{30}{365}\approx \mathrm{\$}32.22$ (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:

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

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

$2,282.22+31.52=\mathrm{\$}2,313.74$

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 $A=P\times \frac{r\times {\left(1+r\right)}^t}{{\left(1+r\right)}^t-1}$. The monthly interest rate is the annual interest rate divided by 12.

5 years = 60 months

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

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.

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.

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.

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.

$I=P\times r\times t$

$I=1,500\times 0.1299\times 6$

The interest is $1,169.10.

Interest on the loan after 25 days:

Subtract the partial interest off the loan payment.

$13,000 – 174.43 = $12,825.57 went toward paying off the loan.

$24,200 – 12,825.57 = $11,374.43 is the remaining balance of the loan still due.

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 $A=P\times \frac{r\times {\left(1+r\right)}^t}{{\left(1+r\right)}^t-1}$. The monthly interest rate is the annual interest rate divided by 12.

4 years = 48 months

$A=9,800\times \frac{\frac{0.138}{12}\times {\left(1+\frac{0.138}{12}\right)}^{48}}{{\left(1+\frac{0.138}{12}\right)}^{48}-1}=\mathrm{\$}266.82$

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.

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

$I=P\times r\times t$

Step 1: End of the first year

$I=5,000\times 0.03\times 1=150$

$T=P+I=5,000+150=\mathrm{\$}5,150$

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

Step 2: End of the second year

$I=5,150\times 0.03\times 1=154.50$

$T=P+I=5,150+154.50=\mathrm{\$}5,304.50$

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.

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

Find the amount if earning simple interest:

$I=P\times r\times t$

$I=5,000\times 0.03\times 4=\mathrm{\$}600$

$T=P+I=5,000+600=\mathrm{\$}5,600$

You earned $27.54 more when compounding 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!

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

$A\approx \mathrm{\$}11,443.81$

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!

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

$A\approx \mathrm{\$}26,551.23$

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!

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

$A\approx \mathrm{\$}5,813.64$

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!

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

$A\approx \mathrm{\$}3,555.88$

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 $PV=\frac{A}{{\left(1+\frac{r}{n}\right)}^{n\times t}}$ where $A$ is the value of the account at the investment’s end. Always round this value up to the next penny.

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

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

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 $PV=\frac{A}{{\left(1+\frac{r}{n}\right)}^{n\times t}}$ where $A$ is the value of the account at the investment’s end. Always round this value up to the next penny.

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

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

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 $PV=\frac{A}{{\left(1+\frac{r}{n}\right)}^{n\times t}}$ 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 $n\times t$ before you plug numbers into your calculator! $n\times t=\left(12\right)\left(35\right)=420$

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

Hajun needs to invest $96,271.23.

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.

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.

Find the effective annual yield of each bank.

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

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! $nt=\left(12\right)\left(15\right)=180$ It will help you get the right results. Use those parentheses in your calculator, too!

$A=8,560{\left(1+\frac{0.0405}{12}\right)}^{180}$

$A=\mathrm{\$}15,698.69$

Future value with simple interest:

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

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

$I=P\times r\times t=10,000\times 0.056\times 5=\mathrm{\$}2,800$

$FV=P+I=10,000+2,800=\mathrm{\$}12,800$

Future value 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! $nt=\left(4\right)\left(5\right)=20$ It will help you get the right results. Use those parentheses in your calculator, too!

$A=10,000{\left(1+\frac{0.056}{4}\right)}^{20}$

$A=\mathrm{\$}13,205.62$

The difference between the two methods: $13205.62-12,800=\mathrm{\$}\mathrm{405.62.}$

You earn $405.62 more by compounding.

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 $PV=\frac{A}{{\left(1+\frac{r}{n}\right)}^{n\times t}}$ where $A$ is the value of the account at the investment’s end. Always round this value up to the next penny.

Hint: Find $n\times t$ before you plug numbers into your calculator! $n\times t=\left(12\right)\left(40\right)=336$

$PV=\frac{75,000}{{\left(1+\frac{0.0325}{12}\right)}^{336}}=\mathrm{\$}30,226.48$

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.

$Y={\left(1+\frac{0.0489}{4}\right)}^4-1=\mathrm{0.049804034....}$

Effective annual yield: 4.98%

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.

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.

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.

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.

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.

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.

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.

Income = $3,450

Expenses $=925+178.54+129.49+117+195+290+21.99+49.99+400=\mathrm{\$}2,307.01$

Budget $=3,450-2,307.01=\mathrm{\$}1,142.99$

He has $1,142.99 after accounting for expenses.

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 is $3,450.

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.

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

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

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.

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

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

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.

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

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

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.

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

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.

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

The future value of an ordinary annuity is $FV=pmt\times \frac{{\left(1+\frac{r}{n}\right)}^{n\times t}-1}{\frac{r}{n}}$, 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 $n\times t$ first.

$n\times t=\left(4\right)\times \left(15\right)=60$

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

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

The future value of an ordinary annuity is $FV=pmt\times \frac{{\left(1+\frac{r}{n}\right)}^{n\times t}-1}{\frac{r}{n}}$, 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 $n\times t$ first.

$n\times t=\left(12\right)\times \left(20\right)=240$

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

To compute the payment to reach a final goal, the payment (pmt) of an ordinary annuity to reach a specific future value (FV) is $pmt=\frac{FV\times \frac{r}{n}}{{\left(1+\frac{r}{n}\right)}^{n\times t}-1}$, 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 $n\times t$ first.

$n\times t=\left(4\right)\left(5\right)=20$

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

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

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!

$nt=\left(12\right)\left(4\right)=48$