Question 1

Find the accumulated amount A if the principal P is invested at an interest rate of r per year for t years. Round your answer to the nearest cent.
​
P = $180,000, r = 6%, t = 3, compounded daily
​
$__________

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Question 2

Find the nominal interest rate that, when compounded monthly, yields an effective interest rate of 16% / year. Round your answer to two decimal places. ​

Accepted Answer

The effective interest rate (EIR) can be calculated from the nominal interest rate (r) compounded monthly using the formula: $EIR = (1 + \frac{r}{12})^{12} - 1$. Given an EIR of 16% (or 0.16), we solve for r: $1.16 = (1 + \frac{r}{12})^{12}$. Solving for r gives approximately 0.1493 or 14.93%.

Question 3

The Estradas are planning to buy a house 4 year(s) from now. Housing experts in their area have estimated that the cost of a home will increase at a rate of 8% / year during that 4-year period. If this economic prediction holds true, how much can they expect to pay for a house that currently costs $40,000? Round your answer to the nearest cent.
​
$__________

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Question 4

How long will it take an investment of $4,000 to double if the investment earns interest at the rate of 7% compounded continuously? Round your answer to the nearest hundredth.
​  __________ year(s)

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Question 5

Julio purchased 700 shares of a certain stock for $22,750 (including commissions). He sold the shares 5 year(s) later and received $32,100 after deducting commissions. Find the effective annual rate of return on his investment over the 5-year period. Round your answer to two decimal places. ​

Accepted Answer

The effective annual rate of return can be calculated using the formula for compound interest: $A = P(1 + r)^n$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (initial investment), $r$ is the annual interest rate, and $n$ is the time the money is invested for in years. Rearranging the formula to solve for $r$, we get $r = \left(\frac{A}{P}\right)^{\frac{1}{n}} - 1$. Substituting the given values: $A = \$32,100$, $P = \$22,750$, and $n = 5$ years, we get $r = \left(\frac{32100}{22750}\right)^{\frac{1}{5}} - 1$. Calculating this gives an effective annual rate of return of approximately 7.13%.

Question 6

Find the interest rate needed for an investment of $3,000 to grow to an amount of $8,500 in 3 year(s) if interest is compounded monthly. Round your answer to the nearest hundredth.
​
__________%

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Question 7

Find the present value of $54,049 due in 5 year(s) at an interest rate of 7% / year compounded continuously. Round your answer to the nearest integer. ​

Accepted Answer

The formula for continuous compounding is:

$A = Pe^{rt}$

where 
A = the future value of the investment
P = the present value of the investment
e = the mathematical constant (approximately equal to 2.71828)
r = the annual interest rate (as a decimal)
t = the time period (in years)

We are given P and t, and we need to solve for A.

$A = Pe^{rt}$

$A = 54,049e^{0.07(5)}$

$A = 54,049e^{0.35}$

$A \approx 79,536.89$

Rounding to the nearest integer gives us $79,537$.

Therefore, the present value of $54,049 due in 5 years with continuous compounding at 7% is approximately $38,088.

Question 8

Bernie invested a sum of money 5 year(s) ago in a savings account, which has since paid interest at the rate of 6% / year compounded quarterly. His investment is now worth $22,927.66. How much did he originally invest? Round your answer to the nearest cent.
​  $__________

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Question 9

Find the accumulated amount after 6 year(s) if $5,000 is invested at 8% compounded continuously. Round your answer to the nearest cent.
​
$__________

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Question 10

Find the nominal interest rate that, when compounded continuously, yields an effective interest rate of 10% / year. Round your answer to three decimal places. ​

Accepted Answer

The effective interest rate (EIR) when interest is compounded continuously can be found using the formula $EIR = e^r - 1$, where $e$ is the base of the natural logarithm (approximately 2.71828), and $r$ is the nominal interest rate. Rearranging the formula to solve for $r$, we get $r = \ln(1 + EIR)$. Given an EIR of 10% (or 0.10), we find $r = \ln(1 + 0.10) = \ln(1.10)$, which approximately equals 0.09531, or 9.531% when converted to a percentage.

Question 11

Having received a large inheritance, a child's parents wish to establish a trust for the child's college education. If 7 year(s) from now they need an estimated $60,000, how much should they set aside in trust now, if they invest the money at the given rate of interest. Round your answer to the nearest cent. ​
13)5% compounded quarterly _________
​
13)5% compounded continuously _________
​

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Question 12

A condominium complex was purchased by a group of private investors for $1.2 million and sold 5 year(s) later for $3.7 million. Find the annual rate of return (compounded continuously) on their investment. Round your answer to two decimal places. ​

Accepted Answer

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount after time $t$, $P$ is the principal amount, $r$ is the rate, and $e$ is the base of the natural logarithm. Rearranging the formula to solve for $r$, we get $r = \frac{1}{t} \ln(\frac{A}{P})$. Substituting $A = 3.7$ million, $P = 1.2$ million, and $t = 5$ years, we get $r = \frac{1}{5} \ln(\frac{3.7}{1.2})$, which calculates to approximately 0.2252 or 22.52% when converted to a percentage.

Question 13

Find the interest rate needed for an investment of $6,000 to grow to an amount of $8,000 in 3 year(s) if interest is compounded continuously. Round your answer to the nearest hundredth.
​
Interest rate is __________% per year.

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Question 14

How long will it take an investment of $8,000 to double if the investment earns interest at the rate of 8% / year compounded monthly? Round your answer to the nearest tenth.
​
__________ year(s)

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Question 15

At an annual inflation rate of 5.5%, how long will it take the Consumer Price Index (CPI) to double? Round your answer to the nearest hundredth.
​
__________ year(s)

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Question 16

Eleni, who is now 55 years old, is employed by a firm that guarantees her a pension of $90,000/year at age 70. What is the present value of her first year's pension if inflation over the next 15 year(s) is at the given rate. Assume that inflation is continuously compounded. Round your answer to the nearest cent. ​
6% _________
​
9% _________
​
18% _________
​

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Question 17

Find the accumulated amount A if the principal P is invested at an interest rate of r per year for t years. Round your answer to the nearest cent.
​
P = $2,400, r = 9%, t = 10, compounded semiannually
​
$__________

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Question 18

An investor purchased a piece of waterfront property. Because of the development of a marina in the vicinity, the market value of the property is expected to increase according to the rule  where V(t) is measured in dollars and t is the time in years from the present. If the inflation rate is expected to be 10% compounded continuously for the next 8 years, what is P(t) expected to be in 4 years? Round your answer to the nearest dollar. ​

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Question 19

A condominium complex was purchased by a group of private investors for $1.3 million and sold 7 year(s) later for $3.9 million. Find the annual rate of return (compounded continuously) on their investment. Round your answer to two decimal places. ​

Accepted Answer

The formula for continuous compounding is:
A = Pe^rt
where A is the ending value, P is the beginning value, e is Euler's number (approximately 2.71828), r is the annual interest rate, and t is the time in years.
We can use this formula to find the annual rate of return:
3.9 million = 1.3 million e^(r*7)
e^(r*7) = 3
r*7 = ln(3)
r = ln(3)/7 ≈ 0.223
The annual rate of return is approximately 22.3%, so the closest answer choice is C: 15.69% per year. However, this is only an approximation, since the problem specifies continuous compounding, while the answer choices are rounded to the nearest hundredth of a percent.

Question 20

How long will it take $3,000 to grow to $8,500 if the investment earns interest at the rate of 18% / year compounded monthly? Round your answer to the nearest tenth.
​
__________ year(s)

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The answer of How long will it take $3,000 to...

Quiz 5: Exponential and Logarithmic Functions

Find the accumulated amount A if the principal P is invested at an interest rate of r per year for t years. Round your answer to the nearest cent. P = $180,000, r = 6%, t = 3, compounded daily $__________

Find the nominal interest rate that, when compounded monthly, yields an effective interest rate of 16% / year. Round your answer to two decimal places.

How long will it take an investment of $4,000 to double if the investment earns interest at the rate of 7% compounded continuously? Round your answer to the nearest hundredth. __________ year(s)

Julio purchased 700 shares of a certain stock for $22,750 (including commissions) . He sold the shares 5 year(s) later and received $32,100 after deducting commissions. Find the effective annual rate of return on his investment over the 5-year period. Round your answer to two decimal places.

Find the interest rate needed for an investment of $3,000 to grow to an amount of $8,500 in 3 year(s) if interest is compounded monthly. Round your answer to the nearest hundredth. __________%

Find the present value of $54,049 due in 5 year(s) at an interest rate of 7% / year compounded continuously. Round your answer to the nearest integer.

Bernie invested a sum of money 5 year(s) ago in a savings account, which has since paid interest at the rate of 6% / year compounded quarterly. His investment is now worth $22,927.66. How much did he originally invest? Round your answer to the nearest cent. $__________

Find the accumulated amount after 6 year(s) if $5,000 is invested at 8% compounded continuously. Round your answer to the nearest cent. $__________

Find the nominal interest rate that, when compounded continuously, yields an effective interest rate of 10% / year. Round your answer to three decimal places.

A condominium complex was purchased by a group of private investors for $1.2 million and sold 5 year(s) later for $3.7 million. Find the annual rate of return (compounded continuously) on their investment. Round your answer to two decimal places.

Find the interest rate needed for an investment of $6,000 to grow to an amount of $8,000 in 3 year(s) if interest is compounded continuously. Round your answer to the nearest hundredth. Interest rate is __________% per year.

How long will it take an investment of $8,000 to double if the investment earns interest at the rate of 8% / year compounded monthly? Round your answer to the nearest tenth. __________ year(s)

At an annual inflation rate of 5.5%, how long will it take the Consumer Price Index (CPI) to double? Round your answer to the nearest hundredth. __________ year(s)

Find the accumulated amount A if the principal P is invested at an interest rate of r per year for t years. Round your answer to the nearest cent. P = $2,400, r = 9%, t = 10, compounded semiannually $__________

A condominium complex was purchased by a group of private investors for $1.3 million and sold 7 year(s) later for $3.9 million. Find the annual rate of return (compounded continuously) on their investment. Round your answer to two decimal places.

How long will it take $3,000 to grow to $8,500 if the investment earns interest at the rate of 18% / year compounded monthly? Round your answer to the nearest tenth. __________ year(s)

Quiz 5: Exponential and Logarithmic Functions

Find the accumulated amount A if the principal P is invested at an interest rate of r per year for t years. Round your answer to the nearest cent. ​ P = $180,000, r = 6%, t = 3, compounded daily ​ $__________

Find the nominal interest rate that, when compounded monthly, yields an effective interest rate of 16% / year. Round your answer to two decimal places. ​

How long will it take an investment of $4,000 to double if the investment earns interest at the rate of 7% compounded continuously? Round your answer to the nearest hundredth. ​ __________ year(s)

Julio purchased 700 shares of a certain stock for $22,750 (including commissions) . He sold the shares 5 year(s) later and received $32,100 after deducting commissions. Find the effective annual rate of return on his investment over the 5-year period. Round your answer to two decimal places. ​

Find the interest rate needed for an investment of $3,000 to grow to an amount of $8,500 in 3 year(s) if interest is compounded monthly. Round your answer to the nearest hundredth. ​ __________%