Question 1

Solve the problem. Round to the nearest cent as needed.
-Samantha's savings account has a balance of $4643. After 25 years, what will the amount of interest be at 6% compounded annually?
A) $15,289.16
B) $15,275.16
C) $15,284.16
D) $2785.80

Accepted Answer

We can use the formula A=P(1+r/n)^(nt) where A is the final amount, P is the initial principal, r is the annual interest rate, n is the number of times compounded per year, and t is the time in years. Plugging in the values, we get A = 4643(1+0.06/1)^(1*25) = $15,284.16 when rounded to the nearest cent. Therefore, the answer is choice C. We can use the formula A=P(1+r/n)^(nt) where A is the final amount, P is the initial balance, r is the interest rate, n is the number of times compounded per year, and t is the number of years. Plugging in the values, we get:
A = 4643(1+0.06/1)^(1*25)
A = $15,284.16 (rounded to the nearest cent) 
Therefore, the correct answer is C.

Question 2

What is the annual percentage yield (APY) for money invested at the given annual rate? Round results to the nearest
 hundredth of a percent.
-3.5% compounded continuously.
A) 3.50%
B) 3.53%
C) 3.55%
D) 3.56%

Accepted Answer

The formula for APY is A = Pe^(rt), where P is the principal, r is the annual rate, and t is the time in years. In this case, P = 1, r = 0.035, and t = 1. Plugging these values into the formula, we get A = e^(0.035) = 1.0356. Therefore, the APY is 1.0356 - 1 = 0.0356, or 3.56%.

Question 3

What is the annual percentage yield (APY) for money invested at the given annual rate? Round results to the nearest
 hundredth of a percent.
-5% compounded semiannually
A) 5.06%
B) 5.13%
C) 5.00%
D) 5.09%

Accepted Answer

The APY can be calculated using the formula for compound interest: APY = (1 + r/n)^(n*t) - 1, where r is the annual interest rate (as a decimal), n is the number of compounding periods per year, and t is the time in years. For 5% compounded semiannually, r = 0.05, n = 2, and t = 1. Plugging these values into the formula gives APY = (1 + 0.05/2)^(2*1) - 1 = 1.025^2 - 1 = 1.050625 - 1 = 0.050625 or 5.06% when converted to a percentage and rounded to the nearest hundredth.

Question 4

Find i (the rate per period) and n (the number of periods) for the annuity.
-Semiannual deposits of $400 are made for 10 years into an annuity that pays 7% compounded semiannually.
A) i = 0.07; n = 10
B) i = 0.035; n = 20
C) i = 0.35; n = 10
D) i = 0.0175; n = 40

Accepted Answer

The interest rate (i) in the question is given as an annual rate, compounded semiannually. So we need to divide the annual rate by 2 to get the semiannual rate. Therefore, i = 0.07/2 = 0.035. The number of periods (n) is given as 10 years, but since deposits are made semiannually, we need to multiply by 2 to get the total number of deposit periods. Therefore, n = 10 * 2 = 20. Choice B matches these values. The interest rate per period is half of the annual interest rate as interest is compounded semiannually. So, the interest rate per period is (7%/2) = 3.5%. The number of periods is the total number of semiannual deposits made, which is (10 years) x (2 semiannual deposits per year) = 20 semiannual deposits. Therefore, the correct choice is B, with i = 0.035 and n = 20.

Question 5

Find i (the rate per period) and n (the number of periods) for the annuity.
-Monthly deposits of $1600 are made for 6 years into an annuity that pays 9% compounded monthly.
A) i = 0.75; n = 12
B) i = 0.0075; n = 72
C) i = 0.75; n = 6
D) i = 0.09; n = 72

Accepted Answer

First, we need to find the effective interest rate per month. We can use the formula:

(1 + i)^m = 1 + r

where i is the nominal interest rate per year, m is the number of compounding periods per year, and r is the effective interest rate per year.

In this case, i = 0.09 (9%), m = 12 (compounded monthly), and we want to solve for r:

(1 + 0.09/12)^12 - 1 = r
1.0075^12 - 1 = r
0.0938 = r

So the effective interest rate per month is:

(1 + 0.0938)^(1/12) - 1 = 0.0075 or 0.75%

Next, we need to find the total number of periods:

n = 6 years x 12 months/year = 72 months

So the correct answer is B, with i = 0.0075 and n = 72. The rate per period (i) is the monthly interest rate, which can be found by dividing the annual interest rate by 12: 
i = 9%/12 = 0.0075 or 0.75%.

The number of periods (n) is the total number of monthly deposits made over the 6-year period: 
n = 6 years × 12 months/year = 72 months.

Therefore, the correct answer is B: i = 0.0075; n = 72.

Question 6

Use the future value formula to find the indicated value. Round to three decimal places.
-n = 10; i = 0.02; PMT = $1; FV = ?
A) $10.95
B) $60.95
C) $12.169
D) $9.755

Accepted Answer

The answer of Use the future value formula to find...

Question 7

Solve the problem. Round to the nearest cent as needed.
-A child receives a $10,000 gift toward a college education from her grandparents on her first birthday. How
much money will it be worth in 17 years if it is invested at 8.25% compounded quarterly? Round your answer
to the nearest cent.

Accepted Answer

The answer of Solve the problem. Round to the nearest...

Question 8

Solve the problem. Round to the nearest cent as needed.
-The bacteria in a 11-liter container double every 2 minutes. After 53 minutes the container is full. How long did it take to fill a quarter of the container?
A) 13.3 min
B) 26.5 min
C) 49 min
D) 39.8 min

Accepted Answer

The bacteria double every 2 minutes, so to fill the container to a quarter of its capacity, it would be full 2 minutes before being completely full. Since it takes 53 minutes to fill, subtracting 2 minutes gives 51 minutes to reach a quarter full. However, the question asks for the time to fill a quarter from the start, not from full. The bacteria's growth is exponential, and to find when the container was a quarter full, we consider the doubling time. If the container is full at 53 minutes, it was half full at 53 - 2 = 51 minutes, and a quarter full at 51 - 2 = 49 minutes.

Question 9

Solve the problem. Round to the nearest cent as needed.
-Cheraw Auto Repair believes that it will need new equipment in 10 years. The equipment will cost $26,000. What lump sum should be invested today at 8% compounded semiannually, to yield $26,000?
A) $21,097.18
B) $17,462.98
C) $11,866.06
D) $17,637.61

Accepted Answer

The answer of Solve the problem. Round to the nearest...

Question 10

Solve the problem. Round to the nearest cent as needed.
-How many months will it take until an account will have $3,500 if $2,500 is invested now at 5% compounded
monthly?

Accepted Answer

The answer of Solve the problem. Round to the nearest...

Question 11

Solve the problem.
-The State Employees' Credit Union offers a 1-year certificate of deposit with an APY (or effective rate) of 5.5%. If interest is compounded quarterly, find the actual interest rate. Round to the nearest tenth of a percent.
A) 5.4%
B) 6.4%
C) 5.1%
D) 5.6%

Accepted Answer

The actual interest rate, or nominal rate, can be found using the formula for the effective annual rate (EAR) or APY: $APY = (1 + \frac{r}{n})^n - 1$, where $r$ is the nominal rate and $n$ is the number of compounding periods per year. Rearranging to solve for $r$, we get $r = n[(1 + APY)^{\frac{1}{n}} - 1]$. Plugging in $APY = 0.055$ and $n = 4$ (since interest is compounded quarterly), we find $r \approx 0.054$, or 5.4%.

Question 12

Solve the problem. Round to the nearest cent as needed.
-A bank has $750,000 to lend for 7 months. It can lend it to a local contractor at a simple interest rate of 12%, or it can lend it to a small business that will pay 12% compounded monthly. If the bank wants to maximize its
Interest earned, who should receive the loan (contractor or business) and what is the additional interest earned?
A) Contractor; $52,500
B) Business; $1601.51
C) Contractor; $1601.51
D) Business; $54,101.51

Accepted Answer

To maximize interest, the bank should lend to the business because compound interest yields more than simple interest over time. The additional interest earned is calculated by comparing the total interest from both options. For simple interest (contractor): Interest = Principal × Rate × Time = $750,000 × 12% × (7/12) = $52,500. For compound interest (business): Interest = Principal × (1 + Rate/n)^(nt) - Principal = $750,000 × (1 + 0.12/12)^(12*(7/12)) - $750,000 ≈ $54,101.51. The additional interest with compounding is $54,101.51 - $52,500 = $1,601.51.

Question 13

Solve the problem. Round to the nearest cent as needed.
-Brandon's savings account has a balance of $4069. After 4 years what will the amount of interest be at 5% compounded quarterly?
A) $899.73
B) $894.73
C) $101.73
D) $885.73

Accepted Answer

We can use the formula: A = P(1+r/n)^(nt) where A is the amount, P is the principal, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

In this problem, P = $4069, r = 5% or 0.05, n = 4 (since it's compounded quarterly), and t = 4.

So, A = 4069(1+0.05/4)^(4*4) = $4963.73

The interest amount is the difference between the final amount and the principal: 
Interest = $4963.73 - $4069 = $894.73

Rounded to the nearest cent, the answer is $894.73, which corresponds to choice B. First we need to calculate the total amount in the account after 4 years using the formula for compound interest:

A = P(1 + r/n)^(nt)
where A is the final amount, P is the initial amount, r is the interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

In this case, P = $4069, r = 0.05 (or 5%), n = 4 (since the interest is compounded quarterly), and t = 4. Plugging in these values, we get:

A = $4069(1 + 0.05/4)^(4*4) = $5063.39

The interest earned over 4 years will be the difference between the final amount and the initial amount:

Interest = $5063.39 - $4069 = $994.39

Rounding this to the nearest cent, we get $994.40. However, the question asks for the amount rounded to the nearest cent, so the correct answer is B) $894.73, which is the rounded value of $994.40.

Question 14

Solve the problem. Round to the nearest cent as needed.
-The rabbit population in a forest area grows at the rate of 6% monthly. If there are 230 rabbits in April, find how many rabbits (rounded to the nearest whole number) should be expected by next April. Use e 
A) 447
B) 457
C) 483
D) 470

Accepted Answer

To solve this problem, we need to use the formula for exponential growth:

A = P(1 + r)n

where: 
A = final amount (number of rabbits)
P = initial amount (number of rabbits in April)
r = growth rate per period (6% per month, or 0.06)
n = number of periods (12 months)

Plugging in the values we know, we get:

A = 230(1 + 0.06)12 
A = 230(1.06)12 
A ≈ 469.95

Rounding to the nearest whole number, we get:

A ≈ 470

Therefore, the expected number of rabbits by next April is approximately 470. The closest choice to this value is D, 470. To solve the problem, we can use the formula:

P = P0 (1 + r)^n

where: 
P0 = initial population (given as 230 rabbits)
r = monthly growth rate (6% = 0.06)
n = number of months (12, since we want to know the population after one year)

Substituting these values:

P = 230 (1 + 0.06)^12 
P ≈ 469.69

Rounding to the nearest whole number: 
P ≈ 470

Therefore, the expected number of rabbits (rounded to the nearest whole number) by next April is 470.

The closest choice is D (470), so it is the best choice.

Question 15

Provide an appropriate response.
-A bank account starts with $1,000 in it. Interest is paid at 6% annual interest, compounded monthly. Graph the
exact function for the amount in the account over the first 12 months. Use vertical scale [900, 1,200].

Accepted Answer

The answer of Provide an appropriate response.
-A bank account starts...

Question 16

Solve the problem. Round to the nearest cent as needed.
-Cara knows that she will need to buy a new car in 3 years. The car will cost $15,000 by then. How much should she invest now at 12%, compounded quarterly, so that she will have enough to buy a new car?
A) $10,520.70
B) $12.594.29
C) $11,957.91
D) $9532.77

Accepted Answer

To find out how much Cara should invest now to have $15,000 in 3 years at a 12% annual interest rate compounded quarterly, we use the formula for compound interest: $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the future value, $P$ is the principal amount (initial investment), $r$ is the annual interest rate (in decimal), $n$ is the number of times the interest is compounded per year, and $t$ is the time the money is invested for in years. Rearranging the formula to solve for $P$, we get $P = \frac{A}{(1 + \frac{r}{n})^{nt}}$. Plugging in the values: $A = 15000$, $r = 0.12$, $n = 4$ (since interest is compounded quarterly), and $t = 3$, we find the initial investment required.

Question 17

Solve the problem. Round to the nearest cent as needed.
-An actuary for a pension fund need to have $14.6 million grow to $22 million in 6 years. What interest rate compounded annually does he need for this investment to growth as specified. Round your answer to the
Nearest hundredth of a percent.
A) 0.07%
B) 7%
C) 7.7%
D) 7.07%

Accepted Answer

We can use the formula for compound interest, A = P(1+r/n)^(nt), where A is the ending amount, P is the principal, r is the annual interest rate, n is the number of times compounded per year, and t is the time in years. We know P = $14.6 million, A = $22 million, n = 1 (compounded annually), and t = 6 years. We can solve for r:

$22 million = $14.6 million (1+r/1)^(1*6)
(22/14.6) = (1+r)^6
1.5068 = (1+r)^6
6(log(1+r)) = log(1.5068)
log(1+r) = log(1.5068)/6
1+r = 1.0707
r = 0.0707 or 7.07%

Therefore, the best choice is D. We can use the formula A=P(1+r/n)^(nt) where A=$22 million, P=$14.6 million, n=1 (compounded annually), t=6 years, and solve for r (interest rate).
$22 million = $14.6 million(1+r/1)^(1*6)
(22/14.6) = (1+r)^6
1.50684932 = (1+r)^6
Taking the sixth root of both sides:
(1+r) = 1.07069998
r = 0.07069998 or 7.07% (rounded to nearest hundredth of a percent)

Question 18

What is the annual percentage yield (APY) for money invested at the given annual rate? Round results to the nearest
 hundredth of a percent.
-6% compounded quarterly
A) 6.14%
B) 6.09%
C) 6.18%
D) 6.00%

Accepted Answer

The APY can be calculated using the formula APY = (1 + r/n)^(n*t) - 1, where r is the annual interest rate (as a decimal), n is the number of compounding periods per year, and t is the time in years. For a 6% rate compounded quarterly, r = 0.06, n = 4, and t = 1. Plugging these values into the formula gives APY = (1 + 0.06/4)^(4*1) - 1 = 1.0614 - 1 = 0.0614 or 6.14%.

Question 19

Solve the problem. Round to the nearest cent as needed.
-How long will it take for $8400 to grow to $14,600 at an interest rate of 9.4% if the interest is compounded continuously? Round the number of years to the nearest hundredth.
A) 0.59 year
B) 0.06 year
C) 5.88 years
D) 58.81 years

Accepted Answer

We can use the formula $A=P e^{rt}$ where $A$ is the final amount, $P$ is the initial amount, $r$ is the annual interest rate as a decimal, and $t$ is the number of years. Plugging in the given values, we get:

$14,600 = 8,400 e^{0.094t}$

Dividing both sides by 8,400 gives:

$1.73809524 = e^{0.094t}$

Taking the natural logarithm of both sides gives:

$0.94314718 = 0.094t$

Dividing both sides by 0.094 gives:

$t \approx 5.88$

Rounding to the nearest hundredth gives a final answer of 5.88 years. Therefore, the best choice is C.

Question 20

Solve the problem.
-Jennifer invested $6000 in her savings account for 7 years. When she withdrew it, she had $9004.82. Interest was compounded continuously. What was the interest rate on the account? Round to the nearest tenth of a
Percent.
A) 5.7%
B) 5.95%
C) 5.9%
D) 5.8%

Accepted Answer

The formula for continuous compounding is A = Pe^(rt), where A is the amount, P is the principal, r is the rate, and t is the time. Plugging in the values: 9004.82 = 6000 * e^(7r). Solving for r gives approximately 0.058, which is 5.8% when rounded to the nearest tenth of a percent.

Quiz 3: Additional Derivative Topics

Solve the problem. Round to the nearest cent as needed. -Samantha's savings account has a balance of $4643. After 25 years, what will the amount of interest be at 6% compounded annually?

What is the annual percentage yield (APY) for money invested at the given annual rate? Round results to the nearest hundredth of a percent. -3.5% compounded continuously.

What is the annual percentage yield (APY) for money invested at the given annual rate? Round results to the nearest hundredth of a percent. -5% compounded semiannually

Find i (the rate per period) and n (the number of periods) for the annuity. -Semiannual deposits of $400 are made for 10 years into an annuity that pays 7% compounded semiannually.

Find i (the rate per period) and n (the number of periods) for the annuity. -Monthly deposits of $1600 are made for 6 years into an annuity that pays 9% compounded monthly.

Use the future value formula to find the indicated value. Round to three decimal places. -n = 10; i = 0.02; PMT = $1; FV = ?

Solve the problem. Round to the nearest cent as needed. -A child receives a $10,000 gift toward a college education from her grandparents on her first birthday. How much money will it be worth in 17 years if it is invested at 8.25% compounded quarterly? Round your answer to the nearest cent.

Solve the problem. Round to the nearest cent as needed. -The bacteria in a 11-liter container double every 2 minutes. After 53 minutes the container is full. How long did it take to fill a quarter of the container?

Solve the problem. Round to the nearest cent as needed. -Cheraw Auto Repair believes that it will need new equipment in 10 years. The equipment will cost $26,000. What lump sum should be invested today at 8% compounded semiannually, to yield $26,000?

Solve the problem. Round to the nearest cent as needed. -How many months will it take until an account will have $3,500 if $2,500 is invested now at 5% compounded monthly?

Solve the problem. -The State Employees' Credit Union offers a 1-year certificate of deposit with an APY (or effective rate) of 5.5%. If interest is compounded quarterly, find the actual interest rate. Round to the nearest tenth of a percent.

Solve the problem. Round to the nearest cent as needed. -Brandon's savings account has a balance of $4069. After 4 years what will the amount of interest be at 5% compounded quarterly?

Solve the problem. Round to the nearest cent as needed. -The rabbit population in a forest area grows at the rate of 6% monthly. If there are 230 rabbits in April, find how many rabbits (rounded to the nearest whole number) should be expected by next April. Use e

Provide an appropriate response. -A bank account starts with $1,000 in it. Interest is paid at 6% annual interest, compounded monthly. Graph the exact function for the amount in the account over the first 12 months. Use vertical scale [900, 1,200].

Solve the problem. Round to the nearest cent as needed. -Cara knows that she will need to buy a new car in 3 years. The car will cost $15,000 by then. How much should she invest now at 12%, compounded quarterly, so that she will have enough to buy a new car?

Solve the problem. Round to the nearest cent as needed. -An actuary for a pension fund need to have $14.6 million grow to $22 million in 6 years. What interest rate compounded annually does he need for this investment to growth as specified. Round your answer to the Nearest hundredth of a percent.

What is the annual percentage yield (APY) for money invested at the given annual rate? Round results to the nearest hundredth of a percent. -6% compounded quarterly

Solve the problem. Round to the nearest cent as needed. -How long will it take for $8400 to grow to $14,600 at an interest rate of 9.4% if the interest is compounded continuously? Round the number of years to the nearest hundredth.

Solve the problem. -Jennifer invested $6000 in her savings account for 7 years. When she withdrew it, she had $9004.82. Interest was compounded continuously. What was the interest rate on the account? Round to the nearest tenth of a Percent.