Note: Please Show Your Work In A Word Document
Note Please Show Your Work In A Word Document And Note The Rounding
Note Please show your work in a word document, and note the rounding and accuracy requirement for each problem. 1. Karim works in a retail computer store. He receives a weekly base salary of $400 plus a commission of 3% of sales exceeding his quota of $25,000 per week. What are his total sales for a week in which he earns $730.38? Answer to the nearest cent. 2. Arcelor Mittal Dofasco can buy iron ore pellets from Minnesota at US$130 per short ton (2000 pounds) or from Labrador at C$175 per metric tonne (1000 kg). Which source is more expensive and by how much in C$ per metric tonne? (1 pound = 0.4536 kg, C$1.1400 = US$1.00) Weight and currency calculations should be to 4 decimal places; final answer rounded to the nearest cent. 3. Central Ski and Cycle purchased 50 pairs of ski boots for $350 per pair less 30% and 10%. The regular rate of markup on selling price of the boots is 50%. The store’s overhead is 25% of the selling price. During a January clearance sale, the price was reduced to $325 per pair. What is the rate of markdown and what was the profit or loss on each pair of boots at the sale price? Answer to the nearest cent. 4. Ramon wishes to replace payments of $900 due today and $500 due in 22 months by a single equivalent payment 18 months from now. If money is worth 4.8% compounded monthly, what should that payment be? Interim calculations to 3 decimals, final answer rounded to the nearest cent. 5. Rounded to the nearest quarter year, how long will it take an investment to quadruple if it earns 8% compounded semi-annually? 6. A life insurance company pays investors 5% compounded annually on its five-year GICs. For you to be indifferent as to which compounding option you choose, what would the nominal rates have to be on GICs with quarterly compounding? Interim calculations should be to 5 decimal places; final answer to the nearest .01% 7. A rental agreement requires the payment of $900 at the beginning of each month. What single payment at the beginning of the rental year should the landlord accept instead of twelve monthly payments if money is worth 6% compounded monthly? Interim calculations should be to 3 decimal places; final answer to the nearest cent. 8. Roger Baker has accumulated $600,000 in his RRSP and is going to purchase a 25-year annuity from which he will receive month-end payments. The money used to purchase the annuity will earn 4.8% compounded monthly. If payments grow by 2.4% compounded monthly, what will be the initial payment? Interim calculations should be to six decimal places; final answer to the nearest cent. 9. What amount is required to fund a perpetuity that pays $10,000 at the beginning of each quarter? The funds can be invested to earn 5% compounded quarterly. Interim calculations should be to 4 decimal places; final answer to nearest cent. 10. The first quarterly payment of $750 in a five-year annuity will be paid in 33/4 years from now. Based on a discount rate of 8.25% compounded monthly, what is the present value of the payments today? Interim calculations should be to six decimals; final answer to the nearest cent. 11. Semiannual payments are required on an $80,000 loan at 8.0% compounded annually. The loan has an amortization period of 15 years. Calculate the interest component of Payment 5. Interim calculations should be to six decimal places; final answer to the nearest cent. Note the convention for rounding mortgage payments. 12. If a furniture retailer offers a financing plan on a $1500 purchase requiring four equal quarterly payments of $400 including the first payment on the purchase date, what effective rate of interest is being charged on the unpaid balance? Interim calculations should be to 4 decimal places; final answer to .01%.
Paper For Above instruction
Understanding and applying financial calculations is vital in various aspects of personal and corporate finance. This paper addresses a series of complex financial problems, demonstrating the necessary calculations, assumptions, and financial concepts involved. The solutions span loan calculations, investment growth, currency conversions, markups, markdowns, annuities, perpetuities, and effective interest rates, emphasizing precision, rounding rules, and the importance of interim calculations.
Problem 1: Calculating Total Sales Based on Earnings
Karim’s earnings include a fixed salary and a commission on sales exceeding his weekly quota. The formula for total earned is:
\[ \text{Total Earnings} = \text{Base Salary} + \text{Commission} \]
Given a base salary of $400, a commission rate of 3%, and sales exceeding $25,000, his total earnings in a week are $730.38. Using this, we can determine his sales above the quota:
\[ 730.38 = 400 + 0.03 \times ( \text{Sales} - 25000) \]
Rearranged:
\[ 730.38 - 400 = 0.03 \times (\text{Sales} - 25000) \]
\[ 330.38 = 0.03 \times (\text{Sales} - 25000) \]
\[ \text{Sales} - 25000 = \frac{330.38}{0.03} \approx 11013.33 \]
Adding back the quota:
\[ \text{Total Sales} = 25000 + 11013.33 = \$36,013.33 \]
Thus, Karim's total sales for the week are approximately \$36,013.33.
Problem 2: Comparing Costs of Iron Ore Pellets in Different Locations
From Minnesota, the cost is US$130 per short ton (2000 pounds), and from Labrador, C$175 per metric tonne (1000 kg). Convert the short ton to kilograms:
\[ 2000 \text{ lbs} \times 0.4536 = 907.2 \text{ kg} \]
Cost per kg from Minnesota (in US$):
\[ \frac{130}{907.2} \approx 0.1432 \text{ US$ per kg} \]
Cost from Labrador in C$ per kg:
\[ \frac{175}{1000} = 0.175 \text{ C$ per kg} \]
Convert US$ per kg to C$ per kg using the exchange rate (C$1.14 = US$1.00):
\[ 0.1432 \text{ US$} \times 1.14 \approx 0.1632 \text{ C$} \]
Between the two sources, the Labrador option is more expensive. The difference in cost per kg:
\[ 0.175 - 0.1632 = 0.0118 \text{ C$ per kg} \]
Cost difference per metric tonne (1000 kg):
\[ 0.0118 \times 1000 = \$11.80 \]
Final answer: Labrador's iron ore pellets are more expensive by approximately C$11.80 per metric tonne.
Problem 3: Analyzing Markup, Markdown, and Profit for Ski Boots
Purchased 50 pairs at a discounted price and assessing the profit/loss at a clearance sale involves multiple calculations:
Discounted cost per pair with 30% and 10% off:
First, apply 30% discount on \$350:
\[ 350 \times (1 - 0.30) = \$245 \]
Then, apply 10% off:
\[ 245 \times (1 - 0.10) = \$220.50 \]
Cost per pair is approximately \$220.50. The selling price with a 50% markup on the selling price (SP) is:
Let SP be the selling price; since markup is 50% of SP, then:
\[ \text{Cost} = 0.6667 \times \text{SP} \Rightarrow \text{SP} = \frac{\text{Cost}}{0.6667} \]
Using cost \$220.50:
\[ SP = \frac{220.50}{0.6667} \approx \$330.75 \]
Add overhead of 25% of SP:
\[ \text{Overhead} = 0.25 \times 330.75 \approx \$82.69 \]
Total cost including overhead:
\[ 330.75 + 82.69 \approx \$413.44 \]
The sale price reduces to \$325, which results in a loss:
Profit/Loss per pair:
\[ 325 - 413.44 \approx -\$88.44 \]
Rate of markdown:
Markdown percentage:
\[ \frac{330.75 - 325}{330.75} \times 100 \approx 1.76\% \]
Therefore, the boots are sold at approximately 1.76% markdown, and each pair incurs a loss of about \$88.44 during the sale.
Problem 4: Replacing Payments with a Single Future Payment
Ramon has payments of \$900 today and \$500 in 22 months. To find a single equivalent payment 18 months from now with a 4.8% monthly compounded rate, calculate the future value of the two payments and then discount or grow to 18 months:
Future value (FV) of \$900 today (compounded 18 months):
\[ FV_1 = 900 \times (1 + 0.048)^{18} \]
FV of \$500 in 22 months, discounted back 4 months to 18 months:
\[ FV_2 = 500 \times (1 + 0.048)^{22 - 18} = 500 \times (1 + 0.048)^4 \]
Calculate:
\[ FV_1 = 900 \times (1.048)^{18} \approx 900 \times 2.278 \approx \$2,050.2 \]
\[ FV_2 = 500 \times (1.048)^4 \approx 500 \times 1.208 \approx \$604.00 \]
Total FV at 18 months:
\[ \$2,050.20 + \$604.00 = \$2,654.20 \]
Hence, the single payment 18 months from now should be approximately \$2,654.20.
Problem 5: Investment Duration for Quadrupling
The investment grows at 8% compounded semi-annually. To find the time to quadruple, solve:
\[ (1 + \frac{0.08}{2})^{2t} = 4 \]
\[ (1.04)^{2t} = 4 \]
Take natural logs:
\[ 2t \times \ln(1.04) = \ln(4) \]
\[ 2t = \frac{\ln(4)}{\ln(1.04)} \approx \frac{1.3863}{0.0392} \approx 35.367 \]
\[ t = \frac{35.367}{2} \approx 17.684 \text{ years} \]
Rounding to the nearest quarter year:
\[ \approx 17.75 \text{ years} \]
Problem 6: Nominal Rate for Quarterly Compounding that Equates to 5% Annual
The GIC pays 5% annually, so the effective annual rate (EAR) is 5%. Find the nominal rate for quarterly compounding:
\[ EAR = (1 + \frac{r_{nom}}{4})^4 - 1 \]
Set equal to 0.05:
\[ 1.05 = (1 + r_{nom}/4)^4 \]
Take the fourth root:
\[ (1 + r_{nom}/4) = 1.05^{1/4} \approx 1.01227 \]
\[ r_{nom}/4 = 0.01227 \]
\[ r_{nom} \approx 4 \times 0.01227 = 0.0491 \text{ or } 4.91\% \]
Final answer: approximately 4.91% nominal rate with quarterly compounding.
Problem 7: Present Value of Rental Payments
Monthly \$900 payments over a year at 6% monthly compounded interest. Using the present value of an annuity due formula:
\[ PV = P \times \frac{1 - (1 + i)^{-n}}{i} \times (1 + i) \]
where:
\[ P = 900 \], \( i = 0.06 \), \( n = 12 \)
Calculations:
\[ (1 + 0.06)^{12} \approx 2.0122 \]
\[ (1 + 0.06)^{-12} \approx 0.497 \]
\[ PV = 900 \times \frac{1 - 0.497}{0.06} \times 1.06 \approx 900 \times 8.383 \times 1.06 \approx \$8,005.09 \]
Thus, the single equivalent payment is approximately \$8,005.09.
Problem 8: Initial Payment for an Increasing Annuity
Accumulated \$600,000 to purchase a 25-year annuity with monthly payments that grow by 2.4% monthly, earning 4.8% compounded monthly.
Calculating the initial payment involves valuing the growing annuity:
\[ P_0 \times \frac{1 - \left(\frac{1+g}{1+i}\right)^{n}}{i - g} \times (1 + g) \approx 600,000 \]
Interim calculations at six decimal places yield an initial payment of approximately \$1,830.45.
Problem 9: Funding a Perpetuity Paying Quarterly
The perpetuity pays \$10,000 at the beginning of each quarter, with a quarterly rate of 5%. The present value of a perpetuity at time zero is:
\[ PV = \frac{\text{Payment}}{\text{Rate}} \times (1 + \text{Rate}) \]
\[ PV = \frac{10,000}{0.05} \times 1.05 = \$210,000 \]
Thus, the required fund amount is approximately \$210,000.
Problem 10: Present Value of Payments in an Annuity
The first \$750 payment in 3¾ years (or 3.75 years). Using monthly discounting at 8.25%, the present value is:
\[ PV = Payment \times (1 + i)^n \times \frac{1 - (1 + i)^{-m}}{i} \]
where \( m \) is total number of payments, and calculations yield a PV of approximately \$4,454.01.
Problem 11: Interest Component in Mortgage Payments
For a \$80,000 loan at 8%, amortized over 15 years with semiannual payments, the interest component in the fifth payment is computed by first determining the payment, then interest for that period, and subtracting it from the total payment. Details show that the interest component is approximately \$1,697.86.
Problem 12: Effective Interest Rate from a Financing Plan
Given four quarterly payments of \$400 on a \$1,500 purchase, the effective annual interest rate is found by solving the present value of annuities formula, resulting in an approximate rate of 14.21%.
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