REV 00 BUSINESS MATHEMATICS

REV 00 BUSINESS MATHEMATICS
QMT 3301 BUSINESS MATHEMATICS QMT BUSINESS MATHEMATICS

WHOLE NUMBERS AND DECIMALS
REV 00 CHAPTER 1 WHOLE NUMBERS AND DECIMALS QMT BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS
1.1 Whole Numbers REV 00 1.1.1 Read and Write Whole Numbers ( ) Hundred Trillions ( ) Ten Trillions ( ) Trillions ( ) Hundred Billions ( ) Ten Billions ( ) Billions Hundred Millions ( ) ( ) Ten Millions ( ) Millions Hundred Thousands ( ) Ten Thousands (10 000) Thousands (1 000) Hundreds (100) (10) Tens Ones (1) QMT BUSINESS MATHEMATICS

REV 00 Example: 1. Write its word name given the number below: 28 346 2 335 11 007 QMT BUSINESS MATHEMATICS

REV 00 Write the number given its word name: Four hundred twelve. Thirty-three thousand, seven. Ninety thousand, one. Seven hundred thousand, eleven. Fifteen million, three hundred sixty-two thousand, five hundred thirty-eight. Eight hundred forty-two billion, five hundred thousand, twenty-nine. QMT BUSINESS MATHEMATICS

REV 00 1.1.2 Round Whole Numbers Example: 1. Round 2748 to the (a) Nearest hundred (b) First digit 2. Round 3596 to the nearest (a) Ten (b) Hundred (c) Thousand QMT BUSINESS MATHEMATICS

REV 00 1.1.3 Add, Subtract, Multiply and Divide Whole Numbers Example: 1. Add and round the answer to the nearest hundred. 2. Subtract 27 from 64. 3. Multiply 127 by 53. divided by 35 (answer in 1 decimal place). QMT BUSINESS MATHEMATICS

REV 00 1.2 Application Problems Many business-application problems require mathematics. You must read the words carefully to decide how to solve the problems. QMT BUSINESS MATHEMATICS

REV 00 Example: 1. At a recent garage sale, the total sales were RM584. If the money was divided among Tom, Mina, Abu and Rashid, how much did each receive? 2. In May, the landlord of an apartment building received RM720 from each of eight tenants. After paying RM2180 in expenses, how much did the landlord have left? QMT BUSINESS MATHEMATICS

1.3 Basics of Decimals REV 00 1.3.1 Read, Write, and Round Decimals Hundreds (100) (10) Tens Ones (1) Decimal Point Tenths (0.1) Hundredths (0.01) Thousandths (0.001) Ten-thousandths (0.0001) Hundred-thousandths ( ) ( ) Millionths Ten-Millionths ( ) Hundred-Millionths ( ) QMT BUSINESS MATHEMATICS

REV 00 Example: Write the word name for the decimals below: 0.582 1.0009 782.07 3.12 $234.93 QMT BUSINESS MATHEMATICS

REV 00 1.3.2 Round to a Specified Decimal Place Example: 1. Round to the nearest Hundredth Thousandth Tenth 2. Round the number to the specified place $ to the nearest dollar $ to the nearest cent QMT BUSINESS MATHEMATICS

REV 00 1.4 Addition, Subtraction, Multiplication and Divisions of Decimals Example: 1. Add 2. Subtract 26.3 from 15.84 3. Multiply 3.5 by 3 4. Divide 95.2 with 14 5. Find the quotient of 37.4  24 to the nearest hundredth. QMT BUSINESS MATHEMATICS

REV 00 CHAPTER 2 FRACTIONS QMT BUSINESS MATHEMATICS

REV 00 2.1 Basics of Fractions In the fraction ; 1 = numerator and 2 = denominator. When a fraction has a value less than one, it is called a proper fraction. If fraction  1, fraction = improper fraction. QMT BUSINESS MATHEMATICS

REV 00 Example: = proper fraction = improper fraction = whole or mixed number Example: 1. Write as a mixed number. 2. Write as an improper fraction. QMT BUSINESS MATHEMATICS

2.1.2 Change a Fraction to an Equivalent Fraction Example: 2.1.3 How to Reduce a Fraction to Lowest Term 1. Reduce to lowest term. 2. Write in lowest term. 3. Rewrite as a fraction with a denominator of 72. REV 00 QMT BUSINESS MATHEMATICS

REV 00 2.2 Addition and Subtraction of Fractions Adding and subtracting fractions with the same denominator: Adding and subtracting fractions with the different denominator: QMT BUSINESS MATHEMATICS

Example: 1. Find the sum 2. Solve
REV 00 QMT BUSINESS MATHEMATICS

2.3 Addition and Subtraction of Mixed Numbers
REV 00 Example: 1. Solve QMT BUSINESS MATHEMATICS

2.4 Multiplication and Division of Fractions
REV 00 Multiplying fractions: Dividing fractions: QMT BUSINESS MATHEMATICS

REV 00 Example: 1. Multiply Divide QMT BUSINESS MATHEMATICS

REV 00 CHAPTER 3 PERCENT QMT BUSINESS MATHEMATICS

REV 00 3.1 Writing Decimals and Fractions as Percents Example: Write those numbers as percent i i. ii ii. iii iii. iv iv. v v. (in 2 decimal places) QMT BUSINESS MATHEMATICS

REV 00 3.1.1 Write a Percent as a Decimal Example: 1. Write the percent as a decimal 37% iv. 26.5% v. 0.9% QMT BUSINESS MATHEMATICS

3.2 Finding the Part REV 00 B = Base – the whole or the total. R = Rate – a number followed by “%” or “percent”. P = Part – the result of multiplying base times rate. QMT BUSINESS MATHEMATICS

REV 00 B = Base – sales R = Rate – sales tax rate P = Part – sales tax Example: 1. If the sales tax rate is 5%, what is the sales tax and total sales on $133 of merchandise? Solution: QMT BUSINESS MATHEMATICS

3.2.1 Identifying the Base and the Part
REV 00 3.2.1 Identifying the Base and the Part Usually the Base Usually the Part Sales Sales Tax Investment Return Savings Interest Retail Price Amount of Discount Value of Real Estate Rents Total Sales Commission Value of Stocks Dividends Earnings Expenditures Original Change QMT BUSINESS MATHEMATICS

3.3 Finding the Base REV 00 Using the Basic Percent Equation to solve for Base. Example: is 30% of _____ Solution: QMT BUSINESS MATHEMATICS

REV 00 Finding sales when sales tax rate is given. Example: 1. The 5% sales tax collected by a store was $380. What was the total amount of sales? Solution: QMT BUSINESS MATHEMATICS

REV 00 Finding the amount of an investment. Example: 1. The yearly maintenance cost of an apartment is 2.5% of its value. If maintenance is $37,000 per year, what is the value of the apartment complex? Solution: QMT BUSINESS MATHEMATICS

REV 00 Finding the base if rate and part are different quantities. Example: 1. United Hospital finds that 25% of its employees are men and 720 employees are women. What is the total number of employees? QMT BUSINESS MATHEMATICS

Solution: REV 00 Men = 25%, then women = (100-25)% = 75% QMT BUSINESS MATHEMATICS

3.4 Finding the Rate Using the percent equation to solve for rate.
REV 00 Using the percent equation to solve for rate. Example: 45 is what percent of 180? Solution: Note: Rate is always expressed as a percent. QMT BUSINESS MATHEMATICS

Finding rate of return when the amount of return and the investment are known. Example: 1. $3400 is invested in a new computer yielding additional income of $1700. What is the rate of return? Solution: REV 00 QMT BUSINESS MATHEMATICS

REV 00 Solving for the percent remaining. Example: 1. A car is expected to last 10 years before it needs replacement. If the car is 7 years old, what percent of the car’s life remains? Solution: Years remain = 10 – 7 = 3 years QMT BUSINESS MATHEMATICS

REV 00 Find the percent of increase/decrease. Example: 1. Sales of digital cameras went from $40,000 to $100,000. Find the percent increase. Solution: Increase = $100,000 - $40,000 = $60,000 QMT BUSINESS MATHEMATICS

REV 00 3.5 Increase and Decrease Problems Increase problems: Original + increase = new value Phrases: after an increase of, more than, greater than Decrease problems: Original – decrease = new value Phrases: after a decrease of, less than, after a reduction of QMT BUSINESS MATHEMATICS

REV 00 Example: 1. The value of a home sold by Sally this year is RM , which is 10% more than last year’s value. Find the value of the home last year. Answer: RM 2. At A company, production last year was 20% more than the year before. This year’s production is chairs, which is 20% more than last year’s. Find the number of chairs produced 2 years ago. Answer: chairs QMT BUSINESS MATHEMATICS

REV 00 3. After Sport About deducted 10% from the price of a tennis racket, Robert paid RM135. What was the original price of the racket? Answer: RM 150 QMT BUSINESS MATHEMATICS

REV 00 CHAPTER 4 MATHEMATICS OF BUYING QMT BUSINESS MATHEMATICS

4.1 Invoices and Trade Discounts 4.1.1 Invoice
REV 00 Invoice – a printed record of a purchase or sale Sales invoice – records a sale Purchase invoice – records a purchase Extension total – the number of items purchased times the price per unit, any discounts, the shipping and insurance charges. Invoice total – the sum of all the extension totals. Total invoice amount – same as invoice total excluding shipping and insurance. QMT BUSINESS MATHEMATICS

Seller pays shipping and insurance Buyer pays shipping and insurance
REV 00 Common shipping terms: Free on Board (FOB) shipping point or destination – seller pays for shipping to this destination. Beyond this point, the purchaser pays for shipping. FOB shipping point Seller pays shipping and insurance Buyer pays shipping and insurance QMT BUSINESS MATHEMATICS

REV 00 Invoice terms: Trade discount: offered to businesses who buy an item that is to be sold. List price: suggested price at which the item is to be sold to the public. Net price: the amount to be paid by the buyer. Net price = List price – Trade discount QMT BUSINESS MATHEMATICS

Example of an invoice: REV 00 QMT BUSINESS MATHEMATICS

4.1.2 Trade Discount REV 00 Trade discount is a reduction from the list price offered by wholesalers to retailers so that they can re-sell the merchandise at a profit. The amount of discount that retailers receive from wholesalers is known as trade discount amount. Trade discount = List price – Net price Formula: NP = L (1 – r) Where: NP = Net price L = List price r = trade discount QMT BUSINESS MATHEMATICS

Example 1: REV 00 The list price of a personal desktop is RM A trade discount of 15% is offered. What is the net price of the personal desktop? Solution: List price = RM 2599 Trade Discount = 15% x RM 2599 = RM NP = List price – Trade discount = RM 2599 – RM = RM QMT BUSINESS MATHEMATICS

Example 2: REV 00 USPD offers a discount of 10% on all the model of cars it sells. What is the net price of a Proton Savvy that is listed at RM 56490? Solution: NP = RM (1 – 0.1) = RM (0.9) = RM 50841 QMT BUSINESS MATHEMATICS

REV 00 Example 3: A bill of RM 3500 including a prepaid handling charge of RM 250 is offered a trade discount of 17%. What is the net price? Solution: List price = RM 3500 – RM 250 = RM 3250 NP = RM 3250 (1 – 0.17) = RM 3250 (0.83) = RM *(It should be noted that the discount is based on the cost of goods, excluding any other cost) QMT BUSINESS MATHEMATICS

4.3 Chain Discount REV 00 A trade discount in a series of two or more successive discounts. Wholesaler lists the chain discount as a group, for example 15%, 10%, 5%. These discounts might also be given in circumstances such as when a large quantity is ordered. Formula: NP = L(1 - r1) (1 – r2) (1 – r3) Where: NP = Net price L = List price QMT BUSINESS MATHEMATICS

Example: REV 00 A jeans is advertised for RM 280 less 10% and 5%. Find, a) The net price, b) The total discount. Solution: a) NP = L(1 - r1) (1 – r2) = RM 280 (1 – 0.1) (1 – 0.05) = RM 280 (0.9) (0.95) = RM QMT BUSINESS MATHEMATICS

REV 00 b) Total discount = RM 280 – RM = RM 40.60 *(It should be noted that the chain discount of 10% and 5% is not the same as 15%) QMT BUSINESS MATHEMATICS

4.4 Single Discount Equivalent
REV 00 A single discount equivalent is a single discount which is equivalent to a chain discount. The single discount equivalent, r for a chain discount of r1, r2 and r3 is given by: r = 1 – (1 - r1) (1 – r2) (1 – r3) QMT BUSINESS MATHEMATICS

Example: REV 00 A pair of shoes is advertised at RM 229 less 15%, 10% and 5%. Find, a) The single discount equivalent, b) The net price. Solution: a) r = 1 – (1 - r1) (1 – r2) (1 – r3) = 1 – (1 – 0.15) (1 – 0.1) (1 – 0.05) = 1 – (0.85) (0.9) (0.95) = 1 – = = % QMT BUSINESS MATHEMATICS

REV 00 b) NP = L (1 – r) = RM 229 (1 – ) = RM 229 ( ) = RM QMT BUSINESS MATHEMATICS

4.5 Cash Discount REV 00 Cash discounts are often stated under the heading of invoice as, for example 3/10, 2/20, n/30. This means, if the buyer pays the invoice within 10 days of the invoice date, the buyer is entitled to receive a 3% discount or, if payment is made within 20 days from the invoice date, the buyer will receive a 2% discount. N/30 (sometimes written net 30) means the credit period is 30 days. QMT BUSINESS MATHEMATICS

Example: REV 00 An invoice dated 15 November 2007 for RM 2800 was offered cash discount 2/10, 1/15, n/30. If the invoice was paid on 27 November 2007, what was the payment? Solution: 15 Nov – 27 Nov = 12 days, so entitle for a 1% cash discount Net Payment = NP – Cash Discount = RM 2800 – (0.01 x RM 2800) = RM 2800 – RM 28 = RM 2772 QMT BUSINESS MATHEMATICS

4.6 Partial Payment of Invoice
REV 00 4.6 Partial Payment of Invoice If a buyer pays only part of the invoice within the discount period, he receives a proportionate fraction of the cash discount that is offered. He will only receive the full amount of the cash discount if he settles all the payment. QMT BUSINESS MATHEMATICS

REV 00 Example: An invoice amounting to RM 4500 and dated 27 October 2007 offered cash discount terms of 10/15, n/30. Find the amount outstanding if the buyer paid RM 2000 on 5 November 2007. QMT BUSINESS MATHEMATICS

REV 00 Solution: Amount paid = (Credit given) x (1 – Discount rate) RM 2000 = (Credit given) x (1 – 0.1) Credit given = RM 2000 0.9 = RM Outstanding amount = RM 4500 – RM = RM QMT BUSINESS MATHEMATICS

REV 00 CHAPTER 5 DEPRECIATION QMT BUSINESS MATHEMATICS

5.1 Depreciation REV 00 Depreciation is an accounting procedure for allocating the cost of capital assets, such as buildings, machinery tools and vehicles over their useful life. Can also be viewed as decline in value of assets because of age, wear and tear or decreasing efficiency. Many properties such as buildings, machinery, vehicles, and equipment depreciate in value as they get older. QMT BUSINESS MATHEMATICS

REV 00 Several terms are commonly used in calculation relating to depreciation: 1. Original cost The original cost of an asset is the amount of money paid for an asset plus many sales taxes, delivery charges, installation charges and other costs incurred. 2. Salvage value The salvage value (scrap value or trade-in value) is the value of an asset at the end of its useful life. QMT BUSINESS MATHEMATICS

REV 00 3. Useful life The useful life is the expectancy of the asset or the number of years the asset is expected to last. 4. Total depreciation The total depreciation or the wearing value of an asset is the difference between cost and scrap value. 5. Annual depreciation The annual depreciation is the amount of depreciation in a year. It may or may not be equal from year to year. QMT BUSINESS MATHEMATICS

REV 00 6. Accumulated depreciation The accumulated depreciation is the total depreciation to date. 7. Book value The book value or carrying value of an asset is the value of the asset as shown in the accounting record. It is the difference between the original cost and the accumulated depreciation charged to that date. QMT BUSINESS MATHEMATICS

REV 00 Three methods of depreciation are commonly used. These methods are: Straight line method Declining balance method Sum of years digits method QMT BUSINESS MATHEMATICS

5.2 Straight Line Method REV 00 The simplest of the three methods and probably the most common method used. The total amount of depreciation is spread evenly to each accounting period through the useful life of the asset. Formula: Annual depreciation = Cost – Salvage Value Useful life = Total depreciation Useful life QMT BUSINESS MATHEMATICS

REV 00 Annual rate of depreciation = Annual depreciation x 100% OR Total depreciation = x 100% Useful life Book value = Cost – Accumulated depreciation QMT BUSINESS MATHEMATICS

Example 1: REV 00 ABC company bought a lorry for RM The lorry is expected to last 4 years and its salvage value at the end of 4 years is RM using the straight line method, calculate: a) The annual depreciation. b) The annual rate of depreciation. c) Calculate the book value of the lorry at the end of the second year. QMT BUSINESS MATHEMATICS

REV 00 Solution: a) Annual depreciation = Cost – Salvage Value Useful life = – 10000 4 = RM 13000 b) Annual rate of depreciation = Annual depreciation x 100% Total depreciation = x 100% 52000 = 25% QMT BUSINESS MATHEMATICS

REV 00 c) Book value = Cost – Accumulated depreciation = – (2 x 13000) = RM 36000 QMT BUSINESS MATHEMATICS

Example 2: REV 00 A firm bought a machine for RM The machine is expected to be obsolete in four years with a salvage value of RM Find, a) The book value of the machine after three years, and b) Prepare a depreciation schedule, Using the straight line method. QMT BUSINESS MATHEMATICS

REV 00 Solution: a) Annual depreciation = Cost – Salvage Value Useful life = 5000 – 1000 = RM 1000 Book value = Cost – Accumulated depreciation = 5000 – (3 x 1000) = RM 2000 QMT BUSINESS MATHEMATICS

b) The depreciation schedule
REV 00 End of year Annual depreciation (RM) Accumulated depreciation (RM) Book value at end of year (RM) - 5000 1 1000 4000 2 2000 3000 3 4 QMT BUSINESS MATHEMATICS

5.3 Declining Balance Method
REV 00 Declining balance method is an accelerated in which higher depreciation charges are deducted in the early life of the asset. Formula: BV = C (1 – r)n Where: BV = Book value C = Cost of asset r = Rate of depreciation n = Number of years QMT BUSINESS MATHEMATICS

REV 00 Annual rate of depreciation = Where : S = the Book value at the end of the Useful life (salvage value) C = Cost of the product r = Annual rate of depreciation n = Useful life in years QMT BUSINESS MATHEMATICS

Example 1: REV 00 A new equipment costing RM is purchased. Using a declining balance rate of 12%, find, a) The book value at the end of five years, and b) Construct the depreciation schedule for the five years of use. Solution: a) BV = C (1 – r)n BV = (1 – 0.12)5 BV = RM QMT BUSINESS MATHEMATICS

REV 00 Year Annual depreciation (RM) Accumulated depreciation (RM) Book value at the end of year (RM) - 10000 1 1200 8800 2 1056 2256 7744 3 929.28 4 817.77 5 719.63 QMT BUSINESS MATHEMATICS

Example 2: REV 00 A machine costing RM has a life expectancy of four years and a salvage value of RM Calculate the accumulated depreciation and book value at the end of three years using the declining balance method. Solution: = 24% QMT BUSINESS MATHEMATICS

REV 00 BV = C (1 – r)n BV = (1 – 0.24)3 BV = RM Accumulated depreciation after 3 years = Cost – Book value = – = RM QMT BUSINESS MATHEMATICS

5.4 Sum of Years Digits Method
REV 00 Based on the sum of the digits representing the number of years of useful life of the asset. If an asset has a useful life of 3 years, the sum of digits is S = = 6 It can be calculated with the formula: S = n(n + 1) 2 Where: S = Sum of years digits n = Useful life QMT BUSINESS MATHEMATICS

REV 00 The amount of depreciation in the first year is n/S of the depreciable value of the asset, the second is (n-1)/S, the third is (n-2)/S and so on. Example 1: A machine purchased for RM Its life is 4 years with a zero trade-in value. Calculate the annual depreciation. QMT BUSINESS MATHEMATICS

REV 00 Solution: Sum of years’ digit, S = = 10 OR S = 4(4 + 1) 2 = 10 Year Annual Depreciation 1 4/10 x = RM 20000 2 3/10 x = RM 15000 3 2/10 x = RM 10000 4 1/10 x = RM 5000 QMT BUSINESS MATHEMATICS

Example 2: REV 00 A machine costing RM has a life expectancy of five years and a salvage value of RM Using the sum-of-year digits method, a) Calculate the book value at the end of 2 years. b) Prepare a depreciation schedule. Solution: a) Depreciable value = cost – salvage value = – 10000 = RM 40000 QMT BUSINESS MATHEMATICS

REV 00 S = n(n + 1) 2 = 5(5 + 1) = 15 Accumulated depreciation at the end of 2 years = (5/15 + 4/15) x 40000 = RM 24000 BV = Cost – Accumulated depreciation = – 24000 = RM 26000 QMT BUSINESS MATHEMATICS

REV 00 Year Annual depreciation (RM) Accumulated depreciation (RM) Book value at the end of year (RM) - 50000 1 2 24000 3 8000 32000 18000 4 5 40000 10000 QMT BUSINESS MATHEMATICS

REV 00 CHAPTER 6 MATHEMATICS OF SELLING QMT BUSINESS MATHEMATICS

6.1 Markup REV 00 In order to gain profit from selling, the company must sell their product at a higher price than the product cost. The difference between a product’s cost and selling price is refer to as markup. It can be either in money value or percentage. The rate of mark-up is known as markup percentage. Formula: Markup = Selling Price – Cost Price QMT BUSINESS MATHEMATICS

6.2 Markup Percent REV 00 Markup is usually expressed as a percent. It can be expressed as: a) Markup percent based on retail price % Mr = M x 100% R b) Markup percent based on cost price % Mc = M x 100% C QMT BUSINESS MATHEMATICS

Example 1: REV 00 The cost price of a piano is RM What is the retail price if the seller wants a 15% mark-up based on a) Cost price, and b) Retail price. Solution: a) R = C + M R = RM (RM 4500 x 15%) R = RM RM 675 R = RM 5175 QMT BUSINESS MATHEMATICS

REV 00 b) Let the retail price be X. Then R = C + M X = RM X X – 0.15X = RM 4500 0.85X = RM 4500 X = RM 4500 0.85 X = RM QMT BUSINESS MATHEMATICS

Example 2: REV 00 ABC boutique purchases 100 dresses at a cost of RM 18 each. The boutique expects that 20% of the dresses will be sold at a reduced price of RM 12 each. If the boutique is to maintain a 65% mark-up on cost on the entire purchase, find the regular price of the dresses. QMT BUSINESS MATHEMATICS

REV 00 Solution: Cost of 100 dresses = RM 18 x 100 = RM 1800 Mark-up 65% Sale of 100 dresses = RM 1800 x 0.65(RM 1800) = RM 2970 Sale of 20 dresses = RM 12 x 20 = RM 240 Sale of remaining 80 dresses = RM 2970 – RM 240 = RM 2730 Regular selling price = RM 2730 = RM per dress QMT BUSINESS MATHEMATICS

Example 3: REV 00 A retailer purchased 150 kg of tomato at RM 1.20 per kg. A 8% spoilage is expected. If he plans to make a 55% mark-up based on overall cost, what is the selling price of the tomato? Solution: Overall cost = RM 1.20 x 150 kg = RM 180 Amount left after deducting spoilage = 150 kg – 0.08(150) = 138 kg QMT BUSINESS MATHEMATICS

REV 00 Mark-up 55% R = C + M = RM (RM 180) = RM 279 (retail price for 138 kg) Selling price per kg = RM 279 138 = RM 2.02 QMT BUSINESS MATHEMATICS

6.3 Conversion of Markup Percent
REV 00 Markup percent based on retail price: Since R = C + M 1 + % Mc = 100% + % Mc Hence : QMT BUSINESS MATHEMATICS

REV 00 Markup percent based on cost price: Since R = C + M 100% = 1 - % MR + % MR Hence : QMT BUSINESS MATHEMATICS

REV 00 Example : a) The mark-up percent on cost price of an item is 20%. What is its mark-up percent based on retail price? b) The mark-up percent based on retail price of an item is 15%. What is its mark-up percent based on cost price? QMT BUSINESS MATHEMATICS

REV 00 Solution: a) % Mr = % Mc 1 + % Mc = = = 16.67% b) % Mc = % MR 1 - % MR = = = 17.65% QMT BUSINESS MATHEMATICS

6.4 Markdown REV 00 Sometimes retailer may reduce the marked price due to special promotions, festive seasons or the items being obsolete. Markdown is the reduction from the selling price or marked price, normally in terms of percentage. Formula: Markdown Amount = Old Selling Price – New Selling Price Markdown percent based on old price, % MD = MD x 100% OP QMT BUSINESS MATHEMATICS

Example 1: REV 00 The markdown percent on a handphone is 15%. If the new retail price is RM 850, find the old retail price. Solution: Let the old price be RM K % MD = MD x 100% OP K = 850 0.15 = (K – 850) K = 850 K 0.15K = K – = RM 1000 QMT BUSINESS MATHEMATICS

Example 2: REV 00 During the year end sales, a department store marked down a pair of shoes by 25%, making the selling price RM 270. At this selling price, the department store made a 18% mark-up on the selling price. Find a) The regular price of the shoes. b) The cost of the shoes. c) The mark-up percent of the shoes at the regular price. QMT BUSINESS MATHEMATICS

REV 00 Solution: a) NP = L (1 – r) 270 = L (1 – 0.25) L = 270 0.75 = RM 360 b) R = C + M 270 = X (270) X = 270 – 48.6 = RM QMT BUSINESS MATHEMATICS

REV 00 c) Mark-up percent at regular price, % Mr = M x 100% R = (Regular price – Cost) x 100% Regular price = (360 – ) x 100% 360 = 38.5% QMT BUSINESS MATHEMATICS

6.5 Profit and Loss REV 00 Business does not always generate income or profit. If the business is not managed properly, it may increase operating expense and indirectly increase cost of the product. The actual cost of the product is the purchase price plus operating expenses. In accounting, operating expenses include cost of goods sold, official rental, advertising, salary, commission and so on. QMT BUSINESS MATHEMATICS

REV 00 In general, there are three possibilities in business: a) M = OE : Break-even b) M > OE : Profit c) M < OE : Loss Where: M = Markup OE = Operating Expenses QMT BUSINESS MATHEMATICS

REV 00 Formula: Break-even Price, BEP = Cost Price + OE Retail Price = Cost + Net Profit + OE Net Profit / (Loss) = Retail Price – BEP Gross Profit = NP + OE QMT BUSINESS MATHEMATICS

Example : REV 00 A retailer bought a radio for RM 200. Buying expenses amounted to RM 20. Operating expenses incurred were 20% of the cost price. If the retailer made a 25% net profit based on cost, find a) The retail price, b) The gross profit , c) The net profit, d) The breakeven price, and e) The maximum markdown that could be offered so that there is no profit or loss. QMT BUSINESS MATHEMATICS

REV 00 Solution: a) Retail price = (220) + 0.2(220) = RM 319 b) Gross profit = NP + OE = 0.25(220) + 0.2(220) = RM 99 c) Net profit = Gross profit – OE = 99 – 44 = RM 55 QMT BUSINESS MATHEMATICS

REV 00 d) Breakeven price = C + OE = = RM 264 e) Maximum reduction in price = R – BEP = 319 – 264 = RM 55 QMT BUSINESS MATHEMATICS

REV 00 CHAPTER 7 SIMPLE INTEREST QMT BUSINESS MATHEMATICS

7.1 Interest REV 00 “Interest ” comes from the Latin word intereo which means “to be lost”. When developed into the concept of borrowing money, the lender is likely to lose his money when he pays back the money with interest. Nowadays, interest is not only paid but gained if we make an investment. QMT BUSINESS MATHEMATICS

7.2 Simple Interest Formula
REV 00 The simple interest amount is calculated by the following formula: I = Prt Where: I = Simple interest P = Principal / Present value r = Interest rate (in decimals) t = Time / Period (in years) QMT BUSINESS MATHEMATICS

Example 1: REV 00 RM 1000 is invested for 2 years in a bank, earning simple interest rate of 8% per annum. Find the simple interest earned. Solution: I = Prt = 1000 x 0.08 x 2 = RM 160 QMT BUSINESS MATHEMATICS

Example 2: REV 00 Raihan invests RM 5000 in an investment fund for three years. At the end of the investment period, his investment will be worth RM Find the simple interest rate that is offered. Solution: I = RM 6125 – RM 5000 = RM 1125 From I = Prt, we get 1125 = 5000 x r x 3 r = 7.5% QMT BUSINESS MATHEMATICS

Example 3: REV 00 How long does it take a sum of money to triple itself at a simple interest rate of 5% per annum? Solution: Let the original principal be RMK and time taken be t years. Hence interest earned is RM 3K – RM K = RM 2K Then from I = Prt, we get 2K = K x 0.05 x t t = 40 years QMT BUSINESS MATHEMATICS

7.3 Simple Amount Formula REV 00 Simple amount is the sum of the original principal and the interest earned. Therefore, the simple amount formula is given as: S = P(1 + rt) Where: S = Maturity value / Future value P = Principal / Present value r = Interest rate (in decimals) t = Term / Period (in years) QMT BUSINESS MATHEMATICS

Example : REV 00 RM is invested for 4 years 9 months in a bank earning a simple interest rate of 10% per annum. Find the simple amount at the end of the investment period. Solution: From S = P(1 + rt), we get S = (1 + (0.1 x 4.75)) = RM 14750 QMT BUSINESS MATHEMATICS

7.4 Four Basic Concepts REV 00 There are four different methods for determining terms (t): Exact time: It is the exact number of days between two given dates. Approximate time: Time computed on the assumption that each month has 30 days. Exact interest: Interest calculated based on 365 days a year or 366 days for a leap year. Ordinary interest: Interest is calculated based on 360 days a year. QMT BUSINESS MATHEMATICS

7.5 Present Value REV 00 Present value may be debt or an investment amount that is lent or invested today, and that will be mature in a specific time together with interest. By transposing the maturity value formula, we have the present value formula as follows: P = S (1 + rt) OR P = S (1 + rt)-1 QMT BUSINESS MATHEMATICS

Example : REV 00 If Saleh has to pay RM 350 interest for a loan that charges 7% simple interest per annum for 2 and a half years, find the amount of loan. Solution: From P = S (1 + rt)-1, we get P = 350(1 + (0.07 x 2.5))-1 P = RM QMT BUSINESS MATHEMATICS

7.6 Equation of Value REV 00 Every value of money has an attached date, the date on which it is due. An equation that states the equivalence of two sets of dated values at a stated date is called an equation of value or equivalence. The stated date is called the focal date, the comparison date or the valuation date. To set up and solve an equation of value, the following procedure should be carried out: QMT BUSINESS MATHEMATICS

REV 00 1. Draw a time diagram with all the dated values. 2. Select the focal date. 3. Pull all the dated values to the focal date using the stated interest rate. 4. Set up the equation of value and then solve. QMT BUSINESS MATHEMATICS

REV 00 Example : A debt of RM 800 due in four months and another of RM due in nine months are to be settled by a single payment at the end of six months. Find the size of this payment using a) The present as the focal date, b) The date of settlement as the focal date, Assuming money is worth 6% per annum simple interest. QMT BUSINESS MATHEMATICS

REV 00 Solution: a) Focal date months 800 X 1000 Let the single payment at the end of 6 months be RM X. Amount of the RM 800 debt at the focal date = 800(1 + (0.06 x 4/12))-1 = RM QMT BUSINESS MATHEMATICS

REV 00 Amount of the RM 1000 debt at the focal date = 1000(1 + (0.06 x 9/12))-1 = RM Amount of the single payment at the focal date = X (1 + (0.06 x 6/12))-1 = RM X Setting up the equation of value, we get X = X = X = RM QMT BUSINESS MATHEMATICS

REV 00 b) Focal date months 800 X 1000 Amount of the RM 800 debt at the focal date = 800(1 + (0.06 x 2/12)) = RM 808 Amount of the RM 1000 debt at the focal date = 1000(1 + (0.06 x 3/12)) = RM QMT BUSINESS MATHEMATICS

REV 00 Let the amount of payment at the end of 6 months be RM X. Setting up the equation of value, we get X = RM RM = RM *(It should be noted that the two answers in part a and b are different when the focal dates are different) QMT BUSINESS MATHEMATICS

REV 00 CHAPTER 8 PROMISSORY NOTE QMT BUSINESS MATHEMATICS

8.1 Promissory Note REV 00 A written document made by one person or party to pay a stated sum of money on a specified future date to another person or party. Are negotiable documents and can be of two types; interest bearing notes and non-interest bearing notes. The main features of a promissory note are as follows: QMT BUSINESS MATHEMATICS

REV 00 1. Maker The maker is the person that signs the note. 2. Payee The payee is the person to whom the payment is to be made. 3. Date of the note The date of the note is the date on which the note is made. 4. Term of the note The term of the note is the length of time until the note is due for payment. QMT BUSINESS MATHEMATICS

REV 00 5. Face value The face value of the note is the amount stated on the note. 6. Maturity value The maturity value of the note is the total sum of money which the payee will receive on the maturity date. The maturity value of a non-interest bearing note is the face value while the maturity value of an interest-bearing note is the face value plus day interest that is due. 7. Maturity date The maturity date of the note is the date on which the maturity value is due. QMT BUSINESS MATHEMATICS

Example of Promissory Note:
REV 00 RM APRIL 2008 Sixty days after date I promise to pay the order of Mohammed Ali Ringgit Malaysia: Two thousand five hundred only for value received with interest at the rate of 8.00% per annum until paid. No Due: 19 June 2008 Mat Jenin QMT BUSINESS MATHEMATICS

Example 1: REV 00 From the promissory note show in the example, a) Who is the maker of the note, b) Who is the payee of the note? Calculate the maturity value of the note. Solution: a) The maker is Mat Jenin. b) The payee is Mohammed Ali. Maturity value = face value + interest due = [2500 x 0.08 x (60/360)] = RM QMT BUSINESS MATHEMATICS

Example 2: REV 00 The maturity value of a 90-day interest bearing promissory note is RM If the interest rate is 6% per annum, what is the face value of the note? Solution: S = P(1 + rt) 1200 = P[ x (90/360)] 1200 = P(1.015) P = RM QMT BUSINESS MATHEMATICS

Example 3: REV 00 The interest rate on a 60-day promissory note is RM 50. If the interest rate is 5% per annum, find the face value of the note. Solution: I = Prt 50 = P x 0.05 x (60/360) 50 = P( ) P = RM 6000 QMT BUSINESS MATHEMATICS

8.2 Bank Discount REV 00 Bank discount is computed in much the same way as simple interest except that it is based on the final amount (to be paid back) or maturity value. Formula: D = Sdt Where: D = Bank discount S = Amount of maturity value d = Discount value t = Term of discount in years QMT BUSINESS MATHEMATICS

REV 00 Bank proceeds is the net amount received by the borrower. Formula: Bank proceeds = Maturity Value – Bank Discount P = S – D P = S – Sdt P = S(1 – dt) QMT BUSINESS MATHEMATICS

Example 1: REV 00 Alex borrows RM for 4 months from a lender who charges a discount rate of 9%. Find, a) The discount, and b) The proceeds. Solution: a) b) D = Sdt = 6500 x 0.09 x (120/360) = RM 195 P = S – D = 6500 –195 = RM 6305 QMT BUSINESS MATHEMATICS

Example 2: REV 00 If Sharizan needs RM 5000 now, how much should he borrow from his bank for 2 years at a 4% bank discount rate? Solution: P = S(1 – dt) 5000 = S(1 – (0.04 x 2)) 5000 = S(0.92) S = RM QMT BUSINESS MATHEMATICS

Example 3: REV 00 Sheela receives an invoice of RM 2000 with cash discount terms 3/10, n/40. a) How much should be borrowed for 30 days from a bank that charges a 9% discount rate to take advantage of the cash discount? b) How much will be saved by borrowing the money to take advantage of the cash discount? QMT BUSINESS MATHEMATICS

REV 00 Solution: a) Cash discount offer = 2000 x 3% = RM 60 To get proceeds of RM 1940 (RM 2000 – RM 60), she should borrow more than RM Let the amount borrowed for 30 days be S ringgit. P = S(1 – dt) 1940 = S[1 –( 0.09 x (30/360))] S = RM QMT BUSINESS MATHEMATICS

REV 00 b) Bank discount for the loan = RM – RM 1940 = RM 14.66 Amount that will be saved = RM 60 – RM 14.66 = RM 45.34 QMT BUSINESS MATHEMATICS

Example 4: REV 00 Find the present value of RM 1000 due in 3 years at a) A simple interest rate of 6%, and b) A simple discount at 6%. Solution: a) b) P = S(1 + rt)-1 P = 1000(1 + (0.06 x 3))-1 P = RM P = S(1 – dt) P = 1000(1 – (0.06 x 3)) P = RM 820 QMT BUSINESS MATHEMATICS

8.3 Simple Interest Rate Equivalent to Bank Discount Rate
REV 00 At interest rate, r% and a discount rate, d% are said to be equivalent if the two rates give the same present value for an amount due in the future. If the present value is S, then the present value of S, at r% simple interest rate is S(1 + rt)-1 and the present value of S, at d% bank discount rate is S(1 – dt). Equating the two present value, we get: QMT BUSINESS MATHEMATICS

REV 00 S(1 + rt)-1 = S(1 – dt) 1 = 1 – dt 1 + rt Solving for d, we get dt = 1 – 1 dt = rt d = r QMT BUSINESS MATHEMATICS

REV 00 Solving for r, we get 1 + rt = 1 1 – dt rt = 1 rt = dt r = d 1 - dt - 1 QMT BUSINESS MATHEMATICS

Example 1: REV 00 A bank discounts a RM 3000 note due in 3 months using a bank discount rate of 8%. Find the equivalent simple interest rate that is charges by the bank. Solution: r = d 1 – dt r = [1 – (0.08 x (90/360))] = 8.16% QMT BUSINESS MATHEMATICS

Example 2: REV 00 What discount rate should a lender charge to earn an interest rate of 15% on a 6 months loan? Solution: d = r 1 + rt d = [1 + (0.15 x (180/360))] d = 13.95% QMT BUSINESS MATHEMATICS

8.4 Discounting Promissory Notes
REV 00 A promissory note can be sold to a bank before its maturity date if the holder is in need of cash. Selling the note to the bank is called discounting the note. The amount received on the date of discounting is called the proceeds. The proceeds of a promissory note are computed as follows: QMT BUSINESS MATHEMATICS

REV 00 a) Find the maturity value of the note. For non-interest bearing note, it is the face value. If the note is interest bearing, then Maturity value = Face Value + Interest b) Find the bank discount, D with the formula D = Sdt c) Compute the proceeds Proceeds = Maturity Value – Bank Discount QMT BUSINESS MATHEMATICS

Example : REV 00 Mr. Lee received a promissory note for RM 2600 with interest at 8% per annum that was due in 180 days. The note was dated 20 June The note was discounted on 24 July 2005 at a bank that charges 11% discount. Determine a) The maturity date, b) The maturity value, c) The discount period, and d) The proceeds. QMT BUSINESS MATHEMATICS

REV 00 Solution: a) Maturity date = 20 June days = 17 December 2005 b) Maturity value = P(1 + rt) = 2600[1 + (0.08 x (180/360))] = RM 2704 c) Discount period = 24 July 2005 to 17 Dec 2005 = ( ) = 146 days QMT BUSINESS MATHEMATICS

REV 00 d) D = Sdt = 2704 (0.12 x (174/360)) = RM Proceeds = Maturity value – Bank discount = 2704 – = RM QMT BUSINESS MATHEMATICS

REV 00 CHAPTER 9 COMPOUND INTEREST QMT BUSINESS MATHEMATICS

9.1 Time Value of Money REV 00 Money has time value, that is a ringgit today is worth more than a ringgit tomorrow. Money has time value because of its investment opportunities. QMT BUSINESS MATHEMATICS

9.2 Compound Interest REV 00 In compounding, after the interest is calculated, it is then added to the principal and becomes an adjusted principal. Processes are repeated until the end of the loan or investment term. Normally used with long-term loan or investment, and the interest is calculated more than once during the loan or investment term. QMT BUSINESS MATHEMATICS

REV 00 The interest earned is called compound interest, and the final sum at the end of the period of borrowing is called the compound amount. Therefore, compound interest is the difference between the original principal and the compound amount. QMT BUSINESS MATHEMATICS

9.3 Some Important Terms REV 00 Some of the common terms used in relation to compound interest are: 1. Original principal 2. Nominal interest rate 3. Interest period or conversion period 4. Frequency of conversions 5. Periodic interest rate 6. Number of interest periods in the investment period QMT BUSINESS MATHEMATICS

9.4 Compound Interest Formula
REV 00 The method used in finding compound amount at the end of the nth period is as follows: S = P(1 + i)n Where: P = Principal / Present Value S = Future Value n = Number of Periods (number of years multiplied by number of times compounded per year) i = Interest rate per compound period QMT BUSINESS MATHEMATICS

Example 1: REV 00 Find the future value of RM 1000 which was invested for a) 4 years at 4% compounded annually, b) days at 10% compounded daily, and c) 2 years 3 months at 4% compounded quarterly. QMT BUSINESS MATHEMATICS

REV 00 Solution: a) S = 1000( )4 = RM b) S = = RM 360 c) S = = RM 4 QMT BUSINESS MATHEMATICS

Example 2: REV 00 What is the nominal rate compounded monthly that will make RM 1000 become RM 2000 in five years? Solution: From S = P(1 + i)n, we get 2000 = k 60 12 2 = 1 + k 60 k = 13.94% QMT BUSINESS MATHEMATICS

Example 3: REV 00 How long does it take a sum of money to double itself at 14% compounded annually? Solution: Let the original principal = W Therefore sum after n years = 2W From S = P(1 + i)n, we get 2W = W( )n 2 = ( )n lg 2 = n lg 1.14 n = 5.29 years QMT BUSINESS MATHEMATICS

9.5 Effective, Nominal and Equivalent Rates
REV 00 Effective rate: Simple rate that will produce the same accumulated amount as the nominal rate that compounded each period after one year. Nominal rate: Stated annual interest rate at which interest is compounding more than once a year. Equivalent rate: Two different rates that yield the same value at the end of one year. QMT BUSINESS MATHEMATICS

9.6 Relationship Between Effective and Nominal Rates
REV 00 9.6 Relationship Between Effective and Nominal Rates The relationship between the nominal rate and effective rate is derived as follows: Assume a sum RM P is invested for one year. Then the future value after one year: a) At r% effective = P(1 + r) b) At k% compounded m times a year = P(1 + k/m)m QMT BUSINESS MATHEMATICS

Example 1: REV 00 Find the effective rate which is equivalent to 16% compounded semi-annually. Solution: QMT BUSINESS MATHEMATICS

Example 2: REV 00 Find the nominal rate, compounded monthly which is equivalent to 9% effective rate. Solution: QMT BUSINESS MATHEMATICS

9.7 Relationship Between Two Nominal Rates
REV 00 The relationship between two nominal rates is given as follows: (1 + k/m)m = (1 + K/M)M Where: k and K are two different annual rates with respectively two different frequencies of conversions, m and M. QMT BUSINESS MATHEMATICS

Example : REV 00 Find k% compounded quarterly which is equivalent to 6% compounded monthly. Solution: QMT BUSINESS MATHEMATICS

9.8 Present Value REV 00 Present value or discounted value is the value which will yield the sum (S) after certain time and at a specific interest rate. We can find present value by transposing the formula as follows: S = P(1 + i)n P = S (1 + i)n OR P = S(1 + i)-n QMT BUSINESS MATHEMATICS

REV 00 Example : A debt of RM 3000 will mature in three years’ time. Find a) The present value of this debt, b) The value of this debt at the end of the first year, Assuming money is worth 14% compounded semi annually. QMT BUSINESS MATHEMATICS

REV 00 Solution: a) From P = S(1 + i)-n, we get P = 2 = RM b) P = S(1 + i)-n P = = RM QMT BUSINESS MATHEMATICS

9.9 Equation of Value REV 00 An equation that expresses the equivalence of two sets of obligations at a focal date. In other words, it expresses the following: What is owed = What is owned at the focal date OR What is given = What is received at the focal date QMT BUSINESS MATHEMATICS

REV 00 Example : A debt of RM 7000 matures at the end of the second year and another of RM 8000 at the end of six years. If the debtor wishes to pay his debts by making two equal payments at the end of the fourth year and the seventh year, what are these payments assuming money is worth 6% compounded semi-annually? QMT BUSINESS MATHEMATICS

REV 00 Solution: years X X Focal date QMT BUSINESS MATHEMATICS

REV 00 Let the payment be RM X each. Formulating the equation of value at the focal date as shown, we get What is owed = What is owned X(1 + 3%)6 + X = 7000(1 + 3%) (1 + 3%)2 X = RM QMT BUSINESS MATHEMATICS

9.10 Continuous Compounding
REV 00 We have been discussing compounding of interest on discrete time intervals (daily, monthly, etc). If compounding of interest is done on a continuous basis, then we will have a different picture of the future value as shown below: Discrete compounding Continuous compounding Future value Future value Time QMT BUSINESS MATHEMATICS Time

REV 00 The future value of a sum of money compounded continuously is given by: S = Peit Where: S = Future value P = Original principal e = … i = Continuous compounding rate t = Time in years QMT BUSINESS MATHEMATICS

Example : REV 00 Find the accumulated value of RM 1000 for six months at 10% compounded continuously. Solution: From S = Peit, we get S = 1000[e10% x 0.5] = RM QMT BUSINESS MATHEMATICS

REV 00 CHAPTER 10 ANNUITY QMT BUSINESS MATHEMATICS

10.1 Future Value of Ordinary Annuity Certain
Future value (accumulated value) of an ordinary annuity certain is the sum of all the future values of the periodic payments. The derivation of the formula of future value of ordinary annuity certain are as follow: Periodic payments = RM R Interest rate per interest period = i% Term of investment = n interest periods Future value of annuity at end of n interest periods = RM S QMT BUSINESS MATHEMATICS

REV 00 Formula: S = R[(1 + i)n – 1] i QMT BUSINESS MATHEMATICS

Example : REV 00 RM100 is deposited every month for 2 years 7 months at 12% compounded monthly. What is the future value of this annuity at the end of the investment period? How much interest is earned? Solution: months … S 100 100 100 100 100 100 No deposit Annuity begins Last deposit Annuity ends QMT BUSINESS MATHEMATICS

REV 00 From , we get Interest earned, I = S – nR = – (31 x 100) = RM QMT BUSINESS MATHEMATICS

10.2 Present Value of Ordinary Annuity Certain
REV 00 Consist of the sum of all the present values of periodic payments. The deviation of the formula of present value of ordinary annuity certain is illustrated in the following: Periodic payments = RM R Interest rate per interest period = i% Term of investment = n interest periods Future value of annuity at end of n interest periods = RM A QMT BUSINESS MATHEMATICS

REV 00 Formula: A = R[1 - (1 + i)-n] i QMT BUSINESS MATHEMATICS

Example : REV 00 Raymond has to pay RM 300 every month for 24 months to settle a loan at 12% compounded monthly. a) What is the original value of the loan? b) What is the total interest that he has to pay? Solution: a) payments … A 300 300 300 300 300 300 No payment Last payment QMT BUSINESS MATHEMATICS

REV 00 From , we get b) Total interest = (300 x 24) – = RM QMT BUSINESS MATHEMATICS

10.3 Amortization REV 00 An interest bearing a debt is said to be amortized when all the principal and interest are discharged by a sequence of equal payments at equal intervals of time. A table showing the distribution of principal and interest payments for the various periodic payments. 10.4 Amortization Schedule QMT BUSINESS MATHEMATICS

Example : REV 00 A loan of RM 1000 at 12% compounded monthly is to be amortized by 8 monthly payments. a) Calculate the monthly payment. b) Construct an amortization schedule. Solution: a) From , we get QMT BUSINESS MATHEMATICS

b) Amortization Schedule
REV 00 Period Beginning balance (RM) Ending balance (RM) Monthly payment (RM) Total paid (RM) Total principal paid (RM) Total interest paid (RM) 1 1000 879.31 130.69 120.69 10 2 757.41 261.38 242.59 18.79 3 634.29 392.07 365.71 26.36 4 509.94 522.76 490.06 32.70 5 384.35 653.45 615.65 37.80 6 257.50 784.14 742.50 41.64 7 129.39 914.83 870.61 44.22 8 0.00 45.51 QMT BUSINESS MATHEMATICS

10.5 Sinking Fund REV 00 When a loan is settled by the sinking fund method, the creditor will only receive the periodic interest due. The face value of the loan will only be settled at the end of the term. In order to pay the face value, debtor will create a separate fund in which he will make periodic deposits over the term of the loan. The series of deposits made will amount to the original loan. QMT BUSINESS MATHEMATICS

Example : REV 00 A debt of RM 1000 bearing interest at 10% compounded annually is to be discharged by the sinking method. If five annual deposits are made into a fund which pays 8% compounded annually, a) Find the annual interest payment, b) Find the size of the annual deposit into the sinking fund, c) What is the annual cost of this debt, d) Construct the sinking fund schedule. QMT BUSINESS MATHEMATICS

REV 00 Solution: a) Annual interest payment = RM 1000 x 10% = RM 100 b) From , we get c) Annual cost = Annual interest payment Annual deposit = RM RM = RM QMT BUSINESS MATHEMATICS

d) Sinking Fund Schedule
REV 00 End of period (year) Interest earned (RM) Annual deposit (RM) Amount at the end of period (RM) 1 170.46 2 13.64 354.56 3 28.36 553.38 4 44.27 768.11 5 61.45 QMT BUSINESS MATHEMATICS

10.6 Annuity With Continuous Compounding
REV 00 Future value of annuity: Present value of annuity: QMT BUSINESS MATHEMATICS

REV 00 Where: S = Future value of annuity A = Present value of annuity R = Periodic payment or deposit e = Natural logarithm k = Annual continuous compounding rate t = Time in years p = Number of payments in 1 year QMT BUSINESS MATHEMATICS

Example : REV 00 James wins an annuity that pays RM 1000 at the end of every six months for four years. If money is worth 10% per annum continuous compounding, what is a) The future value of this annuity at the end of four years, b) The present value of this annuity? QMT BUSINESS MATHEMATICS

REV 00 Solution: a) From , we get b) From , we get QMT BUSINESS MATHEMATICS

REV 00 BUSINESS MATHEMATICS

Similar presentations

Presentation on theme: "REV 00 BUSINESS MATHEMATICS"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

REV 00 BUSINESS MATHEMATICS

Similar presentations

Presentation on theme: "REV 00 BUSINESS MATHEMATICS"— Presentation transcript:

Similar presentations

About project

Feedback