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Lecture

A - Unit Notes U3 - Percent.pdf

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Department
Business
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BUS 100
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Marla Spergel

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Math Skills for Business- Full Chapters 57 U3 Full Chapter Chapter 6 Percent 6.1 Percent is a special ‘fraction’ with denominator of 100. The numerator of a percent can be a whole number or a fraction but the numerator of a fraction is always a whole number. Percent symbol %, is the fraction, , that is, % = . Similarly, a given fraction has an equivalent percent, which is a ‘fraction’ with denominator 100. 6. 2 Decimal Form of Percent Percent can also be written in decimal form by dividing it by 100, which is the same as placing a decimal point two digits to the left. Example 3 5% = 5 ÷ 100 = 0.05 14% = 14 ÷ 100 = 0.14 20% = 20 ÷ 100 = 0.20 35.2% = 35.2 ÷ 100 = 0.352 ½ % = 0.50 ÷ 100 = 0.05 33 1 % = 33.33 ÷ 100 = 0.33 3 Math Skills for Business- Full Chapters 58 6.3 Ratio form of Percent Percent can also be written in ratio form with the percent being the first term and 100 the second term. Example 4 10 % = 10:100; 54.5% = 54.5: 100; 15% = 15:100 ; 75% = 75:100 6.4 Calculating Percentages We should distinguish percent from percentage. Percent is the rate and percentage is the result when the rate is applied. Percent is part of a whole which is 100 rather than 1 in fraction sense. For example, 5% could be interpreted as 5 out of 100. So, when someone promises to give you 5% of his $200 pocket money, he simply means he will give you $10. That is, $5 out of $100 and $5 out of the other $100, making it $10 (5 + 5 = 10). This is the same as 0.05 (5% changed to decimal) multiplied by 200. Therefore, we have the following procedure for calculating percentage: percentage = rate x base. In our example, the $200 is the base, and the rate is 5%. Using letters (Do you remember algebra?), B= base, P= percentage and R= rate of percent, we can shorten the procedure to, P = BR, percentage is equal to the rate of percent multiplied by the base quantity. Thus, we have the following formulas to work with, Percentage = Rate x Base, P = R X B, for finding the percentage Base = , B = for finding the base Rate = , R = , for finding the rate. Therefore, to work with percent the student must be able to identify the rate and the base. Math Skills for Business- Full Chapters 59 Example 1 In 2007, John bought a sofa chair for $299. He had to pay the GST of 6% and PST of 8% on his purchase. a) How much was the GST? B) How much was the PST? c) Will it make any difference if you were to apply a single rate of 14% sales tax? Solution a) GST = R x B = 299 × 0.06 =$17.94 b) PST = R × B = 299 × 0.08 =$23.92 c) No, it will not make any difference since 0.14 × 299 = $41.86. This is the same as $17.94 + 23.92. Example 2 Mary’s annual salary is $54,000. Her raise for this year is 3.5%. How much raise did she get? Solution Percentage (raise) =RB = 54000 × 0.035 =$1890. She got a raise of $1890. Example 3 An airplane was 85% full when it had 425 passengers. What is the full seating capacity of the airplane? Solution It should occur to you immediately you read the question that the airplane is full when it is 100%. This implies that a full-seating capacity occurs when it is 100% full Let y represent the full seating capacity of the airplane. Then 85% of y = $425 = ▯ Divide each side by 0.85 y= = 500. Math Skills for Business- Full Chapters 60 Alternatively, we could proceed to solve the problem as follows: 85% full = 425 1% full = 425 ÷ 85 = 5 100% full = 5 x 100 = 500 The full seating capacity of the plane is 500. The plane has a full-seating capacity of 500. 6.5 The Three-Step Method We can use the three-step method to estimate or calculate percentages mentally. To use this procedure the student should be able to calculate 10% and 1% of a quantity with ease. And use the results to make approximations for percentages. This method involves three steps as the following examples illustrate. Example 1 Francis bought a bicycle for her daughter costing $48.00. He had to pay sales tax of 14% on the $48. He was wondering an easy way to estimate the sales tax he had to pay. How should he go about this? Solution First, we have to figure this out: What is 10% of 48? (48 x 0.10 = 4.80, a matter of one decimal point to the left of 48) Second, what is 1% of 48? (48 x 0.01 = 0.48, just two decimal places to the left of 48) Third, since 10% of 48 is $4.80, we are left with finding 4% of 48. Halving 4.80 is 2.40, which is 5% of 48, but 5% > 4 %. We know that 1% of 48 is 0.48. So 2.40 – 0.48 = 1.92, that is 4% of 48. Therefore, 14% of 48 is 4.80 + 1.92 = 6.72. The sales tax is $6.72. If we were to do this mentally we would have just calculated 10% of 48, which is $4.80 and half this to get $2.40. This gives a total of $7.20, which is closer to $6.72. Math Skills for Business- Full Chapters 61 What you have to bear is mind is that, in using the three-step method you have to take the easier route. For example, suppose you were calculating 6% of a quantity. First, calculate 10%, which is easier to do and half it to get a 5% of the quantity. You are then left with calculating 1% of the quantity. You may approximate this in any way you want. The key is to learn to calculate 10% and 1% of a quantity as fast as you can. After that you are ready to use the three-step method. 6.6 Percent increase and decrease a) Finding Percent Rate of Increase or Decrease Percent is often used to show either increase or decrease in amount. The price of lumber, for example, could increase from $2.40 a metre to $2.50. Hydro rate may jump from $0.0530 per kilowatt hours to $0.0570 per kilowatt hours. A paper manufacturing business may reduce the prices of its computer printing paper products from, say $3.97 per 500 sheets to $3.27. The increase or decrease in the amount can be conveniently expressed in terms of percentages. The following procedures are used. (NA −OA ×)00 Rate of change (Increased) = OA , where NA = new amount and OA =Original amount. Where NA is the new amount, and OA is the original amount. Rate of change (Decrease) = (A − NA )100 OA Example 1 Mary’s hourly wage was increased from $14/h to $15.50/h. What is the percentage? Increase in her wage rate? Solution 1 The original amount is $14 and the new amount is $15.50. Therefore, this is an increase. Math Skills for Business- Full Chapters 62 Change = (15.50 −14)×100=10.7 ⇒ Mary’s hourly rate increased by about 14 11%. Example 2 In a survey of 400 people, it was found that 240 watched Everybody Loves Raymond. What percent does not watch that show? Solution 1 The original amount is 400. And those who watch that show are $240. Change = (400 − 240) ÷ 400 x 100 = (160 ÷ 400) x 100 = 40% 40% of the people surveyed do not watch that show. b) Finding the rate of Percent Change Instead of finding the percentage change, sometimes we have a situation where we are given the percent and our task is to find the new quantity. In this case, we have two procedure options, both of which will give us the same ultimate answers. However, the procedure we should use depends on the situation. 1) We may calculate the percentage increase or decrease and add or subtract it from the original quantity. We have, Original Amount + increase, or Original Amount − decrease 2) We may calculate the new amount straight away by using the given percent. Now let us have an illustration of both procedure options. Example 1 What is the number when $40 is increased by 25%? Option 1 (Calculating the percent increase or decrease separately) The original number is $40 and the change (increase) required is 25% of $40. 40 + (0.25 × 40) =? 40 + 10 = 50. The new amount is $50. Option 2 (Calculating the new amount) The original quantity is $40 and the change (increase) is 25% of $40. Math Skills for Business- Full Chapters 63 The quantity $40 represents 100% and must be increased by 25%. So the new quantity must be 125% (100 + 25). Thus, 40: 100 N: 125 By cross-product, 100N = 40 x 125 100N =5000 (Divide each side by 100) N = 50 (or 40 x 1.25) The new amount is 50. Example 2 What is the amount due when a debt of $350 is reduced by 20%? Option 1 The original amount is $350 and it is to be reduced by 20%. 350 − (20% of 350) =? 350 − (0.20 x 350) =? 350 − 70 = $280 Option 2 Since the original amount of $350 is to be reduced by 20%, 80% (100% − 20%) of the amount will be left. Therefore, 0.80 x 350 = $280 is the new amount. Example 3 After the price of a TV set was reduced by 15%, the sale price paid was $245. a) What was the price of the TV set before the reduction? b) What was the amount of reduction? Solution a) We can use any of the procedural option to answer the questions. Note that we are required to find the original price of the TV. Thus, 100% − 15% = 85%. Since the original amount was multiplied by 85% to get the new amount of $245, we should divide the new amount by 85% in order to get the original amount. 245 ÷ 0.85 =288.24 is the original amount. Math Skills for Business- Full Chapters 64 b) To find the amount of the reduction we may use one of these. 288.24 x 0.15 = $43.24, or 288.24 − 245 = $43.24. 6.7 Commercial Discounts and Mark up i) Commercial Discount Businesses often give discounts to their customers. Discounts are reduction in the price of a product or service, which allows the customers to save money. Discounts are often quoted in percent that allows the purchaser to compare savings in terms of money. Some times a business may give series of discounts rather than one discount. There are many reasons a business offers discounts to its customers. Some of these reasons are stated below: a) To reduce excessive inventories; It costs a lot of to carry inventories in the form of warehousing, insurance, and caretaking. By giving discounts to its customers, a business is able to reduce the amount of inventories it is holding. b) To response to the prices of a competitor; c) To encourage quick turnover or sell fast; and d) To encourage consumer loyalty. Example 1 A portable CD player costing $199.00 is on sale for $99. (a)How much is the discount? (b) What is the rate of discount? Solution a) List Price− Discount = sale price 199 − 99 =100. The discount is $100 b) Discount rate = Example 2 Ahmed wants to buy a new carpet for his house. The new carpet costs $200. One day he saw the carpet being offered for 25% off the purchase price. (a) How much money does he save by buying the carpet? b) What is the net price? Math Skills for Business- Full Chapters65 Solution a) Amount of discount =Rate of discount x List price = 0.25 x 200 =$50, Ahmed saves $50.00 b) Net price = List pri−eDiscount amount = 200 − 50 = $150 Instead of calculating the amount of discount and then deducting it from the list, the net price or net cost can be calculated by using the more efficient formula developed in the illustration below. In example 2, since the discount is given as a percent of the list price, the following is obvious: List……………………$200 − 100% of List price Less discount…………$50 − 25% of List price Net price or Net cost 150 − 75% of List price Note that the 75% is called the net cost factor or net factor. It is abbreviated as NCF and obtained by deducting 75% from 100%. That is, NCF = 100% − 75%. We should let d represent discount and L be the List price. In general, this could be written as, NCF = (1.00− d %) × L or NCF = (1 − d) L, omitting the decimal point and the multiplication sign. Using this efficient procedure, the Net price in example 2 could be calculated as follows. NCF = (1 − 0.25)200 =0.75 × 200 = $150 In the case of series of discounts, that is more than one discount, the formula becomes, NCF = (1 − d )(1 − d )(1 − d …(1 − d )L. In this formula, d is the first discount, d is the second discount, d is the third discount and so on. Example 3 More Store Corner is having a sale event. Ladies’ jeans pants regularly priced at $69 is being sold for 20 off and additional 10% on Saturdays only. Felicia bought 2 jeans on Saturday. How much does he pay for them? Math Skills for Business- Full Chapters 66 Solution NCF = (1 − 0.20) (1− 0.10) 69 = (0.80)(0.90)69 = $49.68 In the above example, we can use rewrite the formula to calculate a single equivalent rate of discount, instead of applying series of discount to the base quantity. To rewrite the formula, we have to drop the L, so that we have, SDR = {1- (1 – d1) (1-d2)(1-d3)…(1-dn). Let’s use the above formula to test the accuracy of the answer in example 3. SDR = {1- (1-0.20) (1- 0.10)} = {1 – (0.80) (0.90)} = (1 – 0.72) = 0.28 The single rate is 28%. If we apply it to $69, we get $19.32 which is the discount amount. To get the net price as follows: NP = 69 – 19.32 = $49.68. The most important thing about this formula is that you should be able to convert percents into decimals. ii) Mark up Businesses are set up for the primary purpose of making profit. So, businesses always add up their expenses, called overhead, and profit margin to the cost price of the good s or services they sell or produce in order to arrive at the selling price. The profit margin may be expressed as percent of the cost price of the goods/service or the selling price. The following relationship is therefore true, Selling price = Cost of buying + Expenses + Profit Using letters, it becomes, SP = CP + E +P, where CP is the cost of buying the product or producing it, SP is the selling price, E is the expenses, and P is the profit. Math Skills for Business- Full Chapters 67 Example 1 A retailer has some skirts that cost her $48 each. Her overhead expense for each shirt is $3. She wants to sell them at a profit of 15% of the cost. What price should she sell each skirt? Solution SP= CP + E + P SP = 48 + 3 + 15% of 48 SP = 48 + 3 + 0.15 × 48 SP = 48 + 3 + 7.20 SP = $58.20. She should sell each skirt at $58.20 in order to realize 15% profit on cost. Example 2 A pair of shoes cost a retailer $32 and he sells each for $44.80, including overhead expenses. What is his rate of profit based on the cost? Solution SP = $44.80, CP = $32, (SP − CP)×100 Rate of Profit on cost = CP (44.80 − )2 ×100 = 32 = 40% (approximately) The retailer’s rate of profit is about 40% on cost. Example 3 A head of lettuce costs a retailer $0.45. (a)At what price should it be sold to make a profit of 40% on the selling price? (b) What is the actual profit on each lettuce head? Solution a) Cost price = $0.45, SP=? P= 40% on selling price SP= CP + E + P SP = 0.45 + 0 + 0.40(SP) SP = 0.45 + 0.40SP SP − 0.40SP = 0.45 0.60SP = 0.45 (divide both sides by 0.60) Math Skills for Business- Full Chapters 68 SP = = $75.00 b) P = 40% of $75 = 0.40 x 75 = $0.30. The retail’s profit is $0.30. 6.8 Special Pricing Consideration for Produce Grocery stores that sell perishable produce such as orange, kiwi, apples, pear, honeydew, pineapple, potato and spinach face a special pricing problem relative to retailers that sell non-perishable items. This is because before such grocers set prices for their produce, they have to take into consideration spoilage, cost as well as their rate of mark up. To take spoilage consideration, the grocer has to draw on his or her experience for the length of time it takes for a particular fruit or vegetable to rot. The following example shows the pricing procedure most grocers used. Example 1 A grocer bought two boxes of cantaloupes (each box contains 200cantaloup) for$50. Overhead expenses related to the produce was $25. Based on experience, the grocer estimated that 25% of the 400 cantaloupes are expected to rot before being sold. The grocer wants 198% rate of profit (mark up) on cost. At what price must the cantaloupes be sold to achieve the desired mark-up? Solution Total profit =Mark up rate x cost = 1.98 x 50 = $99.00 Change the 198% mark up to 1.98 selling price = mark up + cost + expenses = $99.00 + $50 + $25 = $174 Expected quantities to sell = (100 – 25) x 400 Calculate the number of cantaloupes that will be sold. = 0.75 x 400 = 300 is the number that can be sold. Selling price per cantaloupe = = = $0.59 Math Skills for Business- Full Chapters 69 In the example, we used the number of items rather than the weight. The procedure we used in the above pricing process will not change, even if we were dealing with weights (in pound or kilogram). The following example illustrates this. Example 2 A grocer bought 150 pounds of banana at $0.20/pound. The store wants a profit margin of 200% on cost. It estimates that 10% of the banana will rot before they are sold. How much should it sell one pound of the banana? Solution Total profit = (150) (0.20) x 2.00 = $60.00 Total selling price = cost + mark up = (150 x 0.20) + 60 = 30 + 60 = $90.00 Quantities expected to sell = (100% -10%) x 150 = 90% x 150 (then change 90% to decimal) = 0.90 x 150 = 135 pounds will sell Selling price per pound = = $ 0.67 per pound You might have noticed that the quantities that rot affect the unit-selling price. However, in practice, most grocers are compelled to cut down on their profit margin on perishable produce when they see that the produce are showing signs of rotting, or when they want to sell them quickly to prevent them from going bad. Experience is a valuable guide in predicting the perishability of a produce. Math Skills for Business- Full Chapters70 6.9 What are Basis Points? A basis point (bps, for short) is a unit of measure used in finance to describe the smallest change in the value or rate of financial instrument such as stock, bonds and interest rate. One basis point is equal to 0.01% (or of a percent), or 0.001 in decimal form. If the Bank of Canada raises the interest rate by 25 basis points, it means the interest rate has been increased by 0.25% percent If the existing interest rate is 4.20%, then the new rate of interest is 4.45% (4.20% + 0.25%). If it is expected that a stock index has moved up 124 basis points in the day’s trading, this means 1.25% increase in the value of the index. The table below summaries the difference between basis points, percent and decimal. Basis Points Percent Decimal 1 0.01% 0.0001 10 0.10% 0.0010 20 0.20% 0.0020 40 0.40% 0.0040 50 0.50% 0.0050 100 1.00% 0.010 1000 10.00% 0.100 10,000 100.00% 1.000 2 0.02% 0.0002 From the above table, it may be obvious that to convert basis points to a percent divide by 100. Similarly, to convert percent to basis point simply multiply by 100. As well, to convert a percent to decimal form divide by 100. 6.10 Why Percent? It is easier to compare percents in terms of their relative sizes than it is to compare fractions. Comparing percents is about comparing the numbers the percent sign is attached to, since the sign can be viewed as a common unit of measurement. For example, it is easier to compare 75% to 72%, than it is comparto .This is why percent rather than fractions are used more often for comparison. Math Skills for Business- Full Chapters 71 Percents are encountered often in our daily lives in a variety of situations: credit card interest rates, interest rate on OSAP, shopping (calculation of store sales discount), politics (to calculate support for various parties, issues and causes), business (to project increase or loss in volume of sales, comparing sales between two consecutive months), and in general (population growth, comparison among many items, rates of change in between periods). In most cases, percent is used to apportion cost or profit. Indeed, its use for sharing common costs or profit is more convenient compared to ratios or fractions. 6.11 Review, Exercises and Assignments 1. Express each of the following as percent. a) 10 out of 50 b) 5 out of 25 c) 3 out of 10 d) 260 out of 400 e) 21 out of 70 f) 18 out of 60 g) 200 of 1000 h) 12 out of 48 2. The value of a one-bedroom condominium is $80000 and its contents are worth $3000. Express the value of the contents as a percentage of the value of the house. 3. A butter cake weighs 950 grams, of which 20% is sugar, 60% wheat flour, and 20% other ingredients. a) Calculate the weight of the sugar in the cake. b) What is the percentage difference between the amount of wheat flour and sugar in the cake? 4. In 2004, the total wages bill for a plastic manufacturing was $250000 and its total net sales were $2,500,000. a) Express the company’s wages bill as a percentage of the net sales. b) What interpretation will you give to the result in (a)? c) If the company’s wages bill expressed as a percent of its net sales in 2005 was 12%, state three factors that might have caused the increase. 5.Don bought a car with sale tax of 14% tax included for $31,920. In addition, Don bought an extended warranty at 2% of the original purchase price. a)What was the price of the car without the sale tax? b) How much was the extended warranty? Math Skills for Business- Full Chapter72 6. The value of a car decreases as shown in the table below: Vehicle Value New $12,000 After 1 year $10,000 After 2 years $8,800 After 3 years $8000 a) In which year did the value of the car decreased by $2000? b) Calculate the percentage decrease between year 2 and year 3. c) During which year was the percentage decrease in the value of the car the greatest? 7. A gas bill of $46.06 includes GST of 6%. Find the amount of the GST paid. 8. The year-end profit of a company increased this year by 12% to $90944. What was the profit made last year? 9.In a massive sale event, the following items were offered at discount prices as listed Item Sale Price Discount Television $299.00 10% VCR $273 12% Computer $2,200 25% Calculator $24.99 5% Colour Printer $225 10% What were the prices of these items before the sale? 10. After one year, the value of a car has fallen by 15% to $8330. What was the value of the car at the beginning of the year? 11.A real estate agent receives as commissio
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