Equalities. Equalities

Transcription

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2 Equalities Working with Equalities There are no special rules to remember when working with equalities, except for two things: When you add, subtract, multiply, or divide, you must perform the same operation on both sides of the equation. Work slowly and methodically so that you don t make mistakes. Skipping steps or doing calculations in your head is tempting and seems quicker, but often leads to errors. 3 Equalities Working with Equalities Examples of solving equalities with a variable, x: x 1 = 4: Add 1 to both sides: (x 1) + 1 = x + 0 = 5 x = 5 ½x = 4: Multiply both sides by 2: (½x)2 = 4(2) x = 8 10x = 200: Divide both sides by 10: 10x/10 = 200/10 x = 20 x 2 = 16: Take the square root of both sides: x 2 = 16 x = 4 4 Page 2

3 Equalities Working with Equalities An Aside About My Approach: There are often different ways to solve equations or approach a mathematical problem. Throughout these CD-ROM lectures, from the most basic review to more complicated mathematical operations, I will be showing you one way of solving the problem. There may be other equally valid ways that I do not mention. I prefer to show you one way that works all the time, so that you can develop a consistent approach. With consistency will come confidence, as well as speed in working with the problems. 5 Equalities Solving for x The previous examples conveniently had the x variable on the left-hand side of the equation. If x is on the right, the simplest approach is to move things around so that x is on the left-hand side of the equation and everything else is on the right-hand side: 6 = 2x Switch the left and right-hand sides: 2x = 6 Divide both sides by 2: x = 3 6 = -2x Switch the left and right-hand sides: -2x = 6 Divide both sides by 2: x = -3 6 Page 3

4 Equalities Solving for x 36 = 4x 4 Switch the left and right-hand sides: 4x 4 = 36 Add 4 to both sides: 4x = 40 Divide both sides by 10: x = 10 4/x = 32 Take the reciprocal of the whole equation (turn it upside down): x/4 = 1/32 Multiply both sides by 4: x = (1/32)4 Solve: x = 4/32 = Inequalities Rules Inequalities behave exactly the same way as equalities when you are adding, subtracting, multiplying or dividing. The only important difference is that if you multiply or divide by a negative number, the direction of the inequality is reversed. 8 Page 4

5 Inequalities Working With Inequalities Let s start simply, working with numbers and no x s: 3 > 2: Add 1 to both sides: 3+1 > 2+1 or 4 > 3 3 > 2: Subtract 1 from both sides: 3-1 > 2-1 or 2 > 1 3 > 2: Multiply both sides by 2: 3(2) > 2(2) or 6 > 4 3 > 2: Multiply both sides by 2 and reverse the inequality: 3(-2) < 2(-2) or -6 < -4 9 Aside: Working with Negative Numbers The last example brings up an issue for review. The number line drawn below is a reminder that as we move to the right, numbers are larger. This means that, for example, 1 is a larger number than 3, since it is further to the right. Or, as in the last example 6 is less than 4, since it is further to the left. This will come up in a business context later on, when we measure the effect of an increase in price on quantity sold Page 5

6 Inequalities Solving for x Use the rules we have developed to solve for x by isolating x on the left-hand side of the inequality and everything else on the right-hand side. Make sure to preserve the direction of the inequality when you are rewriting the equation. 3 2x Rewrite so x is on left: 2x 3 Divide both sides by 2: 2x/2 3/2 Solve: x 3/2 11 Inequalities Solving for x 3 > -2x + 1 Rewrite so x is on left: -2x + 1 < 3 Subtract 1 from both sides: -2x < 2 Multiply both sides by 1: 2x > -2 Divide both sides by 2: 2x/2 > -2/2 Solve: x > Page 6

7 Equalities & Inequalities Exercises Solve the following equations for x x = = 0.5x x < 20 5x 13 Equalities & Inequalities Exercises: **Answers** Solve the following equations for x x = 30 => x = 30/1.5 => x = = 0.5x 200 => 0.5x 200 = 500 => 0.5x = 700 => x = 700(2) = x 216 => x 216/6 => x < 20 5x => 20 5x > 100-5x > => -5x > 80 => 5x < -80 => x < -80/5 => x < Page 7

8 Absolute Value Notation As you can see from some our earlier work, handling negative numbers can be awkward at times. Therefore, sometimes it is easier to speak of things in absolute value terms. The absolute value of a number is denoted by two vertical lines. When you see this notation, convert the negative number into a positive number. For example, the absolute value of 2 is 2: 2 = 2 15 Definition a b The numerator, a, is above the line. The denominator, b, is below the line. Market share for a firm is the ratio of its own sales divided by industry sales. If the firm s sales are 10 million and industry sales are 100 million, what is the firm s market share? Market share is 10/100 = 1/10 = Page 8

9 Reducing We will often want to simplify or reduce fractions. It is easiest to explain by example: Reduce 15/9 15/9 = (5 * 3)/(3 * 3) The 3 s cancel So, 15/9 = 5/3 Reduce 20/60 20/60 = (2 * 10)/(6 * 10) The 10 s cancel So, 20/60 = 2/6.but you can reduce further 2/6 = (2 * 1)/(2 * 3) the 2 s cancel so, 2/6 = 1/3 17 Rules: Addition & Subtraction You can add or subtract numerators only if the denominators are the same: a b a b + = + c c c The largest firm in the industry has sales of 10 million, the second largest firm has 5 million in sales, and total sales are 100 million. What is the total market share of the two largest firms? 10/ /100 = (10 + 5)/100 = 15/100 = Page 9

10 Rules: Addition & Subtraction If the denominators are not the same, then you have to multiply the top and bottom of the fraction by something that will make them the same: 3/10 + 1/2 = 3/10 + (1*5)/(2*5) = 3/10 + 5/10 or, (3 + 5)/10 = 8/10 Reduce your answer: 8/10 = (2*4)/(2*5) = 4/5 19 Rules: Multiplication To multiply two fractions, simply multiply both the numerators and denominators: a b c d = a b c d Example: = = Page 10

11 Rules: Division Dividing two fractions requires a bit more care. Division involves dealing with double-decker fractions, or fractions in which the numerator and denominator are also fractions. Rule: keep the numerator fraction the same, and then multiply it by reciprocal (the upsidedown version) of the denominator fraction: b a d c = c a d b = / / b a d a d = c b c 21 Rules: Division Rule: (a/b) (c/d) = (a/b) (d/c) = a d/b c Divide and simplify: (5/10) (2/3) (5/10) (3/2) = (5 3) (10 2) = 15/20 = 3/4 Divide and simplify: (2/7) (8) (2/7) (8/1) = (2/7) (1/8) = (2 1) (7 8) = 2/56 = 1/28 22 Page 11

12 Working with Increasing (decreasing) the numerator increases (decreases) the fraction: Your firm s sales increase from 10 to 20, but industry sales stay constant at 100. Your market share used to be 10/100 = 0.1 and now it is 20/100 = 0.2; an increase. Increasing (decreasing) the denominator decreases (increases) the fraction: Your firm s sales stay constant at 10, but industry sales rise to 200. Your market share used to be 10/100 = 0.1 and now it is 10/200 = 0.05; a decrease. 23 Working with Warning: The effect of increasing (decreasing) both the numerator and denominator is ambiguous. Your firm s sales increase from 10 to 20, and industry sales increase from 100 to 160. Your market share used to be 10/100 = 0.1 and now it is 20/160 = 0.125; higher than before. Your firm s sales increase from 10 to 20, and industry sales increase from 100 to 200. Your market share used to be 10/100 = 0.1 and now it is 20/200 = 0.1; exactly the same. 24 Page 12

13 Exercises 1. Add: 8/15 + 2/3 2. Add: 2 + 1/36 3. Divide: (21/5) (105/5) 4. Divide: (7/2) (0.4375) 25 Exercises: **Answers** 1. 8/15 + 2/3 = 8/15 + (2*5)/(3*5) = 8/ /15 = (8 + 10)/15 = 18/15 = (3*6)/(3*5) = 6/5 = /36 = 2/1 + 1/36 = (2*36)/(1*36) + 1/36 = 72/36 + 1/36 = (72 + 1)/36 = 73/36 (which can t be reduced any further) = 2.03, rounding 3. (21/5) (105/5) = (21/5)*(5/105) = (21*5)/(105*5) = 21/105 = 1/5 = (7/2) (0.4375) = (7/2) (0.4375/1) = (7/2)*(1/0.4375) = (7*1)/(2*0.4375) = 7/0.875 = 8 26 Page 13

14 Percents Notation Let s start with basic notation: percent (%) means per one hundred. To express a percent in decimal terms, drop the percent sign and move the decimal points two places to the left. Convert 50% to a decimal Drop the % and move the decimal over two places 50% = 0.50 Convert 2% to a decimal 2% = 0.02 Convert 100.5% to a decimal 100.5% = To express a percent as a fraction, drop the percent sign, divide the number by 100, and then reduce (simplify) the fraction. Convert 50% to a fraction 50% = 50/100 = (5*10)/(10*10) = 5/10 = 1/2 Convert 2% to a fraction 2% = 2/100 = 2/(2*50) = 1/50 Convert 100.5% to a fraction 100.5% = 100.5/100 Percents Notation 28 Page 14

15 Percents Notation You can reverse this process as well, converting a decimal into a percent or converting a fraction into a percent: Convert 0.25 to a percent Move the decimal point two places to the right and add the % sign 0.25 = 25% Convert the fraction 14/40 into a percent Convert the fraction to a decimal and multiply by 100 (14/40)100 = (0.35)100 = 35% 29 Percentage Calculations Identifying the Base In order to answer the question what is 5 percent of 40? you have to first identify the base off of which the percentage is calculated. This is easy when there is only one number to consider. In this case the base is 40. What is 5% of 40? Multiply the percent (5%) by the base (40) 5% of 40 = 0.05(40) = 2 If state sales tax is 7%, what is the dollar amount of the tax on a purchase of $1,250? Multiply the percent (7%) by the base (1,250) 7% of 1,250 = 0.07(1,250) = $ Page 15

16 Percentage Calculations Identifying the Base Once again, you can reverse this process. If you know the base amount and the percentage, you can calculate the percent: What percent of 250 is 400? The base is /base = 400/250 = 1.6 = 160% Another way to think of this is that percent equals some part divided by the whole (or base). What percent of 400 is 250? The base is now 400 part/whole = 250/base = 250/400 = = 62.5% 31 Percentage Change Percentage change = (amount of change)/base For example, if price changes from $2 to $5, what is the percentage increase in price? If, on the other hand, price changes from $5 to $2, what is the percentage decrease in price? We still need to identify the base, but here we have to choose and the choice is important. If the price started at $2, then $2 is our base: Percentage change = (5-2)/2 = 3/2 = 1.5 = 150% If the price started at $5, then $5 is our base: Percentage change = (2-5)/5 = -3/5 = -0.6 = -60% 32 Page 16

17 Percents Exercises 1. 1/40 is what percent? 2. It is often handy to be able to estimate percentages. Estimate 28 as a percent of 120. (Hint: round 28 up to 30) 3. The manufacturer of a snack food has recently reduced the fat content from 10 grams per serving to 8 grams. What claim can they make on the new packages regarding the percentage decrease in fat content? 4. The growth in Japan s Gross Domestic Product (GDP) was approximately 0.8 percent in In 2000, Japan s GDP growth was 1.5 percent. Since = 0.7, does this mean that there was an 0.7 percent increase in GDP between ? 33 Percent Exercises: **Answers** 1. 1/40 =.025 = 2.5% 2. 28/120 is roughly 30/120 = 3/12 = 1/4 = 0.25 or 25% 3. (8 10)/10 = -2/10 = Therefore, the manufacturer can advertise a 20% decrease in fat grams per serving. 4. No, this is a common error. It would be correct to say that there has been an increase of 0.7 percentage points, but that is not the same as 0.7 percent. To calculate the percentage change, use the standard formula: Percentage change = (amount of change)/base = ( )/0.8 = 0.7/0.8 = or 87.5% There was an 87.5% increase in Japan s GDP growth! 34 Page 17

18 Application: Price Elasticity of Demand The Basics In Use How do you measure the price sensitivity of consumers? Every topic covered in this module is used for this critical concept in the Economics of Business course that you will be taking in the first semester of your EMBA program. We will go over this again during the course, but this is a great opportunity to show you how these basic quantitative skills (absolute value, working with fractions and percentage change) come up in practice. Economists call price sensitivity the price elasticity of demand. Demand for a product can be either price insensitive (such as the demand for insulin) or price sensitive (the demand for one brand of paper cup). We call price insensitive products price inelastic, and the price sensitive products price elastic. 35 Application: Price Elasticity of Demand Definition Price elasticity is expressed in percentage terms (so that we don t have to worry about the units) and it measures how consumer demand responds to a change in price. For example, if price increases by 5% and demand declines by 2%, then we say that the price elasticity is 2/5 = Studies of price sensitivity have been done for many products. For example, the price elasticity of demand for steak is about This means that a 1% increase in the price of steak results in about a 1.5 percent decrease in the demand for steak. Contrast this with the price elasticity of gasoline, which ranges from about 0.04 to 0.21 in the United States. The demand for gasoline is extremely price insensitive: a 1% increase in the price of gasoline results in only a.04 to.21 percent decrease in gasoline consumption. 36 Page 18

19 Application: Price Elasticity of Demand Absolute Value Because an increase in price will almost always result in a decrease in demand, and vice versa, the price elasticity of demand is almost always a negative number. Therefore, we will speak of it in absolute value terms. For example, an elasticity of 0.5 is referred to as 0.5 and an elasticity of 3 is referred to as 3. Recall that 3 is less than 0.5 because it is further to the left on the number line. But a demand response of 3% is a larger change in customer behavior than a response of 0.5%. So we use absolute values and forget about the negative signs. An elasticity of 3 is clearly larger than an elasticity of Application: Price Elasticity of Demand Percentage Change Now let s define price elasticity of demand in general notation, using the variable P to represent price and Q to represent quantity sold. This will illustrate how we work with percents and fractions. Written in percentage terms: Price elasticity = % change in Q % change in P Let (delta) represent the word change Price elasticity = % Q % P 38 Page 19

20 Application: Price Elasticity of Demand Percentage Change Price elasticity = % Q % P Now use the fact that a percentage change is the amount of change divided by the base. When price changes from P 1 to P 2, quantity demanded changes from Q 1 to Q 2. This means that P 1 is the base price (starting price) and Q 1 is the base quantity. Price elasticity = % Q % P = [(Q 2 -Q 1 )/ Q 1 ] / [(P 2 -P 1 )/ P 1 ] This is exactly the same as the percentage calculations we did a few slides before, but with variables rather than numbers. 39 Application: Price Elasticity of Demand Rewrite [(Q 2 -Q 1 )/ Q 1 ] / [(P 2 -P 1 )/ P 1 ] with the delta notation: Price elasticity = [ Q/ Q 1 ] / [( P)/ P 1 ] Now we have a double-decker fraction: Price elasticity = [ Q/ Q 1 ] / [( P)/ P 1 ] = [ Q/ Q 1 ] * [P 1 / P ] = [ Q * P 1 ] / [Q 1 * P ] This conforms with what we did before. But we will find it convenient to rearrange terms once more: = [ Q/ P ] * [ P 1 / Q 1 ] 40 Page 20

21 Application: Price Elasticity of Demand Price elasticity = [ Q/ P ] * [ P 1 / Q 1 ] Based on recent experience, you know that when you change price by $0.10, quantity demanded falls by 5 units. Let s say your current price is $1 and your current quantity is 100. Then the price elasticity is, in absolute value terms: (5/0.10)*(1/100) = Application: Price Elasticity of Demand Wrap-up We will use this formula in our discussions of consumer behavior. The reason it is so convenient is that, as we will cover in a later CD- ROM module, the term Q/ P is related to the slope of the demand curve for the product. We will come back to this topic during our Economics of Business course. 42 Page 21