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The Mathematics 11 Competency Test Percent Increase or Decrease The language of percent is frequently used to indicate the relative degree to which some quantity changes. So, we often speak of percent increases in pay, or, for that matter, percent decreases in pay. Similarly, we speak of the percent increase or percent decrease in the number of cases of some medical condition, or the percent increase or percent decrease in the amount of petroleum imported by a certain country, and so on. The basic elements in such language, and the information it represents have already been described. In this context, we use the terms: base to refer to the quantity which will change, either increase or decrease. The “base” quantity often has units. We will sometimes use the term original value to mean the same thing as the “base.” Also, in practice, the “base” value, being a physical quantity of stuff, is almost always a positive number. Amount to refer to the actual amount of change, the amount by which the “base” quantity either increases or decreases. The “Amount” always has the same units of measurement as the “base” quantity. rate is the actual percent value, in this situation representing a percent increase or decrease. In the arithmetic calculations, the decimal equivalent of the percent increase or percent decrease is always used. We’ve previously stated that the fundamental relationship between these three quantities is Amount = base x rate (i) and algebraic rearrangements of this formula. When discussing percent increase or percent decrease, it is convenient to introduce one new term, new value, the value of the quantity in question after the increase or decrease has occurred. Then new value = base + amount (iia) = base + base x rate (iib) = base(1 + rate) (iic) These four formulas (and their algebraic rearrangements) cover all possible situations of percent increase and percent decrease if we note one final convention: ♦ if the “rate” is positive, then • the “Amount” is positive • the “new value” is greater than the “base” or “original value” (That is, a positive “rate” is associated with percent increase.) ♦ if the “rate” is negative then David W. Sabo (2003) Percent Increase or Decrease Page 1 of 6 • • the “Amount” is negative (base x rate in formula (i) is a positive times a negative and so is negative) so, “new value” is less than the “base” or “original value” (That is, a negative “rate” is associated with percent decrease.) The main source of error in solving percent increase and percent decrease problems is failing to correctly identify which numbers in the problem match each of these four fundamental terms. In particular, be very careful to distinguish between the base or original value to which the rate refers, and the new value of that quantity. Example 1: A piece of electronic equipment is priced at $329.50. If the retailer offers a 30% discount, (i) What would be the new price of the item, and, (ii) what would be the total cost of the item when the 14.5% in sales taxes are added on? solution: You know how to solve this problem already, so we will focus more on applying the formal terminology. (i) the 30% discount on the original prices can be considered to be a 30% decrease (rate = -30% or -0.30) in the base price or original price of $329.50. The discounted price is then what we have called the “new value” above: discounted price = new value = (original price)(1 + rate) = ($329.50)(1 + [-0.30]) = ($329.50)(0.70) = $230.65 Thus, with the 30% discount, the new price of the item will be $230.65. (ii) The 14.5% tax represents a percent increase (rate = +14.5% or 0.145) – your cost (the amount of money you must actually pay) is an increase of 14.5% on whatever is the stated price of the item you are buying. So, your final cost here is the “new value” that results when then 14.5% increase is applied to the stated price of the item: total cost = new value = (stated price)(1 + rate) = ($230.65)(1 + 0.145) = ($230.65)(1.145) = $264.09 rounded to the nearest cent. Thus, the actual amount of money you would pay for the item (including sales tax) would be $264.09. David W. Sabo (2003) Percent Increase or Decrease Page 2 of 6 NOTE: The importance of correctly associating rate with the base values to which they refer can be seen even in this example. The work we did above to come up with the final cost of $264.09 is correct – this is the way the calculation would be done in the store. The sales representative would first apply the 30% discount to the original price, to obtain the discounted price of $230.65. Then, the 14.5% sales tax would be applied to this discounted price, resulting in the final cost to you of $264.09. What you absolutely cannot do is to combine the two rates. You might think that a 30% decrease followed by a 14.5% increase amounts to an effective rate of change of -30% + 14.5% = -15.5% and so, we could just apply this net rate to the original price of the item. However, if we were to do that, we would get “supposed final cost” = new value = (original value)(1 + supposed rate) = ($329.50)(1 + [-0.155]) = ($329.50)(0.845) = $278.43 which is incorrect. With this method, you’d end up paying over $13 more than you should. The source of the error here is that the two rates in the problem do not refer to the same base values! The 30% discount is 30% of the original price of $329.50, whereas the 14.5% sales tax is 14.5% of the discounted price of $230.65. This is the reason why you can’t simply add percents together in the same way that you can add quantities of stuff with the same units together to get a total. The extra $13 or so you would pay if the final cost of the item was calculated in this incorrect way amounts to paying tax on the full $329.50 instead of the discounted price of $230.65. So, to avoid error in percent problems, always make sure you have correctly identified the base values to which each percent refers, and also that you correctly distinguish between new values and base values. Example 2: Hank buys a piece of equipment, paying a total of $982.35, which includes 14.5% sales tax. What was the actual price of the item before tax? solution: Obviously, the rate of change here is the 14.5% rate of paying sales tax. This rate is applied to the original price of the item, which is what we are asked to determine. This results in a new value, the $982.35 actually paid. So, starting with new value = base x (1 + rate) we can write $982.35 = base x (1 + 0.145) or $982.35 = 1.145 x base David W. Sabo (2003) Percent Increase or Decrease Page 3 of 6 Thus, base = actual price here = $982.35 = $857.95 1.145 rounded to the nearest cent. Thus, we conclude that the original stated price of the item was $857.95. NOTE: we can double-check our answer to the last example by computing the total cost of an item after 14.5% sales tax is added to its labelled price of $857.95. We should end up with the value $982.35, as stated in the problem. So: assumed original price: plus 14.5% sales tax (=0.145 x $857.95) $857.95 total cost $982.35 $124.40 In this type of problem, students very often make the error of subtracting 14.5% of the final cost of $982.35 from that final cost, thinking this will give the original price: $982.35 – ($982.35)(0.145) ⇒ $982.35 - $142.44 ⇒ $839.91 But you know this is incorrect, because when you buy an item in a store, the sales tax is always computed with the original price, whereas this incorrect calculation applies the sales tax to the final cost which already includes the sales tax. Example 3: Hank buys a house for $212,500, and later sells it for $243,700. By what percent did the price of the house change from when he bought it to when he sold it? solution: Start by sorting out the numbers. We have original price = “original value” = “base” = $212,500 and selling price = “new value” = $243,700 We are asked to find a rate, the percent by which the house price increased between these two transactions. There are at least two ways to do this: (i) the “amount” or increase in the house price = new value – original value = $243,700 - $212,500 = $31,200 Then, using Amount = base x rate we have David W. Sabo (2003) Percent Increase or Decrease Page 4 of 6 $31,200 = $212,500 x rate so that rate = $31200 ≅ 0.1468 $212500 rounded to four decimal places. Thus, the house price increased by 14.68%. (ii) We could also begin with new value = base x (1 + rate) Rearranging 1 + rate = new value base so that rate = new value -1 base Putting the numbers in gives rate = $243700 −1 $212500 = 1.1468 − 1 = 0.1468 again rounding to four decimal places where necessary. This is, of course, exactly the same answer of a 14.68% increase obtained with the first method. Example 4: The Statistics Canada estimate of the population of British Columbia in 2001 was 3,907,738. The population of Newfoundland and Labrador in 2001 was 512,930. If the population of British Columbia grows by 4.9% over the next five years, and the population of Newfoundland and Labrador decreases by 7.0% over those same five years, what will be the populations of the two provinces in 2006? solution: This problem asks us to calculate two “new values.” For British Columbia, the population is stated to have been 3,907,738 in 2001 (the base value). If this population grows or increases by 4.9% over the subsequent five years, then in 2006, we will have a population of new value = base value x (1 + rate) = (3,907,738)(1 + 0.049) = (3,907,738)(1.049) David W. Sabo (2003) Percent Increase or Decrease Page 5 of 6 = 4,099,217 rounded to the nearest whole number. For Newfoundland and Labrador, the base value is 512,930 and we use a decrease of 7.0%, so that the expected population in 2006 will be new value = base value x (1 + rate) = (512,930)(1 + [-0.070]) = (512,930)(0.93) = 477,025 again rounded to the nearest whole number. David W. Sabo (2003) Percent Increase or Decrease Page 6 of 6