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Successive Percentage Change Jeans are on sale for 40% off the retail price. The retail price is $40.00. If you have a coupon, you can receive an additional 20% off the sale price. What is the overall percentage savings? Determine the sale price: Determine the final price with coupon: Determine the overall percentage change: $40 – ($40 * .40) = $24 $24 – ($24 * .20) = $19.20 new – old = $19.20 - $40 = - 0.52 old $40 -0.52 * 100% = Savings of 52% VISUALLY Determine the sale price: Determine the final price with coupon: Determine the overall percentage change: $40 – ($40 * .40) = $24 $24 – ($24 * .20) = $19.20 new – old = $19.20 - $40 = - 0.52 old $40 -0.52 * 100% = Savings of 52% 45.00 40.00 $40.00 x .40 = $16.00 35.00 $40.00 x.52 = $20.80 30.00 25.00 $24.00 x.20 = $4.80 20.00 $40.00 - $16.00 = $24.00 15.00 $24.00 - $4.80 = $19.20 10.00 $40.00 - $20.80 = $19.20 5.00 0.00 Retail Price Sale Price Price Final Price with Coupon Amount Saved Overall Percentage Change The Formula (1 ± P1) * (1 ± P2) – 1 = % where % is written in fractional form (decimal) P1 = First percentage increase/decrease P2 = Second percentage increase/decrease Deriving the Formula • The overall percentage change does NOT depend on the Beginning Value (B). Can we show this by determining a process (formula) that includes just the two percents? Deriving the Formula (cont.) • Variables B = Beginning Value I = Intermediate Value F = Final Value P1 = First Percent Decrease P2 = Second Percent Decrease Deriving the Formula (cont.) First find the intermediate price B – (B * P1) = I Then find the final price I – (I * P2) = F Then find the overall percentage change F - B B Deriving the Formula We can rewrite the first equation: B – (B * P1) = I As B * ( 1 – P1) = I We can rewrite the second equation: I – (I * P2) = F As I * (1 – P2) = F Deriving the Formula (cont.) Substitute these into final equation: F–B B So , since we found that F = I*(1-P2) I*(1-P2) – B B And we also know that I = B*(1-P1) B*(1-P1) * (1-P2) – B B The B’s cancel out to arrive at: (1-P1) * (1-P2) - 1