Successive Percentage Change

Jeans are on sale for 40% off the retail price. The retail price is $ 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:$40 – ($40 *.40) = $24 Determine the final price with coupon: $24 – ($24 *.20) = $19.20 Determine the overall percentage change:new – old = $ $40 = old $ * 100% = Savings of 52%

VISUALLY Determine the sale price:$40 – ($40 *.40) = $24 Determine the final price with coupon: $24 – ($24 *.20) = $19.20 Determine the overall percentage change:new – old = $ $40 = old $ * 100% = Savings of 52%

The Formula (1 ± P 1 ) * (1 ± P 2 ) – 1 = % where % is written in fractional form (decimal) P 1 = First percentage increase/decrease P 2 = 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 P 1 = First Percent Decrease P 2 = Second Percent Decrease

Deriving the Formula (cont.) First find the intermediate price B – (B * P 1 ) = I Then find the final price I – (I * P 2 ) = F Then find the overall percentage change F - B B

Deriving the Formula We can rewrite the first equation: B – (B * P 1 ) = I As B * ( 1 – P 1 ) = I We can rewrite the second equation: I – (I * P 2 ) = F As I * (1 – P 2 ) = F

Deriving the Formula (cont.) Substitute these into final equation: F – B B So, since we found that F = I*(1-P 2 ) I*(1-P 2 ) – B B And we also know that I = B*(1-P 1 ) B*(1-P 1 ) * (1-P 2 ) – B B The B’s cancel out to arrive at: (1-P 1 ) * (1-P 2 ) - 1