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