# Successive Percentage Change ```Successive Percentage
Change
Jeans are on sale for 40% off the retail price.
The retail price is \$40.00. If you have a
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 &plusmn; P1) * (1 &plusmn; 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
```