Successive percent changes use multipliers: each change acts on the amount left by the one before it. A difference in percentage points means subtract the two percent rates, not take a percent of a percent. Multiply the increase and decrease factors to match the final fee, then use a Desmos regression to find the unknown rate. Subtracting the rates misses the second change’s new base.
Hints
- Hint 1
A percentage point is a difference between percent rates: and differ by percentage points. Subtract from the rate, not of it. How would you write the decrease rate?
- Hint 2
A multiplier changes the current amount: a increase uses , while a decrease uses . The second multiplier acts on the new fee. How can you combine the two multipliers?
- Hint 3
A fee that ends greater than the original is of the original, not of it. What should the product of the two change factors equal?
Step-by-step
Multiply the percent-change factors
Step 1Find the decrease rate
The decrease rate is 20 percentage points less than the increase rate . Percentage points compare rates by subtraction, so the decrease is . It isn’t of ; that would mean “ less,” not “20 percentage points less.”
- Step 2
Write the increase factor
An increase keeps the whole fee and adds of it. So the increase multiplier, the number applied to the original fee, is the whole plus the increase:
- Step 3
Write the decrease factor
The decrease removes of the new fee. So its decrease multiplier is
Each percent acts on the fee immediately before that change.
- Step 4
Match the final fee to the original
Let be the original fee. Ending greater means ending at . Multiply the factors to represent the two changes in order:
Divide by the positive fee :
- Step 5
Solve for the percent increase
Type the equation below in Desmos, replacing with and with . The subscript tells Desmos to solve for , and the restriction keeps the later change a decrease. Desmos reports under PARAMETERS. So the service first increases its fee by . Choice C.