When categories change at different rates, track their shares: each category's original share contributes to the new total. Set the original total to and express each starting share using one unknown. Multiply each starting share by its own change factor, then add the pieces. Solve the total equation with a one-variable Desmos regression, and report the share of the original whole, not a smaller subgroup.
Hints
- Hint 1
A share is a part of the original total. Treat week 1 as and call the adult share . Children use , so what share is left for students?
- Hint 2
A change factor tells you how much of a category's original count becomes its new count: up gives , and down gives . What factor applies to the unchanged child count?
- Hint 3
The week 2 total is times the week 1 total, not . Add the three changed category shares and set that sum equal to . Then solve for the adult share.
Step-by-step
Build a share ledger and solve in Desmos
Step 1Find the student share of week 1
Count all week 1 tickets as , and let be the adult share, or fraction of that total. Children account for , so students get what remains. Subtract the adult and child shares:
- Step 2
Add the three contributions to week 2
An adult increase of makes the adult count . A student decrease of leaves . Children stay unchanged, so they still contribute ; don't give them the student factor. Since the whole total increased by , these contributions add to of the week 1 total:
- Step 3
Solve for the adult share
Type in Desmos. Use for and instead of : this asks Desmos for a regression, which finds the value that makes the sides match. Under PARAMETERS, Desmos shows , the adult share of week 1.
- Step 4
Convert the share to the requested percent
Type . Desmos prints ; its fraction button gives . Multiplying by converts the adult share of all week 1 tickets to a percent. Don't divide by : that would use only the non-child tickets as the whole. Adult tickets made up of all tickets sold in week 1. Choice A.