AMC 10 · 2009 · #8
Grade 6 arithmeticPick an answer.
AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Every price is tied to one hidden number: the regular ticket price. We are handed a discounted price (the senior's $6.00) instead of the regular price, so we work backwards through the 25% discount to recover the regular price first. Once the regular price is known, each generation's ticket is a separate subproblem, and adding them gives the total.
Recover the regular price
A 25% discount means the senior pays the other 75%, so undo it: $6.00 ÷ 0.75 puts the regular price at $8.00.
If $6 is only three-quarters of the price, the whole price must be a bit more, and dividing by 0.75 walks back to it.
If the paid amount is only three quarters of the price, dividing by that share walks back to the whole price.
▸ Why?
A share is a count out of a hundred of a stated base, so the base is what has to be recovered.
▸ Why?
Dividing undoes the multiplying that shrank the price, so it lands exactly on the original.
Price a child ticket
A 50% discount is simply half, so each child ticket is half of $8.00, or $4.00.
Fifty percent off just means pay half, and half of 8 is 4.
6.RP.A.3Analyze The UnitsPrice a middle-generation ticket
The middle generation gets no discount, so each of those two tickets stays at the full $8.00.
No discount means the ticket stays at the regular price.
6.RP.A.3Analyze The UnitsAdd up all six tickets
Two of each kind: 2(4) + 2(8) + 2(6) = 8 + 16 + 12, so Grandfather Wen pays $36 — choice (B).
Group the tickets by generation, double each price, then add the three totals.
4.OA.A.3Identify SubproblemsWhen you are given the discounted price, undo the discount first to find the full price, then everything else follows.
- Recover the regular price
- Price a child ticket
- Price a middle-generation ticket
- Add up all six tickets