AMC 10 · 2007 · #1
Grade 6 arithmeticPick an answer.
AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question compares two people, so Tool #7 (Identify Subproblems) splits it into three clean pieces: what Susan pays, what Pam pays, and the difference between them. Inside each piece the work is the same shape — full-price total, then apply that buyer's percent discount. Tool #8 (Analyze the Units) keeps every number in dollars so the discount is taken on a dollar total and the final subtraction compares dollars to dollars.
Full-price cost for each buyer
At $20 a ticket, Susan's 4 tickets come to $80 and Pam's 5 come to $100 before any discount.
Same price on every ticket, so total cost is just price times number of tickets.
4.OA.A.3Identify SubproblemsApply Susan's 25% discount
A 25% discount takes off a quarter, so Susan pays $80 minus $20, or $60.
Percent means per hundred, so 25% off a total is a quarter of that total taken away.
Taking twenty-five percent off a total is taking a quarter of that total away.
▸ Why?
A percent is a count out of a hundred, and twenty-five out of a hundred is one of four equal shares.
▸ Why?
That share is the whole cut into four equal pieces with one taken, so it can be subtracted directly.
Apply Pam's 30% discount
A 30% discount takes $30 off Pam's $100, so Pam pays $70.
A percent of 100 is that many dollars, because a percent is already a count per hundred.
6.RP.A.3Identify SubproblemsCompare the two totals
Pam pays $70 and Susan pays $60, so the gap is $10 — choice (C).
"How much more" is a subtraction of the two final amounts.
4.OA.A.3Identify SubproblemsFind each person's full-price total first, take their percent off, then subtract the two final prices to compare.
- Full-price cost for each buyer
- Apply Susan's 25% discount
- Apply Pam's 30% discount
- Compare the two totals