AMC 10 · 2016 · #3
Grade 7 arithmeticPick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The phrase "for every dollar Ben spent, David spent 25 cents less" is a rate, so Tool #8 (Analyze the Units) turns it into a clean fact: David's total is 12.50 more" becomes one short equation to solve. Tool #3 (Eliminate Possibilities) then checks the size against the choices.
Read the rate
The rate 25 cents less per dollar means David's total is always 0.75 times Ben's total.
A fixed gap per dollar means David always keeps the same 75% share of whatever Ben spends.
A fixed gap per dollar means one person always keeps the same share of what the other spends.
▸ Why?
The rule is quoted per one fixed dollar, so it repeats identically for every dollar spent.
▸ Why?
So the two totals rise in step, and their ratio stays the same whatever the amount.
Solve for Ben's total
Let B be Ben's dollars; David spends 0.75B, so the gap 0.25B = 12.50 gives B=50.
The whole gap of $12.50 is built from 25 cents per Ben-dollar, so it takes 50 dollars to make it.
7.EE.B.4Introduce A VariableAdd the two totals
Ben spent 50 and David 37.50, so together they spent 50+37.50 = 87.50, choice (C).
Once Ben's total is known, David's is just three-quarters of it, and the sum is a quick addition.
6.NS.B.3Introduce A VariableA steady gap per dollar lets you turn the total difference into one easy equation for the bigger amount.
- Read the rate
- Solve for Ben's total
- Add the two totals