AMC 10 · 2003 · #7
Grade 8 algebraPick an answer.
Two facts are given about the same pile — how many coins and how much money — so tool #13 (Convert to Algebra) turns each fact into one equation and lets them be combined. Tool #4 (Introduce a Variable) names all three counts even though only the dimes are asked about, because the nickels are what makes the count work out and the quarters are what pushes the dime count around. Two equations cannot pin down three unknowns, so there is a whole family of legal coin piles; tool #14 (Extreme Principle) asks only for the two ends of that family, which is all the question needs. Tool #8 (Analyze the Units) does the small but essential first move: work in cents, not dollars, so every number in sight is a whole number.
Work in cents, not dollars
Working in cents makes every number whole, so the total reads 835.
Switching to cents makes every quantity a whole number, which is the only reason the counting argument later has any bite.
4.MD.A.2Analyze The UnitsName all three counts
Naming the three counts gives a coin equation and a value equation.
One equation counts coins and the other counts money; the same pile has to satisfy both at once.
7.EE.B.4Introduce A VariableGet rid of the nickels
Dividing by five and subtracting removes the nickels, leaving b + 4c = 67.
Subtracting the coin count from the scaled money total cancels the nickels, leaving one clean relation between dimes and quarters.
Subtracting the coin count from the scaled money total cancels the nickels and leaves one clean relation.
▸ Why?
Both statements describe the same pile, so one may be subtracted from the other and still hold.
▸ Why?
The pile's value is its coins' values added together, so a count and a value are two honest descriptions of one thing.
Check the nickels never go negative
Checking back, the nickels come to 33 plus 3c, always positive, so nothing is blocked.
Eliminating a variable hides it but does not free it — you still have to confirm the hidden count stays legal.
7.EE.A.1Convert To AlgebraPush c to both ends
Pushing the quarters to both ends gives dimes from 3 to 67, a spread of 64, choice (D).
Since dimes drop steadily as quarters rise, the two extremes sit at the smallest and largest quarter counts the whole-number rule allows.
6.EE.B.5Extreme PrincipleWhen two facts cannot pin down three unknowns, boil them down to one relation and then slide the free unknown to each end — the answer lives at the ends, not in the middle.
- Work in cents, not dollars
- Name all three counts
- Get rid of the nickels
- Check the nickels never go negative
- Push c to both ends