AMC 10 · 2021 · #6
Grade 7 probabilityPick an answer.
The question asks for one number, so Tool #4 (Introduce a Variable) names it: let n be the original deck size. Every other quantity in the story — red count, black count, new deck size — is then an expression in n, which is exactly what Tool #13 (Convert to Algebra) needs to turn the two probability sentences into one equation. Tool #6 (Guess and Check) closes the loop by rebuilding the actual deck from the answer and re-reading both probabilities off it. Tool #3 (Eliminate Possibilities) is the backup route: divisibility alone kills four of the five choices.
Name the deck size
Express the red count via the total.
A probability of 1/3 just says one out of every three cards is red, so the red pile is a third of the deck.
7.SP.C.5Introduce A VariableSee what stays fixed
The red count stays put.
The red cards never move, so the drop in the red probability comes entirely from the deck getting bigger underneath them.
The red cards never move, so the drop in the red share comes entirely from the deck growing underneath them.
▸ Why?
Only black cards are added, so nothing is taken from the red pile and nothing is added to it.
▸ Why?
A share is a part compared to the whole, so with the part fixed the whole alone decides it.
Turn the second fact into an equation
The new probability gives an equation.
Two sentences about the same deck become two expressions in n; setting them equal is the only step that uses both facts at once.
7.EE.B.4Convert To AlgebraSolve for the size
Tidying gives the size.
n/3 is the red count, and the equation says that count equals 4 — the four added black cards exactly match the red pile.
6.EE.B.7Convert To AlgebraRebuild and check
Counting it out confirms 12.
An answer to a deck problem should produce a real deck — whole numbers of red and black cards that hit both fractions exactly.
7.SP.C.7Guess And CheckName the deck size n, notice the red cards never change while the total grows by 4, and one equation n/3/(n+4) = 1/4 hands you n = 12.
- Name the deck size
- See what stays fixed
- Turn the second fact into an equation
- Solve for n
- Rebuild the deck and check