AMC 10 · 2020 · #5
Grade 8 algebraPick an answer.
Tool #4 (Introduce a Variable): the fraction 2/3 already forces team A's total to be a multiple of 3, so naming that total 3k makes both of A's numbers whole automatically — 2k wins and k losses, with no fractions to carry. Tool #13 (Convert to Algebra): once A's record is in terms of k, the two "7 more" facts hand over B's whole record for free, and B's win fraction 5/8 becomes one equation in one unknown. Tool #3 (Eliminate Possibilities): kept in reserve as a check — the five choices can be screened by divisibility alone.
Name the first team's record
Make the games a multiple of three.
Naming the total 3k instead of n builds the 2/3 ratio into the name, so nothing later has to be divided.
6.RP.A.3Introduce A VariableWrite the second team's record
Write the second record with the same letter.
Seven extra wins plus seven extra losses means team B simply played 14 more games than team A.
6.EE.B.6Convert To AlgebraBuild the equation
The second team's win rate gives the equation.
A win fraction is just wins compared to total games, so setting those two ratios equal is the whole condition.
A win fraction is wins compared to total games, so setting the two fractions equal is the whole condition.
▸ Why?
A fraction is a count of equal shares of the whole, and here the whole is every game played.
▸ Why?
Two expressions naming the same ratio name the same number, so they can be set equal.
Solve the equation
Tidying fixes the letter.
The k terms on the two sides differ by only 1k, so one subtraction collapses the whole equation.
8.EE.C.7Convert To AlgebraFind the games played
Converting back gives 42 games.
The variable was a stand-in for one third of the games, so the last move is to multiply it back by 3.
6.RP.A.3Introduce A VariableWhen a problem says a team won 2/3 of its games, name the total 3k instead of n — the fraction turns into whole numbers on its own, and the rest of the problem becomes one short equation.