Competition · AMC preparation · step 4 of 4
AMC 8 · 2004 · #6
Grade 6 arithmeticPick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem links two percentages of the same shooter, so the natural move is Tool #3 (Set Up an Equation): write the new success rate as a fraction whose numerator and denominator both depend on what happens in the last 5 shots. Tool #4 (Introduce a Variable) names the unknown count x of made shots among those last 5, so we can turn the percentage sentence into a single linear equation and solve. Once x is isolated, the answer is one subtraction away.
Count the first makes
Taking 55% of the first 20 shots gives Sally's makes so far: 11.
Percent of a whole is the Grade 6 "percent times total" move: 55% of 20 is 11.
6.RP.A.3Eliminate PossibilitiesName the unknown makes
Let x be the last-5 makes, so the total is 11 + x out of 25 shots.
One letter for the unknown count keeps the rest of the work as plain arithmetic.
6.EE.A.2Introduce A VariableWrite the new rate equation
The 56% rate is makes over total, so .
A percent equation is just a fraction set equal to a decimal — exactly what Grade 6 "solve px = q" handles.
After all 25 shots, Sally's 56% success rate is exactly the fraction (11 + x)/25.
▸ Why?
Sally's 56% success rate means she made 56 out of every 100 shots' worth, and that per-hundred count is just the proportion of shots made among shots taken.
▸ Why?
A percent is a count out of one hundred, so 56% names the fraction 56/100 of the shots, and a success rate is precisely that fraction of shots taken that were made.
▸ Why?
The shots made after 25 attempts total 11 + x, since the 11 makes from the first 20 and the x makes from the last 5 are all the makes there are.
▸ Why?
Every made shot belongs to exactly one group, the first-20 makes or the last-5 makes, so those two counts add up to all the makes with nothing missed and nothing counted twice.
▸ Why?
The shots taken total 25, since the first 20 attempts and the 5 later attempts together are every shot she took.
▸ Why?
The whole run of shots splits cleanly into the first 20 and the next 5, so those two counts add back to the full 25.
Solve for x
Multiply by 25 to get 11 + x = 14, so x lands on choice (C).
Clearing the denominator turns the percent equation into one-step arithmetic.
6.EE.B.7Eliminate PossibilitiesPercent problems get easy once you turn the percent into a fraction with the right top and bottom. Here, 56% of 25 shots is exactly 14 makes, and Sally already had 11 — so she needed 14 - 11 = 3 more makes in those last 5 shots.
- Count the first makes
- Name the unknown makes
- Write the new rate equation
- Solve for x
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