AMC 8 · 2004 · #6

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

percentagefraction-arithmeticlinear-equations-one-var identify-subproblemsconvert-to-algebra ↑ Prerequisites: multi-digit-arithmeticfraction-decimal-conversion

📏 Short solution 💡 2 insights

Problem

Sally has made 55% of her first 20 shots. After she takes 5 more shots, her overall success rate rises to 56%. How many of those last 5 shots did she make?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Set Up an Equation

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.

1STEP 1

Taking 55% of the first 20 shots gives Sally's makes so far: 11.

first made = 0.55 × 20 = 11

2STEP 2

Let x be the last-5 makes, so the total is 11 + x out of 25 shots.

total made = 11 + x, total taken = 20 + 5 = 25

3STEP 3

The 56% rate is makes over total, so $\frac{11 + x}{25} = 0.56$ .

\frac{11 + x}{25} = 0.56

4STEP 4

Multiply by 25 to get 11 + x = 14, so x lands on choice (C).

11 + x = 0.56 × 25 = 14 → x = 14 - 11 = 3 → (C)

Answer

Plug x = 3 back in. Total made is 11 + 3 = 14 out of 25, and 14 ÷ 25 = 0.56 = 56%. That matches the problem exactly. The answer is also a whole number from 0 to 5, which it must be since you cannot make a fractional shot. A quick sanity check on the size: the rate went up by only 1 percentage point, so Sally must have made a bit more than 55% of the last 5 shots — and 3 out of 5 is 60%, just a touch above her old rate, which fits.

💡Key takeaway

Percent 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.

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