AMC 8 · 2006 · #8

Grade 6 arithmetic

Archetype Overlap / Complement Counting └ Sub-archetype Two-Attribute Table: Fill or Optimize a Cell

percentagemulti-digit-arithmeticset-partition identify-subproblemscomplementary-counting ↑ Prerequisites: percentagemulti-digit-arithmetic

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A radio station survey is partly recorded in a two-way table. The grand total is 200 people: 136 listen and 64 don't. The female row totals 96, with 58 listeners. The male "Don't Listen" cell is 26. What percent of the males surveyed listen to the station?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make an Organized List/Table

The problem is literally a partly filled table, so Tool #2 (Make an Organized List/Table) is the natural fit — copy the table and fill the blanks. The reason we can fill the blanks at all is Tool #11 (Find an Invariant): every row and every column has to add to its given total, and both row totals and column totals must sum to the same grand total of 200. Those add-up rules are the invariants that pin down each missing cell with a single subtraction. Once the male row is complete, the requested percent is just (male listeners) ÷ (male total) expressed out of 100.

1STEP 1

Both row totals add to 200 and females are 96, so the male total is 104.

Males total = 200 - 96 = 104

2STEP 2

The male row reads listeners + 26 = 104, so male listeners = 78.

Male listeners = 104 - 26 = 78

3STEP 3

78 of the 104 males listen, and $\frac{78}{104}$ = $\frac{3}{4}$ = 75%.

\frac{78}{104}

\frac{3}{4}

\frac{75}{100}

= 75%

4STEP 4

That matches choice (E).

75% → (E)

Answer

Cross-check by completing the rest of the table and seeing that every row and every column adds up. Female non-listeners = 96 - 58 = 38. Now check the columns: listeners = 78 + 58 = 136 (matches), non-listeners = 26 + 38 = 64 (matches). The grand total 104 + 96 = 200 (matches). Every invariant holds, so the male listener count of 78 is right, and

\frac{78}{104}

= 75% stands. A quick sanity look at the choices: only 26 males don't listen out of 104, which is a quarter, so listeners must be three-quarters — choices (A)-(D) all undershoot.

💡Key takeaway

When a table has missing cells, lean on the rule that every row and column must add up to its total. Each missing number costs you one subtraction, and then a percent is just a fraction rewritten with denominator 100.

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