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

The table shows some of the results of a survey by radiostation KACL. What percentage of the males surveyed listen to the station?

$\begin{tabular}{|c|c|c|c|}\hline & Listen & Don't Listen & Total\\ \hline Males & ? & 26 & ?\\ \hline Females & 58 & ? & 96\\ \hline Total & 136 & 64 & 200\\ \hline\end{tabular}$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Grand total: $200$ people surveyed; Total listeners: $136$; total non-listeners: $64$; Females: $58$ listen, total $96$; Males: $26$ don't listen; Answer choices: (A) $39$, (B) $48$, (C) $52$, (D) $55$, (E) $75$

Unknowns: The percentage of males surveyed who listen to the station

Understand

Plan

Primary tool: #2 Make an Organized List/Table

Secondary: #11 Find an Invariant

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) $\div$ (male total) expressed out of $100$.

Execute — Answer: E

#11 Find an Invariant 3.OA.D.8 Step 1

Find the male total.
The male row and the female row together make the grand total of $200$, and the female total is $96$.
Subtract.

$$\text{Males total} = 200 - 96 = 104$$

💡 Row totals must sum to the grand total. That is the invariant the table is built on.

#2 Make an Organized List/Table 3.OA.D.8 Step 2

Find the male listeners.
The male row now reads: listeners $+$ $26$ $=$ $104$.
Subtract to get the missing cell.

$$\text{Male listeners} = 104 - 26 = 78$$

💡 Within a row, the two cells add to the row total — fill the table one subtraction at a time.

#2 Make an Organized List/Table 6.RP.A.3 Step 3

Convert the male listener fraction into a percent.
The male row has $78$ listeners out of $104$ total males.
Write this as a fraction and scale to a denominator of $100$.

$$\dfrac{78}{104} = \dfrac{3}{4} = \dfrac{75}{100} = 75\%$$

💡 Simplify $\tfrac{78}{104}$ by dividing top and bottom by $26$ to get $\tfrac{3}{4}$, then read $\tfrac{3}{4}$ as $75$ per $100$.

#2 Make an Organized List/Table 6.RP.A.3 Step 4

Pick the answer choice that matches.

$$75\% \;\Rightarrow\; \textbf{(E)}$$

💡 $75\%$ matches choice (E) exactly.

[1] #11 3.OA.D.8 Find the male total. The male row and the female row together make the grand tot

[2] #2 3.OA.D.8 Find the male listeners. The male row now reads: listeners $+$ $26$ $=$ $104$. S

[3] #2 6.RP.A.3 Convert the male listener fraction into a percent. The male row has $78$ listene

[4] #2 6.RP.A.3 Pick the answer choice that matches.

Review

Reasonableness: 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 $\tfrac{78}{104} = 75\%$ 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.

Alternative: Tool #9 (Try an Easier Problem): you don't need every cell. Just compute males total $= 200 - 96 = 104$, notice that male non-listeners $= 26$ is exactly a quarter of $104$, and conclude that male listeners are the other three quarters, i.e. $75\%$. This shortcut skips finding $78$ and goes straight from "$26$ is $\tfrac{1}{4}$ of $104$" to the answer.

CCSS standards used (min grade 6)

3.OA.D.8 Solve two-step word problems using the four operations (Subtracting $200 - 96 = 104$ and $104 - 26 = 78$ to recover the missing row total and missing cell of the table.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, including finding a percent of a quantity (Turning the ratio $\tfrac{78}{104}$ of male listeners to male total into the percent $75\%$.)

⭐ 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 $100$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

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