AMC 10 · 2007 · #14

Grade 6 arithmetic

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Ratio Shift — Two State

percentagelinear-equations-one-var convert-to-algebra ↑ Prerequisites: percentage

📏 Short solution 💡 2 insights

Problem

Some boys and girls are having a car wash to raise money for a class trip to China. Initially $40\%$ of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then $30\%$ of the group are girls. How many girls were initially in the group?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2007 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.

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Toolkit + CCSS Solution

Understand

Restated: A group of boys and girls starts with $40\%$ girls. Then two girls leave and two boys arrive, which keeps the total number of people the same but changes the mix to $30\%$ girls. Find how many girls were in the group at the start.

Givens: At the start, $40\%$ of the whole group are girls; Two girls leave and two boys arrive at the same time; After this swap, $30\%$ of the group are girls; Answer choices: (A) 4, (B) 6, (C) 8, (D) 10, (E) 12

Unknowns: The number of girls in the group before the swap

Understand

Plan

Primary tool: #13 Convert to Algebra

Secondary: #4 Introduce a Variable, #3 Eliminate Possibilities

The trick is that the total head count never changes, so it is the natural quantity to name. Introduce a variable for the total (Tool #4), then translate the two percent facts into expressions for the girl count before and after (Tool #13). Because a swap of two-for-two changes the girls by $2$ but not the total, one clean equation pops out and solves in a line. Tool #3 (Eliminate Possibilities) is kept in reserve to sanity-check the result against the five answer choices.

Execute — Answer: C

#4 Introduce a Variable 6.EE.B.6 Step 1

Name the total and the girls

Let $n$ be the total number of people in the group at the start.
Since $40\%$ of them are girls, the starting girl count is $40\%$ of $n$, which is $0.4n$ (the same as $\tfrac{2}{5}n$).
Naming the total is smart here because the swap of two girls for two boys leaves that total unchanged.

$$\text{total}=n,\qquad \text{girls}=0.4n$$

💡 Give the quantity that stays fixed a name, and everything else can be measured against it.

#13 Convert to Algebra 6.RP.A.3 Step 2

Translate the after-swap fact

After two girls leave and two boys arrive, the total is still $n$ people, but the number of girls has gone down by $2$, to $0.4n-2$.
The problem says these girls are now $30\%$ of the group, so the girl count also equals $0.3n$.
Setting the two descriptions of the same girl count equal gives one equation.

$$0.4n-2=0.3n$$

💡 The same number of girls written two ways must be equal, which turns the words into an equation.

#13 Convert to Algebra 6.EE.B.7 Step 3

Solve and read off the girls

Subtract $0.3n$ from both sides to collect the $n$ terms: $0.1n-2=0$, so $0.1n=2$ and $n=20$.
The total group had $20$ people.
The starting girls are $0.4n=0.4\times 20=8$.
So there were $8$ girls to begin with, which is choice $\textbf{(C)}$.

$$0.1n=2\Rightarrow n=20,\qquad \text{girls}=0.4\times 20=8$$

💡 Gathering the variable on one side leaves a simple $px=q$ that hands you the total directly.

[1] #4 6.EE.B.6 Let $n$ be the total number of people in the group at the start. Since $40\%$ of

[2] #13 6.RP.A.3 After two girls leave and two boys arrive, the total is still $n$ people, but th

[3] #13 6.EE.B.7 Subtract $0.3n$ from both sides to collect the $n$ terms: $0.1n-2=0$, so $0.1n=2

Review

Reasonableness: Check the full story with $20$ people and $8$ girls. At the start, $8$ of $20$ is $\tfrac{8}{20}=40\%$ girls, matching the first fact. After two girls leave and two boys arrive, there are $6$ girls out of the same $20$ people, and $\tfrac{6}{20}=30\%$, matching the second fact. Both conditions hold, so $8$ girls is correct.

Alternative: Reason in percentage points. The total stays fixed, and losing $2$ girls drops the girl share from $40\%$ to $30\%$, a fall of $10$ percentage points. So $2$ girls equal $10\%$ of the total, meaning the total is $20$. Then $40\%$ of $20$ is $8$ girls, confirming (C).

CCSS standards used (min grade 6)

6.EE.B.6 Use variables to represent numbers and write expressions to solve problems (Naming the fixed total $n$ and writing the starting girl count as $0.4n$.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Turning the $30\%$-girls fact into the expression $0.3n$ and equating it with $0.4n-2$.)
6.EE.B.7 Solve real-world problems by writing and solving equations of the form px = q (Solving $0.1n=2$ to get $n=20$ and then computing the $8$ starting girls.)

⭐ When a swap keeps the total the same, name that total, write the girl count two ways, and set them equal.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Name the total and the girls

Translate the after-swap fact

Solve and read off the girls

Review

CCSS standards used (min grade 6)

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