AMC 8 · 2014 · #7

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up

ratio-proportionlinear-equations-one-varsystems-of-equations convert-to-algebraratio-proportion ↑ Prerequisites: linear-equations-one-varratio-proportion

📏 Short solution 💡 2 insights

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Problem

Ms. Raub's class has 28 students total, and there are exactly 4 more girls than boys. Find the ratio of girls to boys in lowest terms.

Pick an answer.

(A)

3 : 4

(B)

4 : 3

(C)

3 : 2

(D)

7 : 4

(E)

2 : 1

AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The class has 4 extra girls on top of an otherwise equal split. Tool #7 (Identify Subproblems) lets us peel off those 4 extra girls first, so the remaining 28 - 4 = 24 students split evenly into 12 boys and 12 girls. Adding the 4 back gives the counts in one clean step. We keep Tool #6 (Guess and Check) on hand because it's the natural way to double-check the answer against the multiple-choice ratios: only 16:12 satisfies both "sums to 28" and "differs by 4". Algebra (Tool #13) would also work, but for two unknowns connected by a sum and a difference, splitting off the extra is faster and more intuitive.

1STEP 1

Set aside the 4 extra girls; that leaves 28 - 4 = 24 students who split evenly.

28 - 4 = 24

2STEP 2

Half of 24 is 12, so there are 12 boys (and 12 girls before the extras go back).

24 ÷ 2 = 12 boys

3STEP 3

Put the 4 extra girls back: 12 + 4 = 16 girls.

12 + 4 = 16 girls

4STEP 4

Write girls : boys = 16 : 12, then divide both by gcd(16, 12) = 4 for lowest terms.

girls : boys = 16 : 12 =

\frac{16}{4}

\frac{12}{4}

= 4 : 3 → (B)

Answer

4 : 3

Check both conditions on 16 girls and 12 boys. Total: 16 + 12 = 28 ✓. Difference: 16 - 12 = 4 ✓. The ratio 16:12 = 4:3 also makes sense as a "slightly more girls than boys" answer — not as lopsided as 2:1 (which would mean

\frac{2}{3}

of the class is girls, i.e. about 19 girls) and clearly more than 1:1.

💡Key takeaway

Once you set the 4 extra girls aside, the rest of the class splits in half — and the ratio idea you need is right at the Grade 6 level.

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