AMC 8 · 2014 · #7
Easy mode Grade 6Problem
Ms. Raub's class has 28 students in total.
The class has 4 more girls than boys.
What is the ratio of girls to boys in the class?
Pick an answer.
AMC 8 2014 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.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: Total students $= 28$; Girls $=$ boys $+ 4$; Answer choices: (A) $3:4$, (B) $4:3$, (C) $3:2$, (D) $7:4$, (E) $2:1$
Unknowns: The ratio (girls $:$ boys), simplified
Understand
Restated: 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.
Givens: Total students $= 28$; Girls $=$ boys $+ 4$; Answer choices: (A) $3:4$, (B) $4:3$, (C) $3:2$, (D) $7:4$, (E) $2:1$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #6 Guess and Check
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.
Execute — Answer: B
2.OA.A.1 Step 1 - Set aside the $4$ extra girls.
- Imagine removing $4$ girls from the class so that boys and girls would be equal.
- That leaves $28 - 4 = 24$ students who split evenly.
💡 Subtracting the "extra" first is the classic subproblem move: turn an unequal split into a fair one.
3.OA.A.2 Step 2 - Split the $24$ evenly between boys and girls.
- Half of $24$ is $12$, so there are $12$ boys (and also $12$ girls before we put the extras back).
💡 Dividing a total into two equal groups is a Grade 3 "partitive division" idea.
2.OA.A.1 Step 3 Put the $4$ extra girls back to find the actual number of girls.
💡 Once the equal split is found, the extras only land in the girls' column.
6.RP.A.1 Step 4 Write the ratio girls $:$ boys and simplify by dividing both parts by their greatest common factor, $\gcd(16, 12) = 4$.
💡 A ratio in lowest terms divides both numbers by their greatest common factor, just like reducing a fraction.
2.OA.A.1 Set aside the $4$ extra girls. Imagine removing $4$ girls from the class so that 3.OA.A.2 Split the $24$ evenly between boys and girls. Half of $24$ is $12$, so there are 2.OA.A.1 Put the $4$ extra girls back to find the actual number of girls. 6.RP.A.1 Write the ratio girls $:$ boys and simplify by dividing both parts by their grea Review
Reasonableness: 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 $\tfrac{2}{3}$ of the class is girls, i.e. about $19$ girls) and clearly more than $1:1$.
Alternative: Tool #6 (Guess and Check) on the choices: each ratio $g:b$ with $g + b = 28$ gives a candidate split. (A) $3:4$ would mean fewer girls than boys — wrong direction. (B) $4:3$ scales to $16:12$, total $28$, difference $4$ ✓. (C) $3:2$ scales to $\tfrac{3}{5}(28) : \tfrac{2}{5}(28)$, which isn't a whole number. (D) $7:4$ doesn't divide $28$ evenly either. (E) $2:1$ gives $\tfrac{56}{3}$ girls — not a whole number. Only (B) survives.
CCSS standards used (min grade 6)
2.OA.A.1Use addition and subtraction within $100$ to solve word problems (Subtracting the $4$ extra girls to get $28 - 4 = 24$, and adding them back with $12 + 4 = 16$.)3.OA.A.2Interpret whole-number quotients as partitioning into equal shares (Splitting the $24$ remaining students into $2$ equal groups: $24 \div 2 = 12$.)6.RP.A.1Understand the concept of a ratio and use ratio language to describe a relationship (Forming the ratio girls $:$ boys $= 16:12$ and simplifying it to lowest terms $4:3$.)
⭐ 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.
⭐ 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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