AMC 8 · 2009 · #23
Easy mode Grade 4Problem
On the last day of school, Mrs. Wonderful brought jelly beans for her class.
Here is how she handed them out. She gave every boy the same number of jelly beans as the number of boys in the class. She also gave every girl the same number of jelly beans as the number of girls in the class.
When she was done, she had jelly beans left over. Also, her class had two more boys than girls.
How many students are in her class altogether?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: A teacher gives every boy a number of jelly beans equal to the number of boys, and every girl a number equal to the number of girls. She starts with $400$ jelly beans and has $6$ left, so $394$ were handed out. There are $2$ more boys than girls. How many students are in the class?
Givens: Each boy gets (number of boys) jelly beans, so the boys together get (number of boys)$^2$; Each girl gets (number of girls) jelly beans, so the girls together get (number of girls)$^2$; Total handed out $= 400 - 6 = 394$; Number of boys $=$ number of girls $+\, 2$; Answer choices for total students: (A) $26$, (B) $28$, (C) $30$, (D) $32$, (E) $34$
Unknowns: The total number of students in the class
Understand
Restated: A teacher gives every boy a number of jelly beans equal to the number of boys, and every girl a number equal to the number of girls. She starts with $400$ jelly beans and has $6$ left, so $394$ were handed out. There are $2$ more boys than girls. How many students are in the class?
Givens: Each boy gets (number of boys) jelly beans, so the boys together get (number of boys)$^2$; Each girl gets (number of girls) jelly beans, so the girls together get (number of girls)$^2$; Total handed out $= 400 - 6 = 394$; Number of boys $=$ number of girls $+\, 2$; Answer choices for total students: (A) $26$, (B) $28$, (C) $30$, (D) $32$, (E) $34$
Plan
Primary tool: #6 Guess and Check
Secondary: #2 Make a Systematic List
Each answer choice is a total like $28$. Once we know the total $T$, the "boys $-$ girls $= 2$" condition pins down the split exactly: girls $= (T-2)/2$, boys $= (T+2)/2$. So Tool #6 (Guess and Check) lets us try each choice and see which one makes (boys)$^2 +$ (girls)$^2$ equal $394$. Tool #2 (Make a Systematic List) keeps the five trials organized in a clean table so we don't lose track. This avoids setting up a quadratic equation — we just test five small cases.
Execute — Answer: B
4.OA.A.3 Step 1 - Find how many jelly beans were actually handed out.
- She started with $400$ and ended with $6$, so the rest went to the students.
💡 Subtracting the leftover from the starting amount gives the amount used — the standard "start, change, result" word-problem move.
4.OA.A.3 Step 2 - Turn each total $T$ into a (girls, boys) pair.
- Since boys are $2$ more than girls and together they sum to $T$, girls get half of $T - 2$ and boys get half of $T + 2$.
💡 If two numbers differ by $2$ and add to $T$, the smaller one is the average minus $1$ and the larger is the average plus $1$ — a Grade 4 word-problem habit.
3.OA.C.7 Step 3 - Make a systematic list.
- For each answer choice, compute girls, boys, and (girls)$^2 +$ (boys)$^2$, then compare to $394$.
💡 Squaring small two-digit numbers ($12^2, 13^2, \ldots, 18^2$) uses the multiplication fluency built in Grade 3.
4.NBT.B.4 Step 4 - Check the totals against $394$.
- Only $T = 28$ gives exactly $394$.
- The values jump from $340$ to $394$ to $452$, so the answer is unique — no near-misses to worry about.
💡 Adding two three-digit numbers and matching the target is the Grade 4 multi-digit addition standard, used here as the "check" step of Guess and Check.
4.OA.A.3 Step 5 - Read off the answer.
- The total $T = 28$ matches choice (B), with $13$ girls and $15$ boys.
💡 After Guess and Check confirms one row, that row's total is the answer — the closing move of Tool #6.
4.OA.A.3 Find how many jelly beans were actually handed out. She started with $400$ and e 4.OA.A.3 Turn each total $T$ into a (girls, boys) pair. Since boys are $2$ more than girl 3.OA.C.7 Make a systematic list. For each answer choice, compute girls, boys, and (girls) 4.NBT.B.4 Check the totals against $394$. Only $T = 28$ gives exactly $394$. The values ju 4.OA.A.3 Read off the answer. The total $T = 28$ matches choice (B), with $13$ girls and Review
Reasonableness: With $13$ girls and $15$ boys, the teacher hands out $13 \times 13 = 169$ jelly beans to the girls and $15 \times 15 = 225$ to the boys, totaling $169 + 225 = 394$. Adding the $6$ left over gives $400$, matching the starting amount. The boys outnumber the girls by $15 - 13 = 2$, as required. Every condition checks out.
Alternative: Tool #5 (Look for a Pattern): the totals in the table $340, 394, 452, 514, 580$ jump by $54, 58, 62, 66$ — increasing by $4$ each time. Starting at $T = 26$ with $340$ and needing $394$ means a jump of $54$, which is exactly the first gap. So we know the answer is the second row, $T = 28$, before even reading further. The pattern $g^2 + (g+2)^2 = 2g^2 + 4g + 4$ explains the steady increase.
CCSS standards used (min grade 4)
4.OA.A.3Solve multi-step word problems with whole numbers using the four operations (Reading the situation as "start with $400$, $6$ left over, the rest split between boys and girls," and translating each candidate total into (girls, boys) pairs.)3.OA.C.7Fluently multiply and divide within 100 (Computing the squares $12^2 = 144$, $13^2 = 169$, $14^2 = 196$, $15^2 = 225$, $16^2 = 256$, $17^2 = 289$, $18^2 = 324$ for each trial in the table.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Adding pairs like $169 + 225 = 394$ and $400 - 6 = 394$ to compare each trial against the target.)
⭐ You don't need a quadratic equation — with five answer choices and the "$2$ more boys than girls" rule, Guess and Check with Grade 4 arithmetic finds the answer in one tidy table!
⭐ You don't need a quadratic equation — with five answer choices and the "$2$ more boys than girls" rule, Guess and Check with Grade 4 arithmetic finds the answer in one tidy table!