Competition · AMC preparation · step 4 of 4
AMC 8 · 2009 · #23
Grade 4 algebraPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Subtract the leftover beans
Subtract the 6 left over from the 400 she brought: 394 jelly beans were handed out.
Subtracting the leftover from the starting amount gives the amount used — the standard "start, change, result" word-problem move.
4.OA.A.3Guess And CheckWrite girls and boys from the total
For any total T, the two-more-boys rule splits it: girls get and boys get .
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.
Once you know only the total number of students T, the fact that there are two more boys than girls forces a single split: the girls must be (T-2)/2 and the boys (T+2)/2.
▸ Why?
Girls and boys are the only two parts of the class, with nobody counted twice and nobody left out, so the girl count and the boy count add up to the total T.
▸ Why?
The boy count is the girl count plus 2, so the total T is a girl count plus another girl count plus 2 — two equal girl groups and a spare 2.
▸ Why?
"Boys" and "girls plus 2" name the very same amount, so one may stand in for the other inside the total without changing what the total equals.
▸ Why?
A girl count added to an identical girl count is simply two equal groups of that count.
▸ Why?
Since T is two equal girl groups plus 2, taking the 2 back off and then splitting what remains into its two equal groups recovers one girl count, and the boy count is that plus 2.
Make a systematic list
For each answer choice, compute girls, boys, and girls² + boys², then compare each sum to 394.
Squaring small two-digit numbers (12², 13², …, 18²) uses the multiplication fluency built in Grade 3.
3.OA.C.7Make A Systematic ListCheck which total gives 394
Only T = 28 makes the sum exactly 394; values jump 340 → 394 → 452, so the match is unique.
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.NBT.B.4Guess And CheckRead off the answer
The winning row's total, T = 28, is the answer — choice (B), 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.3Guess And CheckYou 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!
- Subtract the leftover beans
- Write girls and boys from the total
- Make a systematic list
- Check which total gives 394
- Read off the answer
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