AMC 10 · 2021 · #2
Easy mode Grade 4Problem
Picture two high schools. One belongs to Portia, and the other belongs to Lara.
Portia's school has 3 times as many students as Lara's school. Put together, the two schools have 2600 students in total.
How many students go to Portia's school?
Pick an answer.
AMC 10 2021 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: Portia's school has $3$ times the students of Lara's school. Together they have $2600$ students. Find how many students attend Portia's school.
Givens: Portia's students $= 3 \times$ Lara's students; Portia's students $+$ Lara's students $= 2600$; Answer choices: (A) $600$, (B) $650$, (C) $1950$, (D) $2000$, (E) $2050$
Unknowns: The number of students at Portia's school
Understand
Restated: Portia's school has $3$ times the students of Lara's school. Together they have $2600$ students. Find how many students attend Portia's school.
Givens: Portia's students $= 3 \times$ Lara's students; Portia's students $+$ Lara's students $= 2600$; Answer choices: (A) $600$, (B) $650$, (C) $1950$, (D) $2000$, (E) $2050$
Plan
Primary tool: #1 Draw a Diagram
Secondary: #6 Guess and Check, #3 Eliminate Possibilities
"$3$ times as many" begs for a tape diagram (tool #1): draw one box for Lara and three identical boxes for Portia. That makes the four boxes together equal to $2600$, so one box is $2600 \div 4 = 650$ and Portia gets $3$ boxes. Tool #6 (Guess and Check) and tool #3 (Eliminate) confirm: the answer must be more than half of $2600$, knocking out (A), (B); and the matching small box value $650$ shows up as choice (B) for Lara — confirming Portia's $1950$.
Execute — Answer: C
4.OA.A.2 Step 1 - Picture the schools as identical boxes.
- Lara's school is one box.
- Portia's school is three boxes the same size.
- Together that is four equal boxes.
💡 "$3$ times as many" is a Grade 4 multiplicative comparison — the bar model turns the comparison into countable boxes.
4.NBT.B.6 Step 2 - Four equal boxes total $2600$, so each box is $2600 \div 4 = 650$ students.
- That is the size of Lara's school.
💡 Splitting a four-digit total evenly among $4$ groups is Grade 4 long division.
4.NBT.B.5 Step 3 Portia's school is $3$ boxes, so $3 \times 650 = 1950$ students.
💡 Multiplying a three-digit number by a one-digit number is Grade 4 "multi-digit by one-digit".
4.OA.A.3 Step 4 - Check against the choices.
- $1950$ is choice (C) — and it sits comfortably above half of $2600$, matching the constraint that Portia's share is the larger one.
💡 Plugging the candidate back into the two given relationships and seeing them both hold is Grade 4 "multi-step word problem with reasonableness check".
4.OA.A.2 Picture the schools as identical boxes. Lara's school is one box. Portia's schoo 4.NBT.B.6 Four equal boxes total $2600$, so each box is $2600 \div 4 = 650$ students. That 4.NBT.B.5 Portia's school is $3$ boxes, so $3 \times 650 = 1950$ students. 4.OA.A.3 Check against the choices. $1950$ is choice (C) — and it sits comfortably above Review
Reasonableness: Portia's share is the larger one, so the answer must beat half of $2600 = 1300$. Choices (A) $600$ and (B) $650$ are too small. (C) $1950$ sits at three-quarters of $2600$, which is exactly what "$3$ of $4$ equal boxes" predicts. The other big choices (D) $2000$ and (E) $2050$ do not make $4$ equal boxes with the rest.
Alternative: Tool #6 (Guess and Check) directly on the choices: test choice (C). If Portia $= 1950$, then Lara $= 2600 - 1950 = 650$, and $1950 \div 650 = 3$ exactly. Both conditions satisfied — pick (C).
CCSS standards used (min grade 4)
4.OA.A.2Multiply or divide to solve word problems involving multiplicative comparison (Modeling "$3$ times as many" as one box for Lara and three identical boxes for Portia.)4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number (Computing $3 \times 650 = 1950$ to get Portia's enrollment.)4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends (Dividing the total $2600 \div 4 = 650$ to find one box's worth of students.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Verifying $1950 + 650 = 2600$ and $1950 = 3 \times 650$ in one combined check.)
⭐ This AMC 10 problem only needs Grade 4 "$3$ times as many" tape diagrams you already know — split $2600$ into $4$ equal boxes, give Portia $3$ of them, and the answer is $1950$.
⭐ This AMC 10 problem only needs Grade 4 "$3$ times as many" tape diagrams you already know — split $2600$ into $4$ equal boxes, give Portia $3$ of them, and the answer is $1950$.
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