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2021 AMC 10A Problem 2

2021 AMC 10A Problem 2AMC 10 Step by Step

Name: Date: October 28, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 1 min · position P1-10

Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have?

A) $600$
B) $650$
C) $1950$
D) $2000$
E) $2050$

Answer

Key insight

Lara's school is one share and Portia's is three shares, so the 2600 students make four equal shares.

Solution

"Three times as many" is easiest to handle by thinking in equal shares.

Call the size of Lara's school one share. Portia's school is three of those shares.
Together the two schools are $1 + 3 = 4$ shares, and that is $2600$ students.
One share is $2600 \div 4 = 650$ students. This is Lara's school.
Portia's school is three shares: $3 \times 650 = 1950$ students.
Check: $1950 + 650 = 2600$ , and $1950$ is three times $650$ .

The answer is $\boxed{\textbf{(C)}\ 1950}$ .

Why this works

When one amount is a number of times another, the total is made of equal shares: here $1$ share plus $3$ shares. Divide the total by the number of shares to find one share, then multiply. Using a letter works the same way: if Lara's school has $L$ students, then $L + 3L = 4L = 2600$ .

Alternative approach

Test the choices. For each one, subtract it from $2600$ to get Lara's school, and see whether Portia's number is three times Lara's. For $1950$ , Lara's school has $2600 - 1950 = 650$ , and $3 \times 650 = 1950$ . It fits. For $2000$ , Lara's school would have $600$ , and $3 \times 600 = 1800$ , not $2000$ .

The trap

Dividing 2600 by 3 instead of by 4, or giving Lara's number, 650, as the answer.

Concise solution

Let Lara's school have $L$ students. Then Portia's school has $3L$ students, and the total is
$L + 3L = 4L = 2600,$
so $L = 650$ . Portia's school therefore has $3 \cdot 650 = 1950$ students.

The answer is $\boxed{\textbf{(C)}\ 1950}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Prealgebra, Chapter 13: Data and Statistics (§13.1)

Techniques

Set up the equation/formula and compute; no special trick needed