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2001 AMC 10 Problem 3

2001 AMC 10 Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

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

The sum of two numbers is $S$ . Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

A)
$2S + 3$
B)
$3S + 2$
C)
$3S + 6$
D)
$2S + 6$
E)
$2S + 12$

Answer

Key insight

Adding 3 to each of two numbers adds 6 to their sum, and doubling each number doubles the sum, giving 2(S + 6) = 2S + 12.

Solution

Call the numbers $a$ and $b$ , so $a + b = S$ . After adding $3$ to each and doubling, the numbers are $2(a+3)$ and $2(b+3)$ . Their sum is
$2(a+3) + 2(b+3) = 2(a+b) + 12 = 2S + 12.$

The answer is $\boxed{\textbf{(E)}\ 2S + 12}$ .

Why this works

Operations applied to each number affect the sum in a predictable way: adding $c$ to each of $k$ numbers adds $kc$ to the sum, and scaling each number scales the sum. Tracking the sum directly, rather than the individual numbers, is enough here.

Alternative approach

Test a concrete pair: take $a = b = 0$ , so $S = 0$ . Each becomes $3$ , then $6$ , for a final sum of $12$ . Only choice (E) gives $12$ when $S = 0$ .

The trap

Adding 3 only once (as if it were added to the sum) and answering 2S + 6.

Common mistakes

Adding 3 only once (as if it were added to the sum) and answering 2S + 6.
Doubling before adding $3$ , which gives $2S + 6$ as well; the order of operations in the statement matters.

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer