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2023 AMC 10B Problem 5

2023 AMC 10B Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$ . Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$ . How many numbers are written on the blackboard?

A)
$10$
B)
$5$
C)
$6$
D)
$8$
E)
$9$

Answer

Key insight

Adding 3 to each of n numbers adds 3n to the sum, while tripling each number triples the sum, so S + 3n = 45 = 3S.

Solution

Let there be $n$ numbers with sum $S$ .

Lara triples every number, so her sum is $3S = 45$ , giving $S = 15$ .

Maddy adds $3$ to each of the $n$ numbers, which raises the sum by $3n$ : $S + 3n = 45$ . With $S = 15$ this gives $3n = 30$ , so $n = 10$ .

The answer is $\boxed{\textbf{(A)}\ 10}$ .

Why this works

You never need the individual numbers, only how the two operations affect the sum. Multiplying every term by $k$ multiplies the sum by $k$ ; adding $c$ to every term adds $c$ times the number of terms. Lara's clue fixes the sum, and Maddy's clue then converts the sum into a count.

The trap

Treating Maddy's operation as adding 3 to the total once, instead of once per number.

Common mistakes

Treating Maddy's operation as adding 3 to the total once, instead of once per number.
Solving $3S = 45$ and then reporting $S = 15$ or $45/3 = 15$ -related values instead of finishing with $n$ .

Techniques

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