2023 AMC 10B Problem 5

P1-10~2 minPrint

Linear Equations & Word Problems Systems of Equations Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

Tripling every number triples the total, so the original total is 15. Adding 3 to every number adds 3 once for each number, and that extra is 30.

Solution

We never need the actual numbers. We only need to know what each girl's change does to the total.

1. Lara's clue gives the total. If every number is multiplied by $3$ , the total is multiplied by $3$ too. (For example, $2 + 5 = 7$ and $6 + 15 = 21$ .) Lara's total is $45$ , so the original total is
$45 \div 3 = 15.$
2. Maddy's clue gives the count. If every number gets $3$ more, the total goes up by $3$ once for each number. Her total is $45$ , which is $30$ more than the original $15$ .
3. Count the numbers. Each number added $3$ , and together they added $30$ , so there are
$30 \div 3 = 10 \text{ numbers}.$

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

Why this works

Two rules about totals do all the work. Multiplying every number by $k$ multiplies the total by $k$ . Adding $c$ to every number adds $c$ times (how many numbers there are) to the total. Lara's clue fixes the total, and Maddy's clue then turns the total into a count.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(A)}\ 10}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 5: Multi-Variable Linear Equations (§5.2, §5.3, §5.6)
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Intermediate Algebra, Chapter 1: Basic Techniques for Solving Equations (§1.4)