2023 AMC 10B Problem 5

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

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.

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