AMC 10 · 2023 · #5

Grade 6 arithmetic

linear-equations-one-varmean-median-mode-rangesystems-of-equations identify-subproblemswork-backwardsconvert-to-algebra ↑ Prerequisites: linear-equations-one-var

📏 Short solution 💡 2 insights

Problem

A list of numbers has sum S and length n. Adding 3 to each entry produces a list whose sum is 45 (Maddy). Multiplying each entry by 3 also produces a list whose sum is 45 (Lara). Find n.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The problem packs two facts into one paragraph; Tool #7 (Subproblems) splits them: (i) Lara's clue alone tells us the original sum S, and (ii) Maddy's clue then tells us how many numbers added up to give that increase. Tool #11 (Work Backwards) applies in step (i) — Lara's sum was tripled to reach 45, so undo the triple to recover the original. A Tool #9 (Easier Problem) sanity check — "what if the list were {5,5,5}?" — reassures that the structure is right. Full algebra (#13) would work but is overkill for two one-step inversions.

1STEP 1

Lara's clue: tripling every number triples the total, so 3S = 45; divide by 3 to get S = 15.

3S = 45 → S =

\frac{45}{3}

= 15

2STEP 2

Maddy's clue: adding 3 to each of the n numbers adds 3n total, so her sum is S + 3n = 45.

S + 3n = 45

3STEP 3

Substitute S = 15 into Maddy's equation and subtract 15 from both sides: 3n = 30.

15 + 3n = 45 → 3n = 30

4STEP 4

Divide by 3 to get n = 10 — ten numbers on the blackboard, choice (A).

n =

\frac{30}{3}

= 10 → (A)

Answer

Plug back. Pick any list of 10 numbers summing to 15 — for example all 1.5s. Lara multiplies each by 3 to get ten 4.5s, summing to 45. Maddy adds 3 to each 1.5 to get ten 4.5s, summing to 45. Both clues check out, and (A) n = 10 is consistent with both equations.

💡Key takeaway

This AMC 10 problem only needs Grade 6 expression-and-equation thinking — Lara's clue pins the original sum at 15, and Maddy's extra 30 is just +3 per number, so there are 10 numbers.