AMC 8 · 2025 · #4

Grade 4 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation
sequences-arithmeticpattern-recognitionmulti-digit-arithmetic pattern-recognitionformula-substitution ↑ Prerequisites: multi-digit-arithmetic
📏 Medium solution 💡 2 insights
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Problem
Lucius counts backward by 7, starting from 100. So the numbers he says are 100, 93, 86, and so on. What is the 10th number he says?

Pick an answer.

(A)
30
(B)
37
(C)
42
(D)
44
(E)
47

AMC 8 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

The numbers 100, 93, 86, … are a clean repeating pattern: each step subtracts 7. Tool #5 (Look for a Pattern) is the natural fit — list a few more terms to confirm the rule, then jump to the 10th term. Because we are counting from the 1st term, the 10th term is 9 steps away, so we subtract 7 a total of 9 times (equivalently subtract 9 × 7 = 63). Tool #3 (Eliminate) is a fast sanity check on the multiple-choice list — only one of the five choices can be 100 - 63.

1STEP 1

List a few terms — 100, 93, 86, 79, 72 — the gap is always 7, so the rule subtract 7 holds.

100, 93, 86, 79, 72, …
2STEP 2

From term 1 to term 10 there are 10 - 1 = 9 jumps, each removing 7.

jumps = 10 - 1 = 9
3STEP 3

Each jump removes 7, so 9 jumps remove 9 × 7 = 63 in all.

9 × 7 = 63
4STEP 4

Subtract 63 from the starting number: 100 - 63 = 37, the 10th term.

a₁0 = 100 - 63 = 37
5STEP 5

Only choice (B) equals 37; the others fail the rule "100 minus nine 7s".

Match: 37 → (B)
Answer
37
Each step shrinks the number by 7, so after 9 steps we should be well below 100 but still positive (since 9 × 7 = 63 ＜ 100). The answer 37 lies between 30 and 44 — exactly where the choices cluster — and a direct hand-count 100, 93, 86, 79, 72, 65, 58, 51, 44, 37 confirms the 10th number is 37.
💡Key takeaway

This AMC 8 problem only needs Grade 4 number-pattern skills — "subtract the same amount over and over" — that you already know!

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