Competition · AMC preparation · step 4 of 4

AMC 10 · 2021A · #4

Grade 6 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation
sequences-arithmeticpattern-recognitionequal-spacingmean-median-mode-range pattern-recognitionidentify-subproblems ↑ Prerequisites: sequences-arithmetic
📏 Medium solution 💡 2 insights
Problem
A cart rolls 5 inches in the first second, and each later second it goes 7 more inches than the second before. After 30 seconds it hits the bottom. Find the total inches traveled.

Pick an answer.

(A)
215
(B)
360
(C)
2992
(D)
3195
(E)
3242

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Tool #5 (Pattern) spots the steady +7 jumps and the equal-spaced list. Tool #9 (Easier Problem) checks the idea on a tiny version — pair 5 and 208, 12 and 201, …, each pair sums to 213. Tool #7 (Subproblems) then splits the work into two pieces: "what is the last-second distance?" and "how many pairs, each 213?".

1STEP 1

Write out the first few distances

List the first few: 5, then 5 + 7 = 12, then 19, 26 — each second adds the same 7, a steady ladder.

5, 12, 19, 26, … (gap = 7)
2STEP 2

Find the 30th term

From the 1st second to the 30th the +7 jump happens 29 times, so the last second is 5 + 29 × 7 = 208 inches.

a₃₀ = 5 + 29 × 7 = 5 + 203 = 208
3STEP 3

Pair the terms from both ends

Pair first with last, second with second-to-last: each pair sums to 5 + 208 = 213, and 30 terms make exactly 15 pairs.

15 pairs, each summing to 5 + 208 = 213
4STEP 4

Multiply the pair sums

Add all 15 pair sums: 15 × 213 = 3,195 inches — choice (D).

15 × 213 = 3,195 → (D)
Answer
3195
Average per-second distance is roughly 5+2082\frac{5 + 208}{2} ≈ 106.5 inches, so over 30 seconds the total should be about 106.5 × 30 ≈ 3,195. That lands exactly on choice (D). Tiny choices (A) and (B) are obviously wrong — even the last second alone is 208 inches, far more than 215.
💡Key takeaway

This AMC 10 problem only needs Grade 6 "average × count = total" — pair the first second (5) with the last (208), every pair sums to 213, and 15 pairs give 3,195 inches.

  • Write out the first few distances
  • Find the 30th term
  • Pair the terms from both ends
  • Multiply the pair sums

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