AMC 10 · 2022 · #4

Grade 4 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

ratesequences-arithmeticunit-conversionmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticunit-conversion

📏 Short solution 💡 2 insights

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Problem

A donkey's first hiccup happens at exactly 4/:00 in the afternoon and then one hiccup every 5 seconds. At what wall-clock time does the 700th hiccup occur?

Pick an answer.

(A)

$15 \text{ seconds after } 4:58$

(B)

$20 \text{ seconds after } 4:58$

(C)

$25 \text{ seconds after } 4:58$

(D)

$30 \text{ seconds after } 4:58$

(E)

$35 \text{ seconds after } 4:58$

AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

Jumping straight to the 700th hiccup is hard to picture. Tool #9 (Easier Related Problem) says: try n = 2, n = 3, n = 4 first and see the rule that connects the hiccup number to elapsed seconds. Tool #5 (Look for a Pattern) makes the rule explicit — the nth hiccup happens 5(n - 1) seconds after 4{:}00. Tool #8 (Analyze the Units) finishes the job by converting total seconds to minutes-and-seconds for the clock answer.

1STEP 1

Test small cases: the nth hiccup lands 5(n-1) seconds after 4/:00 — that is 0, 5, 10, 15 s for n = 1, 2, 3, 4.

n=1 → 0 s, n=2 → 5 s, n=3 → 10 s, n=4 → 15 s

2STEP 2

Apply the pattern to n = 700: the number of gaps between the 1st and 700th hiccup is 700 - 1 = 699.

gaps = 700 - 1 = 699

3STEP 3

Multiply the gaps by 5 seconds each: 699 × 5 = 3495 seconds total.

699 × 5 = 3495 seconds

4STEP 4

Convert with 60 s = 1 min: 3495 ÷ 60 = 58 R 15, i.e. 58 min 15 s.

3495 ÷ 60 = 58 R 15 → 58 min 15 s

5STEP 5

Add 58 min 15 s to 4/:00/:00 → 4/:58/:15, i.e. 15 seconds after 4/:58 — choice (A).

4{:}00{:}00 + 58 min 15 s = 4{:}58{:}15 → (A)

Answer

15 seconds after 4:58

Order-of-magnitude check. 700 hiccups at one every 5 seconds is roughly 700 × 5 = 3500 seconds, which is just under 3600 = 60 × 60 — so just under one hour. Starting at 4/:00, we land just before 5/:00, near 4/:58 or so. Our computed 4/:58/:15 matches. Also the fencepost adjustment (699 instead of 700) saves us 5 seconds — that's why the answer is 15 s and not 20 s past 4/:58, ruling out choice (B).

💡Key takeaway

This AMC 10 problem only needs Grade 4 "with n events there are n-1 gaps" — the 700th hiccup happens after 699 × 5 = 3495 seconds, which is 58 minutes 15 seconds, landing at 4/:58/:15.

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