AMC 10 · 2009 · #1

Grade 4 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain
unit-conversionmulti-digit-arithmeticdimensional-analysis dimensional-analysisidentify-subproblems ↑ Prerequisites: multi-digit-arithmetic
📏 Medium solution 💡 2 insights
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Problem
A flight leaves in the morning and lands in the afternoon of the same day with no time-zone change. Find the sum of the hours and the minutes in its length.

Pick an answer.

(A)
46
(B)
47
(C)
50
(D)
53
(E)
54
How to solve
Strategy Analyze the Units

Two units are fighting here: hours and minutes, on a clock that restarts at 12 and relabels itself AM/PM. Mixed units are where the mistakes live, so I first drop to a single unit — minutes counted from midnight — where subtraction is ordinary subtraction and the AM/PM relabelling cannot bite. That gives one number of minutes. The second, separate subproblem is turning that number back into h hours and m minutes. That step is the one that actually needs care: h + m has no meaning at all unless exactly one pair (h, m) fits the rule 0 ＜ m ＜ 60, so I show that only one pair does rather than assuming it.

1STEP 1

Check the clock is trustworthy

The afternoon time needs converting before anything else.

10:34 AM → 10:34, 1:18 PM → (1+12):18 = 13:18
2STEP 2

Put both times on one number line

Minutes put both times on one scale.

10:34 → 10 × 60 + 34 = 634, 13:18 → 13 × 60 + 18 = 798
3STEP 3

Subtract to get the flight length

Subtracting gives 164 minutes.

798 - 634 = 164 minutes
4STEP 4

Split 164 into hours and minutes — only one way

Splitting back has only one legal answer.

164 = 60 × 2 + 44, 0 ＜ 44 ＜ 60 ⟹ h = 2, m = 44 (unique)
5STEP 5

Add the two counts

Adding the two counts gives 46, choice (B).

h + m = 2 + 44 = 46
Answer
46
Run the claim forward instead of backward: leave at 10:34 AM, add 2 hours to reach 12:34 PM, then add 44 minutes. 34 + 44 = 78 minutes, which is 60 + 18, so the hour rolls over to 1 PM with 18 minutes left: 1:18 PM. That is the stated arrival, so (h, m) = (2, 44) is confirmed independently of the subtraction. The size is sensible too — Newark to Miami is a bit under three hours. The wrong choices are all reachable by mishandling exactly one of the two counts. Choice (B) 47 is the classic double count: the hour hand moves from 10 to 1, which looks like 3 hours, and adding 44 minutes on top counts the partial hour twice; the true hour count is 2 because the third hour is only 44 minutes deep. Choices (C) 50, (D) 53, and (E) 54 all require h + m ≠ 46, which by the uniqueness argument in step 4 means either an hour count other than 2 or a minute count other than 44 — and both are ruled out by the forward check above.
💡Key takeaway

Turn both clock times into minutes after midnight, subtract, and then take away 60 as many times as you can — the hours you removed and the leftover under 60 are the only pair that fits.

  • Check the clock is trustworthy
  • Put both times on one number line
  • Subtract to get the flight length
  • Split 164 into hours and minutes — only one way
  • Add the two counts

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