AMC 10 · 2014 · #2

Grade 6 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation
fraction-arithmeticrateunit-conversion dimensional-analysisidentify-subproblems ↑ Prerequisites: fraction-arithmetic
📏 Medium solution 💡 2 insights
Problem
Roy's cat eats 13\frac{1}{3} of a can each morning and 14\frac{1}{4} of a can each evening. Before the Monday morning feeding, Roy opens a box holding 6 full cans, and the cat keeps to this schedule until the box runs out. Find the day of the week on which the cat eats the last of the food.

Pick an answer.

(A)
Tuesday
(B)
Wednesday
(C)
Thursday
(D)
Friday
(E)
Saturday

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Analyze the Units

This is a rate problem: cans are used up at a steady amount per day, so Tool #8 (Analyze the Units) turns 'cans' and 'cans per day' into a number of days. Tool #7 (Identify Subproblems) splits the day into its morning and evening pieces so the daily rate can be built first. Tool #5 (Look for a Pattern) uses the repeating 7-day week to convert a day count into an actual weekday.

1STEP 1

Combine one day of eating

One day is both feedings, so over twelfths: 13\frac{1}{3}+14\frac{1}{4}=412\frac{4}{12}+312\frac{3}{12}=712\frac{7}{12} of a can a day.

1/3+1/4=4/12+3/12=7/12
2STEP 2

Turn cans into days

Days = 6÷712\frac{7}{12} = 6×127\frac{12}{7} = 727\frac{72}{7}, more than 10 days but less than 11.

6÷7/12=6×12/7=72/7=10 2/7
3STEP 3

See how much is left after 10 days

Ten whole days eat 7012\frac{70}{12} cans, so day 11 starts with only 212\frac{2}{12}=16\frac{1}{6} of a can left.

6-10×7/12=72/12-70/12=2/12=1/6
4STEP 4

Finish during day 11's morning

That morning wants 412\frac{4}{12} but finds only 212\frac{2}{12}, so the box empties on day 11, with no evening feeding.

2/12 ＜ 4/12=1/3
5STEP 5

Convert day 11 to a weekday

Day 1 is Monday and so is day 8, so days 9, 10, 11 run Tuesday, Wednesday, Thursday — choice (C).

11=7+4 → Mon→Thu, (C)
Answer
Thursday
The estimate 72/7≈10.3 days already points between day 10 and day 11, and the exact leftover of 1/6 can confirms the food runs out early on day 11 rather than lasting into day 11's evening. Counting 11 days from Monday lands on Thursday, matching choice (C).
💡Key takeaway

Add up one day's eating, divide the total by it to get the number of days, then count those days around the week to land on the right weekday.

  • Combine one day of eating
  • Turn cans into days
  • See how much is left after 10 days
  • Finish during day 11's morning
  • Convert day 11 to a weekday

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