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2014 AMC 10A Problem 2

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2014 AMC 10A Problem 2AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

Roy's cat eats 13\frac{1}{3} of a can of cat food every morning and 14\frac{1}{4} of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing 66 cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?

  1. A)

    Tuesday\text{Tuesday}

  2. B)

    Wednesday\text{Wednesday}

  3. C)

    Thursday\text{Thursday}

  4. D)

    Friday\text{Friday}

  5. E)

    Saturday\text{Saturday}

Answer

C

Key insight

The cat eats 7/12 of a can per day, so 6 cans last 10 full days with 2/12 left over, which runs out on the 11th morning.

Solution

Each full day the cat eats
13+14=412+312=712\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}
of a can. Measure everything in twelfths: the box holds 6=72126 = \frac{72}{12} cans.

After 1010 full days the cat has eaten 7012\frac{70}{12} cans, leaving 212\frac{2}{12}. That is less than the morning portion of 412\frac{4}{12}, so the food runs out during the morning of day 1111.

Day 11 is Monday, so day 88 is Monday again and day 1111 is Thursday.

The answer is (C) Thursday\boxed{\textbf{(C)}\ \text{Thursday}}.

Why this works

Combine the two feedings into a single daily rate, then divide the total supply by that rate. The quotient tells you how many whole days are covered; the remainder tells you whether the food runs out in the morning or the evening of the following day. Days of the week are just arithmetic mod 77 once you have the day number.

The trap

Stopping at 'about 10 days' and answering Wednesday, without noticing that a small remainder still has to be eaten on the next morning.

Common mistakes

  • Stopping at 'about 10 days' and answering Wednesday, without noticing that a small remainder still has to be eaten on the next morning.
  • Counting Monday as day 00 instead of day 11, which shifts the answer to Friday.

Techniques

Set up the equation/formula and compute; no special trick needed