AMC 10 · 2014 · #2
Grade 6 arithmeticRoy's cat eats 31 of a can of cat food every morning and 41 of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing 6 cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
Pick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A cat eats $\frac{1}{3}$ of a can each morning and $\frac{1}{4}$ of a can each evening. Starting on Monday morning with a full box of $6$ cans, find the day of the week on which the last of the food is eaten.
Givens: The cat eats $\frac{1}{3}$ of a can every morning; The cat eats $\frac{1}{4}$ of a can every evening; The box holds $6$ full cans; The first feeding is Monday morning; Answer choices: (A) Tuesday, (B) Wednesday, (C) Thursday, (D) Friday, (E) Saturday
Unknowns: The day of the week when the box becomes empty
Understand
Restated: A cat eats $\frac{1}{3}$ of a can each morning and $\frac{1}{4}$ of a can each evening. Starting on Monday morning with a full box of $6$ cans, find the day of the week on which the last of the food is eaten.
Givens: The cat eats $\frac{1}{3}$ of a can every morning; The cat eats $\frac{1}{4}$ of a can every evening; The box holds $6$ full cans; The first feeding is Monday morning; Answer choices: (A) Tuesday, (B) Wednesday, (C) Thursday, (D) Friday, (E) Saturday
Plan
Primary tool: #8 Analyze the Units
Secondary: #7 Identify Subproblems, #5 Look for a Pattern
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.
Execute — Answer: C
5.NF.A.1 Step 1 Combine one day of eating
- First find how much the cat eats in a whole day by adding the morning and evening amounts.
- The denominators $3$ and $4$ both divide $12$, so rewrite each fraction over $12$: $\frac{1}{3}=\frac{4}{12}$ and $\frac{1}{4}=\frac{3}{12}$.
- Adding gives $\frac{7}{12}$ of a can each day.
💡 Two feedings in one day are just one combined feeding, so add them into a single daily amount.
6.NS.A.1 Step 2 Turn cans into days
- The box has $6$ cans and the cat uses $\frac{7}{12}$ of a can per day, so the number of days is $6$ divided by $\frac{7}{12}$.
- Dividing by a fraction means multiplying by its reciprocal: $6\times\frac{12}{7}=\frac{72}{7}=10\frac{2}{7}$.
- So the food lasts a bit longer than $10$ full days but not a full $11$ days.
💡 Total amount divided by amount-per-day tells you how many days it stretches across.
5.NF.B.4 Step 3 See how much is left after 10 days
- After $10$ complete days the cat has eaten $10\times\frac{7}{12}=\frac{70}{12}=5\frac{5}{6}$ cans.
- Subtracting from the $6$ cans, the food left is $6-\frac{70}{12}=\frac{72}{12}-\frac{70}{12}=\frac{2}{12}=\frac{1}{6}$ of a can.
- So a small amount, $\frac{1}{6}$ of a can, remains at the start of day $11$.
💡 Whatever the cat has already eaten, subtract it from the full box to see what is still there.
4.NF.A.2 Step 4 Finish during day 11's morning
- On the morning of day $11$ the cat tries to eat $\frac{1}{3}=\frac{4}{12}$ of a can, but only $\frac{2}{12}$ of a can is left.
- Since $\frac{2}{12}<\frac{4}{12}$, that morning uses up everything that remains, so the box becomes empty during day $11$ (no evening feeding is needed).
💡 If what is left is less than a normal feeding, that very feeding empties the box.
4.OA.C.5 Step 5 Convert day 11 to a weekday
- Day $1$ is Monday, and weekdays repeat every $7$ days, so day $8$ is again Monday.
- Counting on: day $8$ Monday, day $9$ Tuesday, day $10$ Wednesday, day $11$ Thursday.
- The box empties on day $11$, which is Thursday, choice (C).
💡 Because the week repeats every $7$ days, drop full weeks and count only the leftover days from Monday.
5.NF.A.1 First find how much the cat eats in a whole day by adding the morning and evenin 6.NS.A.1 The box has $6$ cans and the cat uses $\frac{7}{12}$ of a can per day, so the nu 5.NF.B.4 After $10$ complete days the cat has eaten $10\times\frac{7}{12}=\frac{70}{12}=5 4.NF.A.2 On the morning of day $11$ the cat tries to eat $\frac{1}{3}=\frac{4}{12}$ of a 4.OA.C.5 Day $1$ is Monday, and weekdays repeat every $7$ days, so day $8$ is again Monda Review
Reasonableness: The estimate $\frac{72}{7}\approx10.3$ days already points between day $10$ and day $11$, and the exact leftover of $\frac{1}{6}$ 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).
Alternative: Build a running total by weekday: after Monday $\frac{7}{12}$, after Tuesday $\frac{14}{12}$, and so on, adding $\frac{7}{12}$ each day. The total first reaches or passes $6=\frac{72}{12}$ during the $11$th day (Thursday), since $\frac{70}{12}$ after day $10$ is still under $6$ but the next morning tips it over. Same answer, Thursday (C).
CCSS standards used (min grade 6)
5.NF.A.1Add and subtract fractions with unlike denominators (Adding $\frac{1}{3}+\frac{1}{4}$ to get the daily amount $\frac{7}{12}$.)6.NS.A.1Interpret and compute quotients of fractions (Dividing $6$ cans by $\frac{7}{12}$ can/day to get $\frac{72}{7}$ days.)5.NF.B.4Multiply a fraction or whole number by a fraction (Computing $10\times\frac{7}{12}$ and subtracting to find $\frac{1}{6}$ can left.)4.NF.A.2Compare two fractions with different numerators and denominators (Seeing that the leftover $\frac{2}{12}$ is less than a morning feeding $\frac{4}{12}$.)4.OA.C.5Generate and analyze a number or shape pattern that follows a given rule (Using the repeating $7$-day week to turn day $11$ into Thursday.)
⭐ 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.
⭐ 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.
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