AMC 10 · 2017 · #4
Easy mode Grade 5There are 30 toys on the floor. Every 30 seconds, Mia's mom puts 3 toys into the toy box. But right after each 30 seconds is up, Mia takes 2 toys back out. How many minutes will it take until all 30 toys are in the box for the first time?
Pick an answer.
AMC 10 2017 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: There are $30$ toys on the floor. Every $30$ seconds Mia's mom puts $3$ toys into the box, and right after each of those $30$-second stretches Mia pulls $2$ toys back out. Find how many minutes pass before the box first holds all $30$ toys.
Givens: There are $30$ toys to put away; Mom adds $3$ toys to the box every $30$ seconds; Immediately after each $30$-second stretch, Mia removes $2$ toys; Answer choices: (A) $13.5$, (B) $14$, (C) $14.5$, (D) $15$, (E) $15.5$
Unknowns: The time, in minutes, until the box first contains all $30$ toys
Understand
Restated: There are $30$ toys on the floor. Every $30$ seconds Mia's mom puts $3$ toys into the box, and right after each of those $30$-second stretches Mia pulls $2$ toys back out. Find how many minutes pass before the box first holds all $30$ toys.
Givens: There are $30$ toys to put away; Mom adds $3$ toys to the box every $30$ seconds; Immediately after each $30$-second stretch, Mia removes $2$ toys; Answer choices: (A) $13.5$, (B) $14$, (C) $14.5$, (D) $15$, (E) $15.5$
Plan
Primary tool: #8 Analyze the Units
Secondary: #5 Look for a Pattern, #14 Extreme Principle
The problem is a rate-of-progress question — toys per $30$ seconds — so Tool #8 (Analyze the Units) anchors it: track the net toys added each $30$-second block. Tool #5 (Look for a Pattern) turns that net change into a repeating $+1$ step we can multiply. The trap is the very last block, where the job finishes before Mia can remove anything, so Tool #14 (Extreme Principle) forces us to inspect that boundary block separately instead of applying the pattern blindly.
Execute — Answer: B
4.OA.A.3 Step 1 Net toys per 30 seconds
- Look at one full cycle.
- In $30$ seconds Mom adds $3$ toys, then Mia takes $2$ out.
- The box gains $3 - 2 = 1$ toy every $30$ seconds.
💡 Adding then removing in the same block is one combined step, so only the leftover $+1$ actually counts.
4.OA.A.3 Step 2 The last push is different
- Mia only removes toys after a stretch finishes.
- The moment Mom's $+3$ first brings the box to $30$, the toys are all in and the task is over — Mia never gets to pull $2$ back out.
- So the final block adds a full $+3$, not a net $+1$.
- That means the $+1$ pattern only has to carry the box up to $27$; Mom's last push of $3$ finishes the job.
💡 The clock stops the instant the box is full, so the last handful of toys never has a chance to come back out.
4.OA.A.3 Step 3 Count the blocks of time
- Reaching $27$ toys at $+1$ per block takes $27$ blocks of $30$ seconds, which is $27 \times 30 = 810$ seconds.
- Then one more $30$-second block holds Mom's final $+3$ that pushes the box from $27$ to $30$.
- Total time is $810 + 30 = 840$ seconds.
💡 Each net toy costs exactly one $30$-second block, plus one extra block for the closing push.
5.MD.A.1 Step 4 Convert seconds to minutes
- Change $840$ seconds into minutes by dividing by $60$, since one minute is $60$ seconds.
- That gives $840 \div 60 = 14$ minutes, which is choice (B).
💡 Seconds and minutes measure the same time, so dividing by $60$ just renames the amount.
4.OA.A.3 Look at one full cycle. In $30$ seconds Mom adds $3$ toys, then Mia takes $2$ ou 4.OA.A.3 Mia only removes toys after a stretch finishes. The moment Mom's $+3$ first brin 4.OA.A.3 Reaching $27$ toys at $+1$ per block takes $27$ blocks of $30$ seconds, which is 5.MD.A.1 Change $840$ seconds into minutes by dividing by $60$, since one minute is $60$ Review
Reasonableness: If you forgot the boundary block and used net $+1$ all the way to $30$, you would get $30 \times 30 = 900$ seconds $= 15$ minutes — that is choice (D), the classic trap. Stopping the pattern at $27$ and adding Mom's final full push shaves off exactly one removal, landing at $14$ minutes, which matches (B) and is a touch under $15$, as expected.
Alternative: Track the box minute by minute. After each $30$-second cycle the count runs $1, 2, 3, \dots$; at $13$ minutes ($26$ cycles) the box holds $26$, at $13.5$ minutes ($27$ cycles) it holds $27$. The next $30$ seconds is Mom's push to $30$, ending at $14$ minutes — confirming (B).
CCSS standards used (min grade 5)
4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Finding the net $+1$ toys per cycle, handling the final full push separately, and multiplying $27$ cycles by $30$ seconds plus one more block.)5.MD.A.1Convert among different-sized standard measurement units within a given system (Converting the total of $840$ seconds into $14$ minutes by dividing by $60$.)
⭐ When something is added then taken away each round, track the net change — but check the final round, because the job can finish before the take-away happens.
⭐ When something is added then taken away each round, track the net change — but check the final round, because the job can finish before the take-away happens.
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