AMC 10 · 2025 · #1

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

floor-functionratemulti-digit-arithmetic dimensional-analysis ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 1 insight

Problem

The instructions on a $350$ -gram bag of coffee beans say that proper brewing of a large mug of pour-over coffee requires $20$ grams of coffee beans. What is the greatest number of properly brewed large mugs of coffee that can be made from the coffee beans in that bag?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2025 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.

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Toolkit + CCSS Solution

Understand

Restated: A bag holds $350$ grams of coffee beans. Each properly brewed large mug uses exactly $20$ grams. Find the greatest number of full mugs you can make from the bag.

Givens: The bag contains $350$ grams of coffee beans; Each properly brewed large mug requires $20$ grams of beans; Answer choices: (A) $16$, (B) $17$, (C) $18$, (D) $19$, (E) $20$

Unknowns: The largest whole number of full mugs the $350$ grams can make

Understand

Restated: A bag holds $350$ grams of coffee beans. Each properly brewed large mug uses exactly $20$ grams. Find the greatest number of full mugs you can make from the bag.

Givens: The bag contains $350$ grams of coffee beans; Each properly brewed large mug requires $20$ grams of beans; Answer choices: (A) $16$, (B) $17$, (C) $18$, (D) $19$, (E) $20$

Plan

Primary tool: #8 Analyze the Units

Secondary: #14 Extreme Principle

The two numbers carry units: $350$ grams total and $20$ grams per mug. Tool #8 (Analyze the Units) shows that grams $\div$ grams-per-mug cancels to give a count of mugs, so a single division does the work. Because the question asks for the greatest number of full mugs and leftover beans cannot fill another mug, Tool #14 (Extreme Principle) says to push to the boundary and round the quotient down to a whole number rather than up.

Execute — Answer: B

#8 Analyze the Units 4.OA.A.3 Step 1

Turn the words into a division

You have $350$ grams and each mug eats up $20$ grams.
To find how many $20$-gram portions fit inside $350$ grams, divide the total by the amount per mug.
The units confirm the setup: grams $\div$ (grams per mug) leaves just mugs.

$$\text{mugs} = \dfrac{350\ \text{grams}}{20\ \text{grams/mug}}$$

💡 Sharing a total into equal-size portions is exactly what division counts.

#8 Analyze the Units 5.NBT.B.6 Step 2

Divide 350 by 20

Carry out the division.
$20\times17=340$ and $20\times18=360$, so $350$ sits between them.
That means $350\div20=17$ with $10$ grams left over: $350-340=10$.

$$350\div20 = 17\ \text{remainder}\ 10$$

💡 Seventeen full scoops of $20$ use $340$ grams and leave a small remainder behind.

#14 Extreme Principle 4.OA.A.3 Step 3

Round down to whole mugs

The leftover $10$ grams is less than the $20$ grams a mug needs, so it cannot make an $18$th mug.
The greatest number of full mugs is the whole part of the quotient, $17$.
Trying $18$ would need $360$ grams, more than the bag holds, so $17$ is the boundary.
The answer is $(\text{B})\ 17$.

$$\left\lfloor \dfrac{350}{20} \right\rfloor = 17\;\Rightarrow\;\textbf{(B)}$$

💡 You cannot brew a fraction of a mug, so any leftover beans are simply wasted.

[1] #8 4.OA.A.3 You have $350$ grams and each mug eats up $20$ grams. To find how many $20$-gram

[2] #8 5.NBT.B.6 Carry out the division. $20\times17=340$ and $20\times18=360$, so $350$ sits bet

[3] #14 4.OA.A.3 The leftover $10$ grams is less than the $20$ grams a mug needs, so it cannot ma

Review

Reasonableness: Check the boundary both ways: $17$ mugs use $17\times20=340$ grams, which fits inside $350$; $18$ mugs would need $18\times20=360$ grams, which overshoots the $350$ available. So $17$ is exactly the largest count that stays within the bag, matching (B). Choices (C)–(E) all demand more than $350$ grams, and (A) $16$ wastes a whole extra mug's worth of beans.

Alternative: Test the choices directly. Multiply $20$ by each: $16\to320$, $17\to340$, $18\to360$. The largest product that does not pass $350$ is $340$, from $17$ mugs, which again gives (B).

CCSS standards used (min grade 5)

5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors (Dividing $350$ grams by the two-digit divisor $20$ grams per mug to get $17$ remainder $10$.)
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Setting up the division from the word problem and interpreting the remainder so the answer is rounded down to whole mugs.)

⭐ When you split a total into fixed portions and only whole ones count, divide and then round down — leftovers that are too small to fill another portion just get left behind.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Turn the words into a division

Divide 350 by 20

Round down to whole mugs

Review

CCSS standards used (min grade 5)

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