Competition · AMC preparation · step 4 of 4

AMC 8 · 2022 · #15

Grade 6 rate-ratio

Archetype Coordinate Plane Figure Computation └ Sub-archetype Coordinate Graph Slope as Rate

graph-readingrateslope-intercept systematic-enumerationidentify-subproblems ↑ Prerequisites: graph-readingrate

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A scatter plot shows 30 black-pepper options, each plotted as a point (w, p) where w is the weight in ounces (the x-axis) and p is the price in dollars (the y-axis). Find the weight (in ounces) of the option whose price-per-ounce

\frac{p}{w}

is smallest. The available weights to choose from are 1, 2, 3, 4, or 5 ounces.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

The five answer choices are exactly the integer x-values in the scatter plot, so Tool #3 (Eliminate Possibilities) turns the whole problem into a tiny finite contest: pick the cheapest dot in each of the five columns, compute its price-per-ounce, and keep the smallest. Tool #8 (Analyze the Units) keeps the rate dollars/ounce honest and explains why we may pick the lowest y in each column. Tool #1 (Draw a Diagram) interprets p/w geometrically as the slope of the line from the origin to (w, p): the flattest such line wins, which is a quick visual sanity check on the scatter plot.

1STEP 1

Set up the unit rate

Read "price per ounce" as a unit rate: for a dot at (w, p) it is $\frac{p}{w}$ , in dollars per ounce.

price per ounce = p/w

2STEP 2

Keep the lowest dot per column

For a fixed weight, a lower price means a lower $\frac{p}{w}$ , so keep only the cheapest dot in each column.

for fixed w: p₁/w ＜ p₂/w ⟺ p₁ ＜ p₂

3STEP 3

Read the five candidate points

Read the lowest dot in each integer column: (1, 1.2), (2, 2.0), (3, 2.5), (4, 3.9), (5, 4.5).

(1, 1.2), (2, 2.0), (3, 2.5), (4, 3.9), (5, 4.5)

4STEP 4

Divide price by weight

Divide price by weight for each candidate: 1.20, 1.00, 0.833, 0.975, 0.90 dollars per ounce.

1.2/1=1.20, 2.0/2=1.00, 2.5/3≈ 0.833, 3.9/4=0.975, 4.5/5=0.90

5STEP 5

Pick the smallest rate

Comparing the five rates, the smallest is 0.833 dollars per ounce, from the w = 3 column.

min(1.20, 1.00, 0.833, 0.975, 0.90) = 0.833 at w = 3 → (C)

Answer

The slope interpretation is a quick sanity check:

\frac{p}{w}

is the slope from the origin to (w, p), so the cheapest-per-ounce option is the dot whose line from (0,0) is the flattest. Among the cheapest dots in each integer column, (3, 2.5) visibly sits below the line y = x the most, while (1, 1.2) sits well above it. The numbers agree: 0.833 ＜ 0.90 ＜ 0.975 ＜ 1.00 ＜ 1.20, so a 3-ounce package at $2.50 truly is the best deal — matching answer (C).

💡Key takeaway

This AMC 8 problem only needs Grade 6 unit-rate thinking — dollars per ounce — that you already know!

Set up the unit rate
Keep the lowest dot per column
Read the five candidate points
Divide price by weight
Pick the smallest rate

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Set up the unit rate

Keep the lowest dot per column

Read the five candidate points

Divide price by weight

Pick the smallest rate

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