← Sensim Lab Sensim Math

AMC 8 · 2022 · #15

Grade 6 rate-ratio
Archetype Coordinate Plane Figure Computation
graph-readingrateslope-intercept systematic-enumerationidentify-subproblems ↑ Prerequisites: graph-readingrate
📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?

(A) 1(B) 2(C) 3(D) 4(E) 5\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5

Pick an answer.

(A)
1
(B)
2
(C)
3
(D)
4
(E)
5
View mode:

Toolkit + CCSS Solution

Understand

Restated: 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 $\dfrac{p}{w}$ is smallest. The available weights to choose from are $1, 2, 3, 4,$ or $5$ ounces.

Givens: $30$ points on a scatter plot, with weight on the $x$-axis (ounces) and price on the $y$-axis (dollars); Points cluster at weights $w = 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5$; Lowest-priced point at each integer weight (read from the plot): $(1, 1.2)$, $(2, 2.0)$, $(3, 2.5)$, $(4, 3.9)$, $(5, 4.5)$; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$ (ounces)

Unknowns: The weight (in ounces) of the pepper option whose price-per-ounce is the smallest

Understand

Restated: 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 $\dfrac{p}{w}$ is smallest. The available weights to choose from are $1, 2, 3, 4,$ or $5$ ounces.

Givens: $30$ points on a scatter plot, with weight on the $x$-axis (ounces) and price on the $y$-axis (dollars); Points cluster at weights $w = 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5$; Lowest-priced point at each integer weight (read from the plot): $(1, 1.2)$, $(2, 2.0)$, $(3, 2.5)$, $(4, 3.9)$, $(5, 4.5)$; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$ (ounces)

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #8 Analyze the Units, #1 Draw a Diagram

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 $\dfrac{\text{dollars}}{\text{ounce}}$ honest and explains why we may pick the lowest $y$ in each column. Tool #1 (Draw a Diagram) interprets $\dfrac{p}{w}$ 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.

Execute — Answer: C

#8 Analyze the Units 6.RP.A.2 Step 1
  • Translate "price per ounce" into a unit rate so each dot becomes a single number we can compare.
  • For a dot at $(w, p)$, the rate is $\dfrac{p \text{ dollars}}{w \text{ ounces}}$, with units of dollars per ounce.
$$\text{price per ounce} = \dfrac{p}{w}$$

💡 Reading "per ounce" as a unit rate is the Grade 6 "unit rate" idea — dollars on top, ounces on the bottom.

#3 Eliminate Possibilities 4.NF.C.7 Step 2
  • Shrink the search to one dot per column.
  • For a fixed weight $w$, the rate $\dfrac{p}{w}$ decreases as $p$ decreases, so in each integer-weight column only the lowest (cheapest) dot can possibly give the column's smallest rate.
  • The other dots in that column are eliminated.
$$\text{for fixed } w: \; \dfrac{p_1}{w} < \dfrac{p_2}{w} \;\Longleftrightarrow\; p_1 < p_2$$

💡 Comparing two decimals with the same divisor reduces to comparing their numerators — Grade 4 decimal comparison.

#1 Draw a Diagram 5.G.A.2 Step 3

Read the lowest dot in each integer-weight column straight off the scatter plot, giving one candidate point per answer choice.

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

💡 Reading $(x, y)$ coordinates of points off the plane is exactly the Grade 5 coordinate-graphing skill.

#8 Analyze the Units 5.NBT.B.7 Step 4

Compute the price-per-ounce for each candidate by dividing the price by the weight, keeping the result as a decimal in dollars per ounce.

$$\dfrac{1.2}{1}=1.20,\;\dfrac{2.0}{2}=1.00,\;\dfrac{2.5}{3}\approx 0.833,\;\dfrac{3.9}{4}=0.975,\;\dfrac{4.5}{5}=0.90$$

💡 Dividing a one-decimal-place price by a one-digit weight is the Grade 5 "divide decimals to hundredths" standard.

#3 Eliminate Possibilities 5.NBT.A.3 Step 5
  • Compare the five rates and pick the smallest.
  • The five values $1.20,\; 1.00,\; 0.833,\; 0.975,\; 0.90$ are smallest at $0.833$ dollars per ounce, which came from the $w = 3$ column.
$$\min(1.20,\,1.00,\,0.833,\,0.975,\,0.90) = 0.833\;\text{at}\;w = 3 \;\Rightarrow\; \textbf{(C)}$$

💡 Lining up decimals to thousandths and picking the smallest is a Grade 5 "compare decimals" task.

[1] #8 6.RP.A.2 Translate "price per ounce" into a unit rate so each dot becomes a single number
[2] #3 4.NF.C.7 Shrink the search to one dot per column. For a fixed weight $w$, the rate $\dfra
[3] #1 5.G.A.2 Read the lowest dot in each integer-weight column straight off the scatter plot,
[4] #8 5.NBT.B.7 Compute the price-per-ounce for each candidate by dividing the price by the weig
[5] #3 5.NBT.A.3 Compare the five rates and pick the smallest. The five values $1.20,\; 1.00,\; 0

Review

Reasonableness: The slope interpretation is a quick sanity check: $\dfrac{p}{w}$ 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).

Alternative: Tool #1 (Draw a Diagram) on its own: lightly draw rays from the origin through each candidate point. The shallowest ray immediately picks out $(3, 2.5)$ without any arithmetic, because slope $= \dfrac{p}{w} = $ price per ounce. This visual check confirms (C) and is the geometric counterpart of the numerical comparison.

CCSS standards used (min grade 6)

  • 6.RP.A.2 Understand the concept of a unit rate and use rate language (Reading "price per ounce" as the unit rate $\dfrac{p \text{ dollars}}{w \text{ ounces}}$ for each scatter-plot point.)
  • 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size (Eliminating all but the lowest-priced dot in each weight column by noticing that with the same divisor, the smaller price gives the smaller rate.)
  • 5.G.A.2 Represent real-world and mathematical problems by graphing points (Reading the $(w, p)$ coordinates of the cheapest dot in each integer-weight column straight off the scatter plot.)
  • 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Computing each candidate's price-per-ounce by dividing the decimal price by the whole-number weight (e.g., $2.5 \div 3 \approx 0.833$).)
  • 5.NBT.A.3 Read, write, and compare decimals to thousandths (Lining up the five rates to thousandths and selecting the smallest, which identifies the $3$-ounce option as the best deal.)

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

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

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