← Sensim Lab Sensim Math

AMC 8 · 2009 · #3

Grade 6 rate-ratio
Archetype Arithmetic Expression Decomposition
rategraph-readingpattern-recognitionunit-conversion pattern-recognitiondimensional-analysis ↑ Prerequisites: multi-digit-arithmeticunit-conversion
📏 Medium solution 💡 3 insights 📊 Diagram

Problem

The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?

Pick an answer.

(A)
5
(B)
5.5
(C)
6
(D)
6.5
(E)
7
View mode:

Toolkit + CCSS Solution

Understand

Restated: A graph shows Suzanna's biking distance over time: at $5, 10, 15, 20$ minutes she has gone $1, 2, 3, 4$ miles respectively. If she keeps that same constant speed for a full half hour ($30$ minutes), how many miles will she have ridden in total?

Givens: Plotted points on the graph: $(5\text{ min}, 1\text{ mi})$, $(10, 2)$, $(15, 3)$, $(20, 4)$; Her speed is constant (the points lie on a straight line through the origin); Total riding time = half an hour = $30$ minutes; Answer choices: (A) $5$, (B) $5.5$, (C) $6$, (D) $6.5$, (E) $7$ (miles)

Unknowns: The total distance in miles after $30$ minutes at the same constant speed

Understand

Restated: A graph shows Suzanna's biking distance over time: at $5, 10, 15, 20$ minutes she has gone $1, 2, 3, 4$ miles respectively. If she keeps that same constant speed for a full half hour ($30$ minutes), how many miles will she have ridden in total?

Givens: Plotted points on the graph: $(5\text{ min}, 1\text{ mi})$, $(10, 2)$, $(15, 3)$, $(20, 4)$; Her speed is constant (the points lie on a straight line through the origin); Total riding time = half an hour = $30$ minutes; Answer choices: (A) $5$, (B) $5.5$, (C) $6$, (D) $6.5$, (E) $7$ (miles)

Plan

Primary tool: #5 Look for a Pattern

Secondary: #8 Analyze the Units

The graph hands us a clean numerical pattern: every extra $5$ minutes adds exactly $1$ mile. Tool #5 (Look for a Pattern) lets us extend that rule from the graph's last point at $20$ min straight up to $30$ min without any algebra. Tool #8 (Analyze the Units) is the safety check — we are mixing "minutes" with "half an hour", so we make sure both are in the same unit before extending the pattern.

Execute — Answer: C

#8 Analyze the Units 5.MD.A.1 Step 1
  • Convert the riding time into the same unit the graph uses.
  • The graph is labeled in minutes, and half an hour is $30$ minutes.
$$\tfrac{1}{2} \text{ hour} \times \tfrac{60 \text{ min}}{1 \text{ hour}} = 30 \text{ min}$$

💡 Lining up units (minutes with minutes) is a Grade 5 measurement-conversion move.

#5 Look for a Pattern 4.OA.C.5 Step 2
  • Read the pattern off the four plotted points: $(5, 1), (10, 2), (15, 3), (20, 4)$.
  • Each $+5$ minutes adds exactly $+1$ mile.
$5 \to 1,\;\; 10 \to 2,\;\; 15 \to 3,\;\; 20 \to 4$ \;(every $+5$ min gives $+1$ mi)

💡 Spotting the constant "$+5$ min, $+1$ mi" repetition is exactly the Grade 4 pattern-rule skill.

#5 Look for a Pattern 4.OA.C.5 Step 3
  • Extend the pattern past the graph.
  • After $20$ min Suzanna has gone $4$ miles.
  • Add two more $5$-minute chunks to reach $30$ min, gaining $1$ mile each time.
$$20 \to 4,\;\; 25 \to 5,\;\; 30 \to 6$$

💡 Continuing the rule beyond the given data points is the second half of "find and use a pattern".

#8 Analyze the Units 6.RP.A.3 Step 4
  • Cross-check with the rate.
  • The pattern $5$ min $\to 1$ mi means a unit rate of $\tfrac{1}{5}$ mile per minute.
  • Multiplying by $30$ minutes gives the same answer.
$$\tfrac{1}{5} \tfrac{\text{mi}}{\text{min}} \times 30 \text{ min} = 6 \text{ mi} \;\Rightarrow\; \textbf{(C)}$$

💡 Distance $=$ rate $\times$ time, with units canceling cleanly, is Grade 6 rate reasoning.

[1] #8 5.MD.A.1 Convert the riding time into the same unit the graph uses. The graph is labeled
[2] #5 4.OA.C.5 Read the pattern off the four plotted points: $(5, 1), (10, 2), (15, 3), (20, 4)
[3] #5 4.OA.C.5 Extend the pattern past the graph. After $20$ min Suzanna has gone $4$ miles. Ad
[4] #8 6.RP.A.3 Cross-check with the rate. The pattern $5$ min $\to 1$ mi means a unit rate of $

Review

Reasonableness: At $20$ minutes she has $4$ miles, so her speed averages $\tfrac{4}{20} = 0.2$ mile per minute, which is $0.2 \times 60 = 12$ mph — a realistic easy bike pace. In $30$ minutes at $12$ mph she covers $\tfrac{1}{2} \times 12 = 6$ miles. That matches the answer (C) $6$ and rules out the close decoys $5.5$, $6.5$.

Alternative: Tool #3 (Eliminate Possibilities) with the rate $\tfrac{1}{5}$ mi/min: in $30$ min she must cover $30/5 = 6$ mi exactly, so (A) $5$ (only $25$ min worth), (B) $5.5$, (D) $6.5$, (E) $7$ are all off by a whole-number ratio and get crossed out, leaving (C). Tool #1 (Diagram) also works: just continue the straight line on the graph two more dots to $(25, 5)$ and $(30, 6)$.

CCSS standards used (min grade 6)

  • 5.MD.A.1 Convert among different-sized standard measurement units within a given system (Turning "half an hour" into $30$ minutes so the time unit matches the graph's $x$-axis.)
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule (Reading the rule "$+5$ min $\to +1$ mi" off the four plotted points and extending it to $(25, 5)$ and $(30, 6)$.)
  • 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Verifying the answer with the unit rate $\tfrac{1}{5}$ mile per minute: $\tfrac{1}{5} \times 30 = 6$ miles.)

⭐ If you can read "every $5$ minutes she gains $1$ mile" off a graph, you can already finish this AMC 8 problem — that's Grade 4 pattern-spotting plus a quick Grade 6 rate check.

⭐ If you can read "every $5$ minutes she gains $1$ mile" off a graph, you can already finish this AMC 8 problem — that's Grade 4 pattern-spotting plus a quick Grade 6 rate check.

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