AMC 8 · 2023 · #15

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

rateunit-conversionfraction-arithmetic dimensional-analysisidentify-subproblems ↑ Prerequisites: rateunit-conversion

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes $1$ minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to make a detour, walking $3$ blocks of equal length instead of $1$ block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time?

Pick an answer.

(A)

(B)

4.2

(C)

4.5

(D)

4.8

(E)

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Toolkit + CCSS Solution

Understand

Restated: Viswam normally walks $10$ equal city blocks (totaling half a mile) to school, taking $1$ minute per block. After $5$ blocks today, he hits a detour: the next corner (originally $1$ block away) now requires walking $3$ blocks. From the moment he starts the detour, what walking speed in mph will still let him arrive at school at his usual time?

Givens: Total route = $0.5$ mile = $10$ blocks of equal length; Normal pace = $1$ minute per block; Walked $5$ blocks before the detour at the normal pace; Detour: $3$ blocks replace what would have been $1$ block; Answer choices: (A) $4$, (B) $4.2$, (C) $4.5$, (D) $4.8$, (E) $5$ (mph)

Unknowns: The speed in mph he must walk, starting from the detour, to reach school at his usual time

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

This is a classic rate problem: $\text{speed} = \text{distance} / \text{time}$, with the catch that distance is in "blocks" and time is in minutes, but the answer wants miles per hour. Tool #8 (Analyze the Units) keeps the bookkeeping honest: convert blocks to miles ($1$ block $= 0.05$ mile) and minutes to hours ($5$ min $= \tfrac{1}{12}$ hr) before dividing. Tool #7 (Identify Subproblems) splits the journey into two clean pieces — the $5$ pre-detour blocks (uses up some time) and the new post-detour leg (the $3$-block detour plus the remaining $4$ original blocks) — so we can find the remaining time and remaining distance separately.

Execute — Answer: B

#8 Analyze the Units 4.MD.A.2 Step 1

Find the total time budget.
The usual walk is $10$ blocks at $1$ minute each, so Viswam's full schedule allows $10$ minutes from start to school.

$$10 \text{ blocks} \times 1 \tfrac{\text{min}}{\text{block}} = 10 \text{ min}$$

💡 Multiplying blocks by minutes-per-block cancels "blocks" and leaves "minutes" — a Grade 4 distance/time word-problem skill.

#7 Identify Subproblems 4.MD.A.2 Step 2

Subtract the time already used.
He has already walked $5$ blocks at the normal pace, so $5$ minutes are gone.
That leaves $10 - 5 = 5$ minutes to cover everything from the start of the detour to school.

$$10 - 5 = 5 \text{ min remaining}$$

💡 Splitting the journey into "already walked" and "still to walk" is the Tool #7 subproblems move.

#8 Analyze the Units 5.NBT.B.7 Step 3

Count the remaining blocks.
Originally $5$ blocks were left (blocks $6, 7, 8, 9, 10$).
The detour replaces block $6$ with a $3$-block path, so the post-detour walk is $3 + 4 = 7$ blocks.
Convert to miles using the fact that $10$ blocks $= 0.5$ mile, so $1$ block $= 0.05$ mile.

$$7 \text{ blocks} \times 0.05 \tfrac{\text{mile}}{\text{block}} = 0.35 \text{ mile}$$

💡 Multiplying $7$ by the decimal $0.05$ is a Grade 5 "decimals to hundredths" calculation.

#8 Analyze the Units 5.MD.A.1 Step 4

Convert the remaining time from minutes to hours so the speed will come out in miles per hour.
$5$ minutes is $\tfrac{5}{60} = \tfrac{1}{12}$ of an hour.

$$5 \text{ min} \times \tfrac{1 \text{ hr}}{60 \text{ min}} = \tfrac{1}{12} \text{ hr}$$

💡 Converting minutes into hours within the same time system is exactly the Grade 5 "convert standard measurement units" standard.

#8 Analyze the Units 6.RP.A.3 Step 5

Apply the rate relationship $\text{speed} = \dfrac{\text{distance}}{\text{time}}$ with consistent units (miles and hours).
Dividing by $\tfrac{1}{12}$ is the same as multiplying by $12$.

$$\text{speed} = \dfrac{0.35 \text{ mi}}{\tfrac{1}{12} \text{ hr}} = 0.35 \times 12 = 4.2 \text{ mph} \;\Rightarrow\; \textbf{(B)}$$

💡 Computing a unit rate (miles per hour) from a distance and a time is Grade 6 rate reasoning.

[1] #8 4.MD.A.2 Find the total time budget. The usual walk is $10$ blocks at $1$ minute each, so

[2] #7 4.MD.A.2 Subtract the time already used. He has already walked $5$ blocks at the normal p

[3] #8 5.NBT.B.7 Count the remaining blocks. Originally $5$ blocks were left (blocks $6, 7, 8, 9,

[4] #8 5.MD.A.1 Convert the remaining time from minutes to hours so the speed will come out in m

[5] #8 6.RP.A.3 Apply the rate relationship $\text{speed} = \dfrac{\text{distance}}{\text{time}}

Review

Reasonableness: Walking $5$ blocks at $1$ min/block is $0.25$ mi in $5$ min, which is $0.25 \times 12 = 3$ mph — a normal walking pace. The detour stretches the remaining trip from $5$ blocks to $7$ blocks but the time budget stays at $5$ minutes, so the speed must go up by a factor of $\tfrac{7}{5}$: $3 \times \tfrac{7}{5} = 4.2$ mph. That matches the answer (B) and is a sensible "brisk walk" speed, not an unrealistic run.

Alternative: Tool #6 (Guess and Check) on the choices: each candidate mph means a distance in $5$ min $= \tfrac{1}{12}$ hr equal to (mph)$/12$. The needed distance is $0.35$ mi, so the right mph is $0.35 \times 12 = 4.2$, which is exactly choice (B). The other choices give $\tfrac{4}{12} \approx 0.33$, $\tfrac{4.5}{12} = 0.375$, $0.4$, $\tfrac{5}{12} \approx 0.417$ — none equal $0.35$.

CCSS standards used (min grade 6)

4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money (Computing the total time for the usual route ($10 \times 1 = 10$ min) and the remaining time budget ($10 - 5 = 5$ min) from the distance/time word-problem setup.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Multiplying $7 \times 0.05 = 0.35$ to convert the remaining $7$ blocks into miles.)
5.MD.A.1 Convert among different-sized standard measurement units within a given system (Converting $5$ minutes into $\tfrac{1}{12}$ hour so the final speed can be expressed in miles per hour.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Computing the required walking speed as the unit rate $\text{speed} = \text{distance} / \text{time} = 0.35 / \tfrac{1}{12} = 4.2$ mph.)

⭐ This AMC 8 problem only needs Grade 6 rate reasoning — distance divided by time — that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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