AMC 8 · 2023 · #15

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

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

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

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?

Pick an answer.

(A)

(B)

4.2

(C)

4.5

(D)

4.8

(E)

AMC 8 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

This is a classic rate problem: speed = distance / 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 = $\frac{1}{12}$ 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.

1STEP 1

The usual walk is 10 blocks at 1 minute each, so the whole trip must fit inside 10 minutes.

10 blocks × 1 min/block = 10 min

2STEP 2

The first 5 blocks used 5 of the 10 minutes, so only 5 minutes remain to reach school from the detour.

10 - 5 = 5 min remaining

3STEP 3

The detour swaps 1 block for 3, so 7 blocks remain; at 0.05 mile each that is 0.35 mile.

7 blocks × 0.05 mile/block = 0.35 mile

4STEP 4

Convert the 5 minutes to hours so the answer lands in mph: $\frac{5}{60}$ = $\frac{1}{12}$ hour.

5 min × (1 hr)/(60 min) =

\frac{1}{12}

5STEP 5

Divide 0.35 mile by $\frac{1}{12}$ hour (same as × 12): speed = 4.2 mph → (B).

speed = (0.35 mi)/(

\frac{1}{12}

hr) = 0.35 × 12 = 4.2 mph → (B)

Answer

4.2

Walking 5 blocks at 1 min/block is 0.25 mi in 5 min, which is 0.25 × 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

\frac{7}{5}

: 3 ×

\frac{7}{5}

= 4.2 mph. That matches the answer (B) and is a sensible "brisk walk" speed, not an unrealistic run.

💡Key takeaway

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

More like this