AMC 8 · 2022 · #7

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

rateunit-conversionmulti-digit-arithmetic dimensional-analysisidentify-subproblems ↑ Prerequisites: unit-conversionmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

In the 1990s the fastest dial-up internet sent data at about 56 kilobits per second. A song is 4.2 megabytes large, and 1 megabyte is 8000 kilobits. Roughly how many minutes would the song have taken to download at 56 kilobits per second?

Pick an answer.

(A)

0.6

(B)

(C)

1800

(D)

7200

(E)

36000

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

How to solve

Strategy Analyze the Units

The core relationship is time = size / speed, but the size is in megabytes, the speed is in kilobits per second, and the answer is in minutes. Tool #8 (Analyze the Units) is the natural primary tool: track "MB", "kb", "s", and "min" through every operation so the wrong unit can never sneak into the final answer. Tool #7 (Identify Subproblems) splits the calculation into three clean pieces — (1) convert the file size to kilobits, (2) divide by the rate to get seconds, (3) convert seconds to minutes — so each step does exactly one job and the arithmetic stays small.

1STEP 1

Multiply the size in MB by the given 8000 kb/MB factor, so the song is 33,600 kb.

4.2 MB × 8000 kb/MB = 33,600 kb

2STEP 2

Divide by the rate: since 56 × 6 = 336, (33,600 kb)/(56 kb/s) = 600 s.

(33,600 kb)/(56 kb/s) = (56 × 6 × 100)/56 s = 600 s

3STEP 3

Divide the seconds by 60 to get minutes: 600 s ÷ 60 = 10 min.

600 s × (1 min)/(60 s) = 10 min → (B)

4STEP 4

Cross-check with time = size / speed: units cancel MB → kb → s → min, leaving only minutes.

(4.2 MB × 8000 kb/MB)/(56 kb/s) × (1 min)/(60 s) = 10 min

Answer

10 minutes feels right for a song on 1990s dial-up: today the same 4.2 MB song downloads in well under a second over broadband, but 56 kbps is about 1000 times slower than a basic modern connection, so multiplying a fraction of a second by roughly 1000 lands in the ten-minute range. The other choices fail an order-of-magnitude check: (A) 0.6 min = 36 s is way too fast, while (C) 1800 min = 30 hr and (D) 7200 min = 5 days are way too slow for a single song.

💡Key takeaway

This AMC 8 problem only needs Grade 6 rate reasoning — chaining unit conversions until just "minutes" is left — that you already know!

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