AMC 8 · 2022 · #7

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition

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

📏 Short solution 💡 2 insights

Problem

When the World Wide Web first became popular in the $1990$ s, download speeds reached a maximum of about $56$ kilobits per second. Approximately how many minutes would the download of a $4.2$ -megabyte song have taken at that speed? (Note that there are $8000$ kilobits in a megabyte.)

$\textbf{(A) } 0.6 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 1800 \qquad \textbf{(D) } 7200 \qquad \textbf{(E) } 36000$

Pick an answer.

(A)

0.6

(B)

(C)

1800

(D)

7200

(E)

36000

View mode:

Toolkit + CCSS Solution

Understand

Restated: In the $1990$s 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?

Givens: Download speed = $56$ kilobits per second (kbps); Song size = $4.2$ megabytes (MB); Unit conversion: $1$ MB $= 8000$ kilobits (kb); Answer choices (minutes): (A) $0.6$, (B) $10$, (C) $1800$, (D) $7200$, (E) $36000$

Unknowns: The approximate download time of the song, expressed in minutes

Understand

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

The core relationship is $\text{time} = \text{size} / \text{speed}$, 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.

Execute — Answer: B

#8 Analyze the Units 5.NBT.B.7 Step 1

Convert the song's size from megabytes to kilobits.
The given conversion factor is $8000$ kb per $1$ MB, so multiply the size in MB by $8000$.

$$4.2 \text{ MB} \times 8000 \tfrac{\text{kb}}{\text{MB}} = 33{,}600 \text{ kb}$$

💡 Multiplying a decimal ($4.2$) by a whole number ($8000$) is exactly the Grade 5 "decimals to hundredths" skill, and the "MB" units cancel to leave "kb".

#7 Identify Subproblems 5.NBT.B.6 Step 2

Divide the file size in kilobits by the download rate to get the time in seconds.
Spotting that $56 \times 6 = 336$ makes the arithmetic clean: $33{,}600 = 336 \times 100 = 56 \times 6 \times 100$.

$$\dfrac{33{,}600 \text{ kb}}{56 \tfrac{\text{kb}}{\text{s}}} = \dfrac{56 \times 6 \times 100}{56} \text{ s} = 600 \text{ s}$$

💡 Dividing a four-digit number by a two-digit number is the Grade 5 long-division standard; the "kb" cancels and only "s" is left, which Tool #8 confirms is correct.

#8 Analyze the Units 5.MD.A.1 Step 3

Convert the download time from seconds to minutes.
Since $1$ minute $= 60$ seconds, divide the total seconds by $60$.

$$600 \text{ s} \times \tfrac{1 \text{ min}}{60 \text{ s}} = 10 \text{ min} \;\Rightarrow\; \textbf{(B)}$$

💡 Converting between standard units of time within one system (seconds $\to$ minutes) is the Grade 5 measurement-conversion standard.

#8 Analyze the Units 6.RP.A.3 Step 4

Cross-check the result against the overall rate $\text{time} = \text{size} / \text{speed}$.
The full chain of units is $\text{MB} \to \text{kb} \to \text{s} \to \text{min}$, every "unwanted" unit cancels, and only "min" is left, which matches what the problem asks for.

$$\dfrac{4.2 \text{ MB} \times 8000 \tfrac{\text{kb}}{\text{MB}}}{56 \tfrac{\text{kb}}{\text{s}}} \times \tfrac{1 \text{ min}}{60 \text{ s}} = 10 \text{ min}$$

💡 Treating $56$ kb/s as a unit rate and using it together with another rate ($8000$ kb/MB) is Grade 6 rate reasoning across multiple unit conversions.

[1] #8 5.NBT.B.7 Convert the song's size from megabytes to kilobits. The given conversion factor

[2] #7 5.NBT.B.6 Divide the file size in kilobits by the download rate to get the time in seconds

[3] #8 5.MD.A.1 Convert the download time from seconds to minutes. Since $1$ minute $= 60$ secon

[4] #8 6.RP.A.3 Cross-check the result against the overall rate $\text{time} = \text{size} / \te

Review

Reasonableness: $10$ minutes feels right for a song on $1990$s 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.

Alternative: Tool #3 (Eliminate Possibilities) on the answer choices: convert each candidate back into seconds and check whether $33{,}600$ kb fits into that time at $56$ kb/s. Only (B) $10$ min $= 600$ s satisfies $56 \times 600 = 33{,}600$ kb, so all other choices are eliminated without a fresh calculation.

CCSS standards used (min grade 6)

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Multiplying the decimal file size by the whole-number conversion factor: $4.2 \times 8000 = 33{,}600$ kilobits.)
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors (Dividing the file size in kilobits by the download speed: $33{,}600 \div 56 = 600$ seconds.)
5.MD.A.1 Convert among different-sized standard measurement units within a given system (Converting the download time from seconds to minutes: $600 \text{ s} \div 60 = 10$ minutes.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Combining two rates ($8000$ kb per MB and $56$ kb per second) through unit-cancellation reasoning to chain MB $\to$ kb $\to$ s $\to$ min and obtain $10$ minutes.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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