Compare Rates with Different Totals

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Printer A makes 6 pages in 2 minutes; B makes 16 in 5 minutes. Which statement follows from these constant-rate models? Candidate answers: (A) A is faster because its time is smaller (B) B is faster: 16/5 versus 3 pages/minute (C) The page totals alone determine the faster rate (D) The rates are equal.
• Independent check inputs: Printer A makes 9 pages in 3 minutes; B makes 20 in 6 minutes. Which statement follows from these constant-rate models? Candidate answers: (A) A is faster because its time is smaller (B) B is faster: 10/3 versus 3 pages/minute (C) The page totals alone determine the faster rate (D) The rates are equal.

Learn the idea

Proportional relationships compare multiplicative changes rather than fixed additions. Rates connect two labeled quantities, and a proportional graph must be consistent with a zero starting amount as well as a constant ratio. Rates can be compared fairly after both are expressed per one identical unit. A larger total can still come from a slower process if it had more time.

Here is the method on the supplied teaching case. A has 6/2=3 pages/minute. B has 16/5=16/5, which exceeds 3. Comparing equally labeled unit rates supports B as faster.

Use the same time unit and the same output unit for both processes. These supplied constant-rate models support a rate comparison, not a promise that real machines never pause.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Printer A makes 27 pages in 9 minutes; B makes 44 in 12 minutes. Which statement follows from these constant-rate models? Candidate answers: (A) A is faster because its time is smaller (B) B is faster: 11/3 versus 3 pages/minute (C) The page totals alone determine the faster rate (D) The rates are equal. Show the mathematical reason, not only the selected answer. Expected result: B is faster: 11/3 versus 3 pages/minute. A has 27/9=3 pages/minute. B has 44/12=11/3, which exceeds 3. Comparing equally labeled unit rates supports B as faster.

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