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All courses Math I · N-Q.2 48 of 59
Define appropriate quantities for descriptive modeling.

Identify independent and dependent quantities in a context

Problem
In this situation, identify the independent quantity and the dependent quantity: A taxi fare depends on the number of miles traveled.
Your answer
Show answer choicesAnswer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
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Hint

Look at the word \(depends\) to see which quantity changes because of the other one.

The independent quantity is the input or cause, and the dependent quantity is the result that changes in response.

Solution walkthrough

01

Identify the stated dependency

\[\text{taxi}~\text{fare}~\text{depends}~\text{on}~\text{miles}~\text{traveled}\]

The sentence explicitly says that the taxi fare depends on the number of miles traveled. Therefore miles traveled is the quantity used to determine the fare, and fare is the quantity that responds.

02

Apply the input-output roles

\[\text{independent}~\text{input}~->~\text{dependent}~\text{output}\]

The independent quantity is the input chosen or recorded first. The dependent quantity is the output that changes in response to that input.

03

Map and check the two quantities

\[\text{miles}~\text{traveled}~->~\text{taxi}~\text{fare}\]

Map miles traveled to the independent input and taxi fare to the dependent output. If the miles change, the fare changes because of that new mileage; this directional check confirms the roles.

04

State both quantity roles

\[\text{dependent}:~\text{taxi}~\text{fare};~\text{independent}:~\text{miles}~\text{traveled}\]

The dependent quantity is taxi fare, and the independent quantity is miles traveled. This exactly matches choice A and answers both parts of the prompt.

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Another way

  1. Ask, 'When the miles traveled changes, what changes as a result?' The taxi fare changes, which identifies fare as dependent.

!

Common mistake

Do not reverse the variables just because fare may be chosen or paid by a rider. In this stated model, fare changes based on miles traveled, so fare is the dependent quantity.