Math Problem Statement

Select all of the equations below in which t is inversely proportional to w.

Solution

To determine which equations show that tt is inversely proportional to ww, we need to understand what it means for two variables to be inversely proportional. Inverse proportionality between tt and ww means that tw=kt \cdot w = k, where kk is a constant. This can also be written as t=kwt = \frac{k}{w}.

Let’s examine each equation:

  1. t=3wt = 3w: This shows a direct proportionality, not an inverse one.
  2. t=w+3t = w + 3: This is a linear equation with ww added to a constant, not inverse proportionality.
  3. t=w3t = \frac{w}{3}: This is still direct proportionality because tt is proportional to ww, even though it's divided by 3.
  4. t=3wt = \frac{3}{w}: This shows that tt is inversely proportional to ww, as it matches the form t=kwt = \frac{k}{w}.
  5. t=w3t = w - 3: This is another linear equation and does not represent inverse proportionality.

Answer: The equation that shows tt is inversely proportional to ww is:

  • t=3wt = \frac{3}{w}.

Would you like more details or have any questions?

Here are some follow-up questions to consider:

  1. What is the difference between direct and inverse proportionality?
  2. Can you provide another example of an equation that shows inverse proportionality?
  3. How can we identify inverse proportionality from an equation?
  4. What are real-life examples of inverse proportionality?
  5. What happens to tt as ww increases in an inverse proportional relationship?

Tip: For two quantities to be inversely proportional, their product should always be a constant.

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Math Problem Analysis

Mathematical Concepts

Inverse Proportion
Direct Proportion
Algebra

Formulas

Inverse proportion: t = k / w, where k is a constant

Theorems

Inverse Proportion Theorem

Suitable Grade Level

Grades 7-9