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Use proportional relationships unit test

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Proportional relationships are mathematical relationships where two quantities vary directly with each other. These relationships can often be represented by equations of the form \( y = kx \), where \( k \) is a constant known as the constant of proportionality.

Here’s how you can structure a unit test on proportional relationships:

Proportional Relationships Unit Test

Part A: Multiple Choice

  1. Which of the following equations represents a proportional relationship? a) \( y = 3x + 5 \)
    b) \( y = \frac{1}{2}x \)
    c) \( y = x^2 \)
    d) \( y = 7 - 2x \)

    Answer: b) \( y = \frac{1}{2}x \)

  2. If \( y \) is directly proportional to \( x \) and \( y = 12 \) when \( x = 4 \), what is the constant of proportionality \( k \)? a) 2
    b) 3
    c) 4
    d) 6

    Answer: a) 3 (since \( k = \frac{y}{x} = \frac{12}{4} = 3 \))

  3. Which of the following graphs represents a proportional relationship? a) A straight line passing through the origin
    b) A parabola
    c) A line that does not pass through the origin
    d) A horizontal line

    Answer: a) A straight line passing through the origin

Part B: Short Answer

  1. Write an equation for the relationship between \( y \) and \( x \) if \( y \) is proportional to \( x \) and \( y = 20 \) when \( x = 5 \).

    Answer: \( y = 4x \) (since \( k = \frac{20}{5} = 4 \))

  2. If \( 3x = 12 \), what is the value of \( x \)?

    Answer: \( x = 4 \)

Part C: Problem Solving

  1. A recipe requires 2 cups of flour for every 3 cups of sugar. How much flour is needed for 9 cups of sugar?

    Solution: Set up a proportion: \[ \frac{2 \text{ cups of flour}}{3 \text{ cups of sugar}} = \frac{x \text{ cups of flour}}{9 \text{ cups of sugar}} \] Cross-multiply and solve for \( x \): \[ 2 \times 9 = 3 \times x \implies 18 = 3x \implies x = 6 \text{ cups of flour} \]

  2. A car travels 150 miles using 5 gallons of gas. How far can it travel using 8 gallons of gas?

    Solution: First, find the constant of proportionality: \[ \frac{150 \text{ miles}}{5 \text{ gallons}} = 30 \text{ miles per gallon} \] Then, multiply by 8 gallons: \[ 30 \text{ miles/gallon} \times 8 \text{ gallons} = 240 \text{ miles} \]

Part D: True or False

  1. If two quantities are in a proportional relationship, then doubling one quantity will always double the other.
    Answer: True

  2. Proportional relationships can be represented in any shape of graph, including curves.
    Answer: False (They are represented as straight lines through the origin.)

Part E: Extended Response

  1. Describe the differences between a proportional relationship and a non-proportional relationship. Provide examples of each.

Answer: A proportional relationship is one where two quantities increase or decrease together at the same rate, and can be represented by the equation \( y = kx \) where \( k \) is a constant. An example is the relationship between distance and time for an object moving at a constant speed. In contrast, a non-proportional relationship does not maintain a constant rate of change, and its graph does not pass through the origin. An example of a non-proportional relationship is the equation \( y = 2x + 3 \).

By structuring the unit test this way, students can demonstrate their understanding of proportional relationships, apply it to problems, and reflect on the nature of these relationships.