Question

Use proportional relationships unit test

Answer 1

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

4. 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 \( k = \frac{20}{5} = 4 \))

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

**Answer:** \( x = 4 \)

#### Part C: Problem Solving

6. 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 \( x \):
\[
2 \times 9 = 3 \times x \implies 18 = 3x \implies x = 6 \text{ cups of flour}
\]

7. 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

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

9. 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

10. 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.