Exercises: Proportion vs Inproportional Relationship Solutions

Exercise 1: Identifying Proportional Relationships

1. Proportional: The cost of buying multiple loaves of bread is proportional because if you double the number of loaves, you double the cost (e.g., 2 loaves cost $4, 3 loaves cost $6, etc.).

2. Inversely Proportional: The speed of a car and the time taken to travel a fixed distance are inversely proportional. If you increase the speed, the time decreases (e.g., driving faster means less time to cover the same distance).

3. Proportional: The length of a shadow is proportional to the height of the object casting it, assuming the light source remains constant. If the height doubles, the shadow length doubles.

4. Inversely Proportional: The time required to fill a swimming pool at a fixed capacity is inversely proportional to the rate of water flow. If the flow rate increases, the time to fill the pool decreases.

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Exercise 2: Proportional Relationship Table

Using the equation \( y = 3x \):

\( x \)	\( y \)
1	3
2	6
4	12
5	15
10	30

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Exercise 3: Inversely Proportional Relationship Table

Using the equation \( y = \frac{12}{x} \):

\( x \)	\( y \)
1	12
2	6
3	4
6	2
12	1

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Exercise 4: Word Problem (Proportional Relationship)

The equation representing the total cost \( C \) as a function of the number of days \( d \) is:

\[ C = 50d \]

To find the cost to rent the car for 7 days:

\[ C = 50 \times 7 = 350 \]

Total Cost for 7 days: $350

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Exercise 5: Word Problem (Inversely Proportional Relationship)

From the information given, if 5 people take 10 hours, we can express it as:

\[ T \times p = k \quad \text{(constant)} \]

\[ 10 \times 5 = k \Rightarrow k = 50 \]

Now, for 2 people:

\[ T \times 2 = 50 \Rightarrow T = \frac{50}{2} = 25 \]

It would take 2 people 25 hours to complete the project.