What is a Proportional Relationship? 

Yes, the relationship in this table is proportional. The ratio of $y$ to $x$ ($\frac{3}{1}$, $\frac{6}{2}$, $\frac{9}{3}$) is constant at $3$, indicating a proportional relationship where $y$ increases by a consistent amount as $x$ increases.

No, the relationship in this table is not proportional. The ratio of $y$ to $x$ changes with each row ($\frac{4}{1}$, $\frac{7}{2}$, $\frac{11}{3}$), indicating that $y$ does not increase by a consistent amount relative to $x$, which is a requirement for a proportional relationship.

Yes, this relationship is proportional. The line passes through the origin, and the ratio of $y$ to $x$ $(\frac{y}{x})$ remains constant at every point ($\frac{2}{1}$, $\frac{4}{2}$, $\frac{6}{3}$). This constant ratio and the line passing through the origin are key indicators of a proportional relationship.

No, this relationship is not proportional. Even though the line is straight, it does not pass through the origin. The graph starting at $(0,2)$ indicates an initial value or y-intercept, which is not characteristic of a proportional relationship.

Yes, the relationship described by this equation is proportional. It follows the form $y = kx$ with $k = 5$, indicating a direct proportionality between $y$ and $x$. As $x$ increases, $y$ increases five times the amount of $x$, maintaining a constant ratio.

No, the relationship described by this equation is not proportional. The presence of a constant term $+3$ in addition to the $2x$ term violates the direct proportionality condition of $y = kx$. This constant term indicates that $y$ does not change at the same rate as $x$, thus not maintaining a constant ratio.

Yes, the relationship is proportional. The equation follows the form $E = kh$ with $k = 10$, showing a direct proportionality between earnings and hours worked. As the hours increase, earnings increase at a constant rate, maintaining a consistent ratio of $\$$10 per hour.

Yes, this relationship is proportional. The ratio of distance to hours is constant ($\frac{60}{1}$, $\frac{120}{2}$, $\frac{180}{3}$), all equaling $60$. This indicates that the train travels at a constant speed of 60 km/h, illustrating a proportional relationship between time and distance.

Yes, the relationship is proportional. The graph shows a straight line passing through the origin, and the ratio of pages read per hour ($\frac{30}{1}$, $\frac{60}{2}$, $\frac{90}{3}$) remains constant at $30$ pages per hour. This indicates a direct proportional relationship between time spent reading and the number of pages read.

Yes, it's proportional. The equation $C = 0.05t$ shows a direct proportionality between the cost and the number of texts sent. For each text message, the cost increases by $\$$0.05, maintaining a constant ratio.

Yes, the relationship is proportional. The ratio of water consumed to days ($\frac{2}{1}$, $\frac{4}{2}$, $\frac{6}{3}$) is constant at $2$ liters per day. This indicates that the plant's water consumption is directly proportional to the number of days.

No, the relationship is not proportional. While the line is straight, the ratio of money saved to weeks ($\frac{10}{1}$, $\frac{15}{2}$, $\frac{20}{3}$) is not constant. Additionally, the line does not pass through the origin, indicating that the savings do not increase at a consistent rate per week.

A proportional relationship is one where two quantities maintain a constant ratio.

By checking if the ratio of one quantity to another remains constant across the table.

It appears as a straight line that passes through the origin (0,0).

Yes, an example is $y = 3x$. Here, $y$ is directly proportional to $x$ with a constant ratio of $3$.

Understanding these relationships helps in solving real-world problems involving ratios and rates, such as speed, distance, and time, as well as in various fields of science and mathematics.

In a proportional relationship, as one quantity increases, the other increases at a constant rate. In an inverse relationship, as one quantity increases, the other decreases.

Yes, the constant of proportionality can be negative, indicating an inverse relationship between the two quantities.

The constant of proportionality is the slope of the line in a graph of a proportional relationship.

No, not all linear relationships are proportional. Proportional relationships are specific linear relationships where the line passes through the origin (0,0).

No, if one quantity is always zero, they cannot be proportional since there is no constant ratio between zero and any other number.

A ratio is a comparison of two quantities, while a proportion states that two ratios are equal.

Direct variation is a type of proportional relationship where one variable is a constant multiple of another.

Yes, proportions are widely used in various real-life applications like cooking, construction, map reading, and more.

In science, proportional relationships are used in formulas and laws to describe how quantities change in relation to each other.