Select all the true statements about the table.

$x$ $y$

10.5 3.5

15.9 5.3

22.5 7.5

27 9

The table shows a proportional relationship.
All the ratios $\frac{y}{x}$ for related pairs of $x$ and $y$ are equivalent to 7.5.
When $x$ is 13.5, $y$ is 4.5.
When $y$ is 12, $x$ is 4.
The unit rate of $\frac{y}{x}$ for related pairs of $x$ and $y$ is $\frac{1}{3}$.

Question

05-11-2026
Mathematics

Answered

Select all the true statements about the table.

$x$ $y$

10.5 3.5

15.9 5.3

22.5 7.5

27 9

The table shows a proportional relationship.
All the ratios $\frac{y}{x}$ for related pairs of $x$ and $y$ are equivalent to 7.5.
When $x$ is 13.5, $y$ is 4.5.
When $y$ is 12, $x$ is 4.
The unit rate of $\frac{y}{x}$ for related pairs of $x$ and $y$ is $\frac{1}{3}$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T22:06:14-05:00

The ratio $\frac{y}{x}$ is constant at $\frac{1}{3}$ for all given pairs, indicating a proportional relationship.
Statement 1 is true because the ratio is consistent.
Statement 3 is true as $\frac{1}{3} \times 13.5 = 4.5$.
Statement 5 is true as the unit rate $\frac{y}{x}$ is indeed $\frac{1}{3}$.
Statements 2 and 4 are false based on calculations.
Final selection: $\boxed{\text{The table shows a proportional relationship, When } x \text{ is } 13.5, y \text{ is } 4.5, \text{The unit rate of } \frac{y}{x} \text{ for related pairs of } x \text{ and } y \text{ is } \frac{1}{3}}$

Explanation

Analyzing the Relationship and Ratio

To determine the validity of the statements provided, we first analyze the relationship between the variables $x$ and $y$ given in the table. We calculate the ratio $\frac{y}{x}$ for each pair $(x, y)$ presented:

For $(10.5, 3.5)$: $\frac{3.5}{10.5} = \frac{1}{3} \approx 0.333$
For $(15.9, 5.3)$: $\frac{5.3}{15.9} = \frac{1}{3} \approx 0.333$
For $(22.5, 7.5)$: $\frac{7.5}{22.5} = \frac{1}{3} \approx 0.333$
For $(27, 9)$: $\frac{9}{27} = \frac{1}{3} \approx 0.333$

Since the ratio $\frac{y}{x}$ is constant ($k = \frac{1}{3}$) for all given pairs, the table represents a proportional relationship where $y = \frac{1}{3}x$.

Evaluating Statements

Now we evaluate each statement:

"The table shows a proportional relationship": Since the ratio $\frac{y}{x}$ is constant, this is True.
"All the ratios $\frac{y}{x}$ are equivalent to 7.5": The calculated ratio is $\frac{1}{3}$, not $7.5$. This is False.
"When $x$ is 13.5, $y$ is 4.5": Using $y = \frac{1}{3}x$, we get $y = \frac{1}{3} \times 13.5 = 4.5$. This is True.
"When $y$ is 12, $x$ is 4": Using $x = 3y$, we get $x = 3 \times 12 = 36$, not $4$. This is False.
"The unit rate of $\frac{y}{x}$ is $\frac{1}{3}$": As calculated in the first step, the constant ratio is $\frac{1}{3}$. This is True.

Conclusion

Based on our analysis, the true statements are:

The table shows a proportional relationship.
When $x$ is $13.5$, $y$ is $4.5$.
The unit rate of $\frac{y}{x}$ for related pairs of $x$ and $y$ is $\frac{1}{3}$.

Examples

In real-life scenarios, proportional relationships are essential for tasks like unit conversions and scaling recipes. For instance, if a recipe requires $3$ cups of flour for every $1$ cup of sugar, this is a proportional relationship where the ratio is always $3:1$. Understanding this allows you to adjust ingredients for any amount of sugar, ensuring the ratio remains consistent for perfectly balanced results!