Comparing Proportional Relationships: Tables, Graphs, and Equations

You get two job offers. Job A pays $60 for 4 hours. Job B pays $72 for 6 hours. Which one pays better per hour?

Both are proportional relationships — each follows $y = kx$ with no added fees or base pay. But they have different $k$ values. Your job is to find which $k$ is larger and decide.

Here's how to compare proportional relationships in any format.

The Rule: Bigger $k$ Wins

Every proportional relationship has a constant of proportionality $k$. In a comparison, the relationship with the larger $k$ is always faster, steeper, pricier, or denser depending on the context.

$y = k_1x \quad \text{vs} \quad y = k_2x$

If $k_1 > k_2$, then for the same $x$, $y_1 > y_2$. That's the entire idea.

Comparing from Two Tables

Find $k$ for each table by dividing $y$ by $x$. Then compare.

| $x$ | Job A: $y$ | | :-: | :-: | | 4 | $60 | | 6 | $90 | | 8 | $120 |

| $x$ | Job B: $y$ | | :-: | :-: | | 4 | $48 | | 6 | $72 | | 8 | $96 |

Job A: $k = 60 \div 4 = 15$. Job B: $k = 48 \div 4 = 12$.

$k_A = 15, \quad k_B = 12$

$15 > 12$, so Job A pays more per hour ($15/hr vs $12/hr).

Check the units. $k$ is in dollars per hour. That tells you what you're actually comparing.

Another Table Example

Compare these two speed relationships:

| Time (hr) | Car X: Distance (mi) | | :-: | :-: | | 2 | 120 | | 5 | 300 |

| Time (hr) | Car Y: Distance (mi) | | :-: | :-: | | 3 | 135 | | 7 | 315 |

Car X: $k = 120 \div 2 = 60$ mph. Car Y: $k = 135 \div 3 = 45$ mph.

Car X is faster. Its $k$ is larger.

Comparing from Two Graphs

The steeper line has the larger $k$.

A graph of a proportional relationship is a straight line through $(0, 0)$. The slope of that line is $k$. Steeper slope = larger $k$ = faster change.

Compare two lines:

• Line A passes through $(2, 6)$ → $k = 6 \div 2 = 3$
• Line B passes through $(2, 4)$ → $k = 4 \div 2 = 2$

Line A is steeper. Its $k$ is larger.

If both lines are on the same set of axes, the one that rises faster (is more vertical) wins. No calculation needed — just look.

One warning: the axes must use the same scale. A graph with a compressed $x$-axis can make a slow relationship look steep.

Comparing from Two Equations

This is the easiest case. $k$ is the coefficient of $x$.

| Equation | $k$ | | --- | :-: | | $y = 4x$ | 4 | | $y = 2.5x$ | 2.5 | | $y = \frac{5}{2}x$ | 2.5 |

Which equation represents the greater constant of proportionality?

$y = 8x \quad \text{vs} \quad y = \frac{1}{2}x$

$8 > \frac{1}{2}$. First equation wins.

What about fractions and decimals?

$y = \frac{7}{3}x \quad \text{vs} \quad y = 2.3x$

Convert or compute: $\frac{7}{3} \approx 2.33$. $2.33 > 2.3$. The first wins — barely.

Comparing Across Formats: Table vs Graph vs Equation

You won't always get two tables or two graphs. Sometimes you get one of each. The method is always the same: find $k$ and compare.

| Source | Data | $k$ | | --- | --- | :-: | | Table | $x = 4, y = 20$ | $20 \div 4 = 5$ | | Graph | Line through $(3, 12)$ | $12 \div 3 = 4$ | | Equation | $y = 4.5x$ | 4.5 |

Ranking from largest $k$ to smallest: table ($k = 5$) > equation ($k = 4.5$) > graph ($k = 4$).

The process never changes:

1. Extract $k$ from each representation.
2. Put them side by side.
3. Pick the largest.

Real-World Comparisons

Job offers. Two friends compare summer jobs. Miguel earns $y = 14x$. Priya's earnings table shows $44 for 4 hours ($k = 11$). Miguel earns more per hour ($14 > 11$).

Grocery prices. Store A: $3 for 12 ounces ($k = 0.25$ per ounce). Store B: $y = 0.30x$ ($k = 0.30$ per ounce). Store A is cheaper.

Typing speeds. A graph shows Kara's line passing through $(5, 200)$ — she types 40 words per minute. Lin's equation is $y = 55x$ — 55 words per minute. Lin is faster.

Key Insight

The constant of proportionality $k$ converts "how many units of $x$" into "how many units of $y$." A larger $k$ means more $y$ per $x$. That translates to:

• Higher pay per hour
• Faster speed per hour
• Higher cost per ounce
• More production per batch

Always ask: "What does $k$ measure here?" Then larger = more of that thing.

Practice Problems

1. Compare these two tables. Which has the larger $k$?

| $x$ | Relationship P | Relationship Q | | :-: | :-: | :-: | | 3 | 18 | 12 | | 7 | 42 | 28 |

Find $k$ for each. Hint: divide $y$ by $x$ for either row.

2. Two graphs show proportional lines. Line M passes through $(8, 4)$. Line N passes through $(4, 8)$. Which has the larger $k$?

Think about slope: $y$ divided by $x$.

3. Which is the more expensive per ounce: a $y = 0.22x$ equation or a table showing $2.50 for 10 ounces?

Find both unit rates. Compare.

FAQ

How do you compare proportional relationships from a table? Find $k$ for each table by dividing $y$ by $x$ using any row where $x \neq 0$. The table with the larger $k$ has the stronger relationship.

What does it mean if one graph is steeper than another? The steeper line has a larger $k$ — a higher constant of proportionality. For the same $x$, the steeper line produces more $y$.

Can you compare a proportional and a nonproportional relationship? Yes, but only for specific $x$ values. A proportional relationship follows $y = kx$; a nonproportional one follows $y = mx + b$. At $x = 0$, the proportional gives $y = 0$ while the nonproportional gives $y = b$. For large $x$, the relationship with the larger coefficient eventually wins.

What if $k$ is negative in one relationship? A negative $k$ means $y$ decreases as $x$ increases. A positive $k$ (even a tiny one like $0.01$) is "larger" than any negative $k$.

How do you compare proportional relationships in a word problem? Extract the $k$ from each situation. A job paying $18/hour ($k = 18$) pays more than one paying $15/hour ($k = 15$). A car going 70 mph ($k = 70$) is faster than one going 55 mph ($k = 55$).

Do the axes need to be the same scale when comparing graphs? Yes. Different axis scales can make a slower relationship appear steeper. Always check the units and scale before comparing steepness visually.

If two proportional relationships have the same $k$, are they the same? They have the same rate, but they could represent different contexts. $k = 5$ could mean 5 dollars per hour or 5 miles per hour. The math is identical, but the real-world meaning differs.

What's the quickest way to compare two proportional relationships? Convert each to $y = kx$ form. The equation gives you $k$ directly. For tables and graphs, compute $k$ from one data point. Then compare the numbers.

Related: Study the constant of proportionality in detail, or see how slope connects to proportional relationships. Use the unit rate calculator or constant of proportionality calculator to check any comparison.