A table shows direct variation when every y-value divided by its corresponding x-value gives the same number. That constant ratio is the key. If even one pair produces a different ratio, the table does not represent direct variation.
This is one of the most common questions in algebra homework, where you’re given two or three tables and asked to pick the one that fits. Here’s exactly how to test any table quickly and confidently.
The One Test That Always Works
Direct variation follows the equation y = kx, where k is a constant called the constant of proportionality. In plain terms, y is always the same multiple of x. To check whether a table fits this pattern, divide y by x for every row. If you get the same result each time, the table shows direct variation. If the results differ, it doesn’t.
Take this example table:
- x = 2, y = 6: 6 ÷ 2 = 3
- x = 5, y = 15: 15 ÷ 5 = 3
- x = 9, y = 27: 27 ÷ 9 = 3
Every ratio equals 3, so this table shows direct variation with k = 3. The relationship is y = 3x.
Now compare it to a table that fails the test:
- x = 2, y = 7: 7 ÷ 2 = 3.5
- x = 5, y = 16: 16 ÷ 5 = 3.2
- x = 9, y = 28: 28 ÷ 9 ≈ 3.1
The ratios are close but not equal. This table does not show direct variation. Close doesn’t count here. The ratios must be exactly the same.
Why the Ratio Matters More Than the Pattern
A common mistake is looking at a table and noticing that y increases as x increases, then assuming that’s enough. It isn’t. Plenty of relationships increase together without being direct variation. The equation y = 2x + 1, for instance, produces a table where y goes up every time x goes up, but dividing y by x gives different results each time (3, 2.5, 2.33…). That extra “+1” breaks the pattern.
Direct variation requires the line to pass through the origin, the point (0, 0). If x were zero, y would also have to be zero. Tables that include a (0, 0) pair give you a quick visual confirmation, but most homework tables skip that row. The ratio test works regardless of whether zero appears in the table.
Step-by-Step: Checking Multiple Tables
When a problem gives you several tables and asks which one shows direct variation, work through them systematically:
- Step 1: For each table, divide the y-value by the x-value in every row.
- Step 2: Write down each ratio. You can leave them as fractions if the division isn’t clean.
- Step 3: Compare the ratios within each table. The table where all ratios match is your answer.
You only need to find one mismatched ratio to eliminate a table. So if the first two rows of a table give you different results, skip the rest and move on to the next table. This saves time on tests.
Finding the Constant of Proportionality
Once you’ve identified the correct table, many problems then ask you to state the constant of proportionality, k. You already have it from the ratio test. It’s the number you got when you divided y by x. If every row gave you 4, then k = 4 and the equation is y = 4x.
The constant can also be a fraction or a decimal. A table with x = 4, y = 3 and x = 8, y = 6 gives k = 3/4, making the equation y = (3/4)x. That’s still direct variation. The constant just needs to be the same nonzero value in every row.
What Direct Variation Looks Like in Context
Real-world direct variation tables show up any time one quantity is a fixed multiple of another. Hourly wages are a classic example: if you earn $15 per hour, a table of hours worked and dollars earned will always produce a ratio of 15. Two hours gives $30, five hours gives $75, eight hours gives $120. Every ratio is 15.
Unit pricing works the same way. If apples cost $2 per pound, then 3 pounds costs $6, 7 pounds costs $14, and dividing cost by weight always returns 2. Fuel consumption at a steady rate, distance traveled at constant speed, and recipe scaling all follow this pattern.
The practical takeaway: if doubling x always doubles y, tripling x always triples y, and zeroing out x gives you zero y, you’re looking at direct variation. The ratio test simply confirms this with arithmetic instead of intuition.