How do I check whether two ratios are actually proportional?
Quick answer
Write the ratios in the same order, such as 3/5 and 12/20. Make sure both denominators are nonzero, then compare cross products: multiply the first numerator by the second denominator, and the first denominator by the second numerator. If the products match, the ratios are equal; if they differ, they are not proportional. This tests the ratios as written, not whether unlike quantities should be compared in that order.
The question, the details, the next step.
How can I check whether two ratios are actually proportional, instead of guessing from how the numbers look?
Treat a proportion as a claim that two ratios are equal. First, write both ratios as fractions in the same order. For a/b and c/d, compare a×d with b×c. If those cross products are equal, the ratios are equal; if not, they are not. The denominators must be nonzero, and the quantities must be arranged consistently before you use the test. [S1]
For example, compare 3/5 and 12/20. The cross products are 3×20 = 60 and 5×12 = 60, so the ratios are equal. You can also see that multiplying both parts of 3/5 by 4 gives 12/20, but cross multiplication gives a direct test.
What does a failed test look like? What if the numbers seem close?
Compare 2/3 and 8/11. Their cross products are 2×11 = 22 and 3×8 = 24. Because 22 and 24 differ, these ratios are not equal. The fact that some numbers are close—or that both fractions are less than 1—does not make the ratios proportional. The test is whether the products match, not whether the original numbers look similar.
The source gives another failed test: 4/9 and 12/28 have cross products of 112 and 108, so the ratios are not equal. [S1] That example illustrates the test; it does not tell you how to choose a meaningful ratio for every real-world situation.
How do I avoid getting a misleading result when I set up the ratios?
Check what each part represents before multiplying. If the question compares apples to oranges in one group with apples to oranges in another, write apples/oranges for both groups—not apples/oranges for one and oranges/apples for the other. Reversing just one ratio changes the comparison, even though it uses the same numbers.
Also check that neither denominator is zero. Cross multiplication is a test for two ratios written as fractions with nonzero denominators; it is not a shortcut to apply to any pair of expressions. [S1] If the quantities or their order are unclear, label them first. For instance, write “red beads / blue beads” above each fraction rather than relying on memory.
Can you give me a quick routine I can use on homework?
Try this routine:
- Name what the numerator and denominator count in each ratio.
- Put the ratios in the same order and write them as fractions.
- Check that both denominators are nonzero.
- Multiply diagonally: first numerator × second denominator, then first denominator × second numerator.
- Compare the products. Equal means the ratios are equal; unequal means they are not.
Copyable working:
a/b and c/d → compare a×d with b×c.
Example: 3/5 and 12/20 → 3×20 = 60, 5×12 = 60 → equal.
Example: 2/3 and 8/11 → 2×11 = 22, 3×8 = 24 → not equal.
This establishes whether the written ratios are equal. It does not by itself establish that the comparison is appropriate for a word problem; that depends on what the quantities mean.
A practical checklist
Label what each numerator and denominator represents.
Write both ratios in the same order.
Confirm neither denominator is zero.
Calculate both cross products carefully.
Compare the products and state whether the ratios are equal.
A few more useful answers
Do equal cross products prove the two ratios are equal?
Yes, when the ratios are written as fractions with nonzero denominators and the comparison is set up correctly. Unequal cross products mean the ratios are not equal. [S1]
Can I compare ratios that use different units?
The arithmetic test can check equality of the fractions you wrote, but it cannot decide whether comparing those quantities makes sense. Identify what each part measures and follow the order required by the problem.
Does a matching result mean the quantities are proportional in every situation?
It means the two written ratios are equal. Whether that relationship is useful or appropriate in a real-world setting depends on the quantities and how they were selected.
Sources
References used for this guide.
- [S1] OpenStax Prealgebra 2e: Solve Proportions and Their Applications openstax.org · Checked