How can I tell whether a ratio compares parts or a part with the whole?
Quick answer
Name both quantities in the comparison. With 3 red objects and 2 blue objects, red to blue is 3:2, a part-to-part ratio. There are 5 objects altogether, so red to total is 3:5, or 3/5, a part-to-whole ratio. A fraction bar shows the notation; the labels and context tell you what the quantities mean.
The question, the details, the next step.
How can I tell whether a ratio compares parts or compares a part with the whole?
Ask what the first quantity and the second quantity count. If each names one category within a group, it is part-to-part. If one quantity names a category and the other names the entire group, it is part-to-whole. Ratios can be written with a colon, with the word “to,” or as a fraction, but the notation alone does not name the quantities being compared. [S1]
For example, imagine a collection with 3 red objects and 2 blue objects. The comparison red to blue is 3:2: both numbers count parts of the collection. The comparison red to all objects is 3:5: the first number counts the red part, and the second counts the whole.
How do I find the whole in that example, and what would the two comparisons look like as fractions?
Add the counts of all the categories to get the total: 3 red + 2 blue = 5 objects. Then label each comparison before writing it:
- Red compared with blue: red/blue = 3/2. This is part-to-part because the denominator counts only the blue objects, not the whole collection.
- Red compared with all objects: red/total = 3/5. This is part-to-whole because the denominator counts every object.
The fraction bar is a way to write the ratio. It does not automatically make a comparison part-to-whole: a ratio written as a fraction can compare two parts, as 3/2 does here. [S1] The labels explain its meaning.
If I reverse the order, does the kind of ratio change too?
Reversing the order changes which quantity is first, so say the labels aloud both ways. Blue to red is 2:3, still part-to-part. All objects to red is 5:3, a whole-to-part comparison. It is not the same as red to all objects, 3:5.
A useful check is to ask what the second number counts. In 2:3, it counts red objects; in 5:3, it counts red objects too, but the first number now counts the whole. For a part-to-whole comparison of red objects, the whole must be the second quantity: red to all is 3:5. [S1]
Can you give me a quick routine I can use on a worksheet so I do not mix them up?
Yes. Write the comparison in words before reducing or changing its notation. For the red-and-blue collection, the copyable setup is:
- Red to blue: 3 red : 2 blue = 3:2 (part-to-part)
- Red to all objects: 3 red : 5 total = 3:5 = 3/5 (part-to-whole)
- All objects to red: 5 total : 3 red = 5:3 (whole-to-part)
The key is to keep the order of the named quantities and check whether the total is actually in the comparison. A fraction-style ratio such as 3/2 can still compare two parts; it does not mean that the comparison is part-to-whole just because it uses a fraction bar. [S1]
A practical checklist
Write the name of each quantity being compared, in order.
Check whether the second quantity is one category or the total of all categories.
If needed, find the total by adding the counts of all parts.
Label the comparison part-to-part, part-to-whole, or whole-to-part.
Keep the labels when rewriting the ratio with a colon, “to,” or a fraction bar.
A few more useful answers
Is 3/2 always a part-to-part ratio?
No. The numbers alone do not identify what is being compared. In the example, 3/2 is red to blue, so it is part-to-part; label the quantities to show the meaning.
Does 3/5 mean red objects are 3 of every 5 objects?
In this example, yes: 3 counts the red objects and 5 counts the total. That makes it a part-to-whole comparison.
Can a ratio compare a whole with a part?
Yes. For the same collection, all objects to red objects is 5:3, a whole-to-part comparison. Keep the order clear so it is not mistaken for red to all, which is 3:5.
Sources
References used for this guide.
- [S1] 5.4 Ratios and Proportions — Contemporary Mathematics openstax.org · Checked