How can I tell whether a word problem calls for a ratio or a difference?

Start with the question being asked, not just the numbers. A difference tells you how far apart two quantities are; when the quantities use the same unit, subtraction gives that gap. A ratio compares quantities in relation to each other, often by expressing one relative to the other. The order matters: a red-to-blue ratio puts red first and blue second. [S1]

Try spotting the phrase that signals the goal. “How many more?” or “How many fewer?” usually asks for a difference. “How many times as many?” asks for a multiplicative comparison, so division is the useful operation. “What is the ratio of red to blue?” asks you to state the comparison in that order. These are clues, not a substitute for reading the whole question.

Suppose a box has 8 red counters and 5 blue counters. Assume both counts refer to individual counters, and the question compares red with blue.

1. If the question is “How many more red counters are there than blue counters?”, subtract: 8 − 5 = 3. The answer is 3 counters. This tells you the size of the gap.
2. If the question is “What is the ratio of red counters to blue counters?”, write 8:5. That compares red with blue in the order named. A ratio compares two quantities, and its order should follow the wording. [S1]
3. If the question is “How many times as many red counters are there as blue counters?”, divide: 8 ÷ 5 = 1.6. This says the red count is 1.6 times the blue count.

Notice that the answers are different because the questions ask for different comparisons. The difference is measured in counters; the ratio 8:5 is a relationship between the two counts. The decimal 1.6 is another way to express red compared with blue as a factor. [S1]

Rewrite the question in your own words before calculating. Ask: “Do I need the gap, or do I need one amount compared with the other?” Then identify which quantity should come first.

For the same counts, “What is the ratio of blue counters to red counters?” reverses the order, so write 5:8, not 8:5. The quantities have not changed; the comparison being requested has. Writing the ratio in the order named in the question is an important check. [S1]

If a problem merely says “compare the counts” without saying how, do not guess which answer is wanted. Check the surrounding instructions or ask for clarification. If the problem asks both “how many more” and “what is the ratio,” answer both parts: a single pair of quantities can support an additive comparison and a multiplicative comparison.

Yes. Here is a reusable routine:

1. Circle the words that state what to find.
2. If they ask “how many more/fewer” or “what is the difference,” subtract the smaller amount from the larger amount.
3. If they ask “what is the ratio,” write the quantities in the order named.
4. If they ask “how many times as many,” divide the first named amount by the second.
5. Label the answer so it is clear whether it is a gap or a comparison.

For the counters, a copyable response could be: “Red minus blue: 8 − 5 = 3, so there are 3 more red counters. Red to blue: 8:5. Red compared with blue as a factor: 8 ÷ 5 = 1.6.”

These cues work for the particular comparison being asked; they do not establish that a problem has only one part. Read all of its instructions, and answer each requested comparison.