Ratios, proportions and unit rates
Separate part-to-part ratios from part-to-whole fractions and test when proportional reasoning is valid.
In this guide
Read the order of the comparison
A ratio compares quantities in a stated order. If blue folders to green folders is 3:5, every matched group contains three blue folders and five green folders. Blue folders are 3/8 of the combined collection, not 3/5 of it. The number 3/5 instead compares blue directly with green. Label both positions before setting up a calculation. A ratio can stay unchanged while the actual quantities grow: 3:5, 6:10 and 15:25 describe the same relative comparison.
Worked example: divide a total into ratio parts
A school club has 64 folders, with blue to green in the ratio 3:5. The total number of ratio parts is 3 + 5 = 8. Each part therefore represents 64 ÷ 8 = 8 folders. Blue folders number 3 × 8 = 24 and green folders number 5 × 8 = 40. Check both conditions: 24 + 40 = 64, and 24:40 simplifies to 3:5. A result that matches only the total or only the ratio is not enough.
Use a unit rate to compare offers
For a fictional printing job, one provider charges PHP 84 for 12 identical pages, while another charges PHP 110 for 20. The unit rates are PHP 7 per page and PHP 5.50 per page. If quality and other conditions are identical, the second has the lower per-page rate. The comparison assumes there is no separate setup charge and that the quoted rate applies to the required quantity. Comparing only the package prices would miss the fact that the packages contain different numbers of pages.
Direct and inverse relationships
If one machine produces 18 identical cards in 6 minutes at a constant rate, it produces 3 cards per minute and needs 15 minutes for 45 cards. More cards require more time: this is direct proportion under the stated fixed-rate assumption. In a different idealized problem, three equally productive workers need eight hours for a divisible task. The total is 24 worker-hours, so six workers would need four hours. That inverse relationship requires equal productivity, no interference and work that can actually be shared. Real tasks need not satisfy those assumptions.
Units expose incorrect proportions
Write comparable quantities in corresponding positions. If pages are in the numerator on one side, use pages in the numerator on the other. Convert mixed units first: 1.5 hours is 90 minutes, not 150 minutes. Cross multiplication is a way of solving a proportion after it has been correctly formed; it cannot repair a relationship that was never proportional.
- Part-to-part ratios become part-to-whole fractions by adding all ratio parts.
- A fixed fee plus a per-item charge is not directly proportional to item count.
- Check the proposed result in the original context, including units and whole-object constraints.
Try it yourself
The ratio of red to yellow notebooks is 4:7, with 55 notebooks altogether. Find each color's count. Separately, a constant-rate printer makes 28 cards in seven minutes. How long will 48 cards take, and what is the total job time if setup takes five additional minutes?
Reveal the worked answer
Red notebooks to yellow notebooks is 4:7, with 55 notebooks altogether. There are 11 ratio parts and five notebooks per part, giving 20 red and 35 yellow. A separate constant-rate printer makes 28 cards in seven minutes. It needs 12 minutes for 48 cards because its rate is four cards per minute. If the printer also requires five minutes of setup, total job time becomes 17 minutes. That extra condition shows why reading all the information matters more than recognizing a familiar-looking ratio.
Red notebooks to yellow notebooks is 4:7, with 55 notebooks altogether. There are 11 ratio parts and five notebooks per part, giving 20 red and 35 yellow. A separate constant-rate printer makes 28 cards in seven minutes. It needs 12 minutes for 48 cards because its rate is four cards per minute. If the printer also requires five minutes of setup, total job time becomes 17 minutes. That extra condition shows why reading all the information matters more than recognizing a familiar-looking ratio.
Put one idea into practice.
Try a short mathematics session, then use the explanations to choose what to review next.
Start a short sessionSources & further reading
OpenStax Prealgebra 2e: Proportions and their applicationsSomething unclear or incorrect? Suggest a correction to this guide.
Original FreeReviewer.ph guidance, informed by the linked sources. Exam requirements and procedures should always be checked with the examining body.