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Numbers and proportions

From counting objects to comparing quantities and rates.

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ConceptMathematicsPhysics

Ratios and rates

Distinguish part-to-part ratios, fractions of a whole and rates with units. Scale quantities together, compare unit rates, and explain when a relationship is proportional. Check zero cases, starting amounts and whether a result fits its context.

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From this articleRatios, rates and proportional reasoning

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In brief

  • A ratio has an order and named quantities. A 3:2 part-to-part ratio gives shares 3/5 and 2/5 of the whole, not 3/2 of the whole.
  • Equivalent ratios scale corresponding quantities by the same nonzero factor. An empty 0:0 selection has no unique actual ratio or shares, even when its recipe is known.
More key ideas
  • Use shared units for same-kind measurement ratios. Unit rates retain compound units; convert those units before comparing rates.
  • Direct proportionality follows y = k × x and has constant y/x when x is nonzero. A nonzero starting amount changes the total rule even if the rate of increase stays constant.
  • Cross-products require nonzero denominators; inverse proportionality keeps a product fixed. Check whole-count constraints and the conditions supporting any prediction.
Worked example

Build a 3:2 recipe from a total count

A group of 35 indivisible counters must contain A- and B-counters in the ratio 3:2. Find each count and its share of the total.

Show the reasoning
  1. One complete bundle has 3 + 2 = 5 counters. The total 35 contains 35 ÷ 5 = 7 complete bundles, so the whole-count requirement is satisfied.
  2. Multiply both recipe counts by 7: A = 3 × 7 = 21 and B = 2 × 7 = 14. Check the sum 21 + 14 = 35.
  3. Simplify 21:14 by dividing both entries by 7 to recover 3:2. The part-to-whole fractions are 21/35 = 3/5 and 14/35 = 2/5, which add to one.

21 A-counters and 14 B-counters, with shares 3/5 and 2/5. The part-to-part quotient is 3/2, a different comparison from either share.

Article sources (6)

References

Sources for the linked article, with its original context and qualifications.

  1. OpenStax2 sources
    1. Prealgebra 2e · Ratios and RateOpenStaxTextbook chapteropenstax.orgRead source
    2. Prealgebra 2e · Solve Proportions and their ApplicationsOpenStaxTextbook chapteropenstax.orgRead source
  2. Australian Mathematical Sciences Institute2 sources
    1. TIMES · Rates and RatioAustralian Mathematical Sciences InstituteTeaching resourceamsi.org.auRead source
    2. TIMES Module 35 · ProportionAustralian Mathematical Sciences InstituteTeaching resourceamsi.org.auRead source
  3. Understanding multiplicative relationshipsNCETMTeaching resourcencetm.org.ukRead source
  4. Grade 6 Mathematics · Educational Package (2024)Palestinian Ministry of EducationTextbook chaptermoe.edu.psRead source