A little room to wonder

Ratio & Proportion

Scale two quantities together, and distinguish a comparison between parts from a share of the total.

What must stay fixed when a pattern grows?

Dark blue waves and pale foam create strongly separated areas around small boats.
Hokusai’s approved print uses contrasting blue and pale areas. It offers a context for discussing colour arrangements, not the fictional blue:white mosaic counts.Katsushika Hokusai · Source · Public domain

Keep the comparison in order

A ratio compares quantities in a stated order. Blue to white counters in the ratio 2:3 means two blue for every three white, not two blue out of three counters altogether. The combined batch has five parts, so blue is 2/5 of the total and white is 3/5. Reversing the named order gives white to blue as 3:2.

Use compatible units before comparing lengths or masses. A ratio without labels can be ambiguous: 2:3 might compare lengths, counts or something else. The same written ratio can describe many total sizes because it specifies a relationship rather than a fixed amount.

Worked example: enlarge a paper pattern

A fictional mosaic uses blue and white squares in the ratio 2:3. To use fifteen squares altogether, each of the five parts must contain three squares. The design therefore needs six blue and nine white. The checks are 6 + 9 = 15 and 6:9 simplifies to 2:3.

If a different version contains eight blue squares, the blue count has been multiplied by four from the original two. Multiply the white count by four as well: twelve white, twenty altogether. The same factor applies to both quantities; the total is an outcome, not an extra ratio term.

Recognise a change of rule

Adding two counters to each original count gives 4:5, which is not 2:3. Equal additions preserve the difference, while equal scaling preserves the ratio. These are different relationships. Saying that both quantities increased is not enough to establish proportion.

For this fixed pattern, white count is 1.5 times blue count. A graph of ideal quantities follows a line through zero. Whole-square designs allow only suitable whole-number batches, so a continuous line describes the proportional model without making half a physical counter part of the actual design.

Explain rather than guess

A useful ratio solution states the ordered comparison, one-part value, scale factor and total check. It should also say what is being held fixed. A designer can deliberately change the ratio for a new effect; that is a new pattern rather than a failed calculation.

Ratio reasoning appears in maps, images and repeated motifs. It does not guarantee that every property scales in the same way: doubling a rectangle’s side lengths doubles its perimeter but multiplies its area by four. Identify the quantity before applying a familiar factor.

Only if you feel like trying

Plan two fictional mosaic batches

  1. Write the blue: white ratio and divide a fifteen-square total into five equal parts.
  2. Calculate the version with eight blue squares, checking both ratio and total.
  3. Compare multiplying both counts by two with adding two to each; describe the different rules.

Optional. Use paper or talk through the supplied fictional example. No purchase, personal information, account, public sharing or practical experiment is needed.