7.9 Summary
Key Points
- Two quantities change proportionally when both are multiplied by the same factor, not when the same amount is added to or subtracted from them.
- A ratio a : b says: for every a units of the first quantity there are b units of the second. The numbers a and b are its terms.
- Reduce a ratio to simplest form by dividing both terms by their HCF.
- Two ratios are in proportion (written a : b :: c : d) when their simplest forms agree, equivalently when ad = bc (cross multiplication).
- The Rule of Three (Trairāśika) finds a missing fourth term: if a : b :: c : d, then d = \dfrac{bc}{a}, “multiply the phala by the icchā and divide by the pramāṇa.”
- To divide a quantity x in the ratio m : n, give the first part m \times \dfrac{x}{m+n} and the second n \times \dfrac{x}{m+n}.
- Always compare quantities in the same units; conversions of length, area, volume and temperature make this possible.
- Not every “looks-like-a-proportion” situation is a direct proportion: when one quantity rises as the other falls (like speed and time), the Rule of Three does not apply, that is inverse proportion, studied in Proportional Reasoning-2.