# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Ratio and proportion: solving 3:4=x:10, checking back, and testing data for direct proportion](/en/derivations/derivation-mko-ratio-proportion-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-ratio-proportion)

Over the rationals: (1) prove that for b≠0 and d≠0 the proportion a:b=x:d has exactly one rational solution, x=ad/b; (2) use it to solve 3:4=x:10 and check back; (3) show that 4:3=x:10 has a different solution, so the order of a ratio matters; (4) decide whether the data (2,5), (4,10), (6,15), (8,21) fit a single direct proportion y=kx.

Ratio and proportion

derivation-mko-ratio-proportion-01





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