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(1−s)(10)+s(1.00)(1−s)(10)+s(0.25)=10−9s10−9.75s\frac{(1-s)(10) + s(1.00)}{(1-s)(10) + s(0.25)} = \frac{10 - 9s}{10 - 9.75s}

Why this formula appears here

Call the cache-hit share s — the fraction of input tokens that are cache reads rather than fresh reads. The blended input-token cost per model, in dollars per million tokens, is (1-s) ×\times 10 + s ×\times r , where r is each model’s cache-read rate: $1.00 for Astra, $0.25 for Fable 5.1. The ratio of Astra’s blended input cost to Fable 5.1’s, at a given cache-hit share, is: (1−s)(10)+s(1.00)(1−s)(10)+s(0.25)=10−9s10−9.75s\frac{(1-s)(10) + s(1.00)}{(1-s)(10) + s(0.25)} = \frac{10 - 9s}{10 - 9.75s}. Run that ratio across a stated bracket rather than picking one number and presenting it as fact:

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(1−s)(10)+s(1.00)(1-s)(10) + s(1.00)

Numerator: (1-s)(10) + s(1.00)

The complete quantity above the fraction bar.

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(1−s)(10)+s(0.25)(1-s)(10) + s(0.25)

Denominator: (1-s)(10) + s(0.25)

The complete quantity below the fraction bar; it must be nonzero for this division.

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10−9.75s10 - 9.75s

Denominator: 10 - 9.75s

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

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(1−s)(10)+s(1.00)(1−s)(10)+s(0.25)=10−9s10−9.75s\frac{(1-s)(10) + s(1.00)}{(1-s)(10) + s(0.25)} = \frac{10 - 9s}{10 - 9.75s}

Equation 4 · AI Industry

Astra's 63%-Cheaper Claim Has to Beat a Price Disadvantage Nobody Mentions

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Call the cache-hit share s — the fraction of input tokens that are cache reads rather than fresh reads. The blended input-token cost per model, in dollars per million tokens, is (1-s) ×\times 10 + s ×\times r , where r is each model’s cache-read rate: $1.00 for Astra, $0.25 for Fable 5.1. The ratio of Astra’s blended input cost to Fable 5.1’s, at a given cache-hit share, is: (1−s)(10)+s(1.00)(1−s)(10)+s(0.25)=10−9s10−9.75s\frac{(1-s)(10) + s(1.00)}{(1-s)(10) + s(0.25)} = \frac{10 - 9s}{10 - 9.75s}. Run that ratio across a stated bracket rather than picking one number and presenting it as fact:

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