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Equation 4 · Part 3 · Astra's 63%-Cheaper Claim Has to Beat a Price Disadvantage Nobody Mentions

fraction

(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}
fraction

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

The passage around this formula

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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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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