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Equation 4 · Part 2 · Astra's 63%-Cheaper Claim Has to Beat a Price Disadvantage Nobody Mentions

=

(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}
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

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

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

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