1. **State the problem:** We want to find the number of 5-star evaluations $x$ needed to raise the current CSAT score $a$ (based on $b$ evaluations) to a target score of 4.70. 2. **Define variables:** - $a$: current average CSAT score - $b$: current number of evaluations - $c = 5$: the score of each new evaluation (5 stars) - $x$: number of new 5-star evaluations needed 3. **Formula setup:** The new average after adding $x$ evaluations of score $c$ is: $$\frac{a \times b + c \times x}{b + x} = 4.70$$ 4. **Solve for $x$:** Multiply both sides by $(b + x)$: $$a b + c x = 4.70 (b + x)$$ Expand the right side: $$a b + c x = 4.70 b + 4.70 x$$ Rearrange terms to isolate $x$: $$c x - 4.70 x = 4.70 b - a b$$ Factor $x$: $$x (c - 4.70) = b (4.70 - a)$$ Finally, solve for $x$: $$x = \frac{b (4.70 - a)}{c - 4.70}$$ 5. **Interpretation:** This formula calculates how many 5-star evaluations are needed to raise the average from $a$ to 4.70 given $b$ current evaluations. 6. **Important notes:** - The denominator $c - 4.70$ must be positive, which it is since $c=5$. - If $a$ is already 4.70 or higher, $x$ will be zero or negative, meaning no additional 5-star evaluations are needed. **Final formula:** $$x = \frac{b (4.70 - a)}{5 - 4.70}$$