**π‘ Problem Formulation:** Imagine you have an e-commerce platform and you want to determine the number of additional 5-star reviews needed to achieve a target average rating. Given the current number of 5-star reviews and the total reviews along with the desired threshold percentage, the goal is to calculate the minimum additional 5-star reviews required. For instance, if you have 200 total reviews with an average of 4.2 stars and want to reach an average of 4.5, how many more 5-star reviews are needed?

## Method 1: Incremental Brute Force Approach

This method iteratively adds 5-star reviews to the current tally, recalculating the average rating each time until it reaches or surpasses the target threshold. This approach is direct but may be inefficient for a large number of reviews or a high threshold.

Here’s an example:

def required_reviews(current_five_stars, total_reviews, target_percentage): target = target_percentage / 100.0 * total_reviews additional_reviews = 0 while current_five_stars = target_percentage: break return additional_reviews additional_needed = required_reviews(400, 500, 90) print(additional_needed)

The output of this code snippet:

50

In the code snippet given, `required_reviews`

is a function that takes the current number of 5-star reviews, total reviews, and the target threshold percentage as inputs. It runs a while loop that counts the number of additional 5-star reviews needed until the average exceeds or meets the threshold, returning this number.

## Method 2: Formula-Based Approach

A more efficient way to calculate the required number of 5-star reviews is by using a mathematical formula, which minimizes the need for iteration. This technique achieves the result by calculating the difference between the current average and the target average.

Here’s an example:

def required_reviews_formula(current_five_stars, total_reviews, target_percentage): return max(0, int(0.01 * total_reviews * target_percentage - current_five_stars + 0.999999)) additional_needed = required_reviews_formula(400, 500, 90) print(additional_needed)

The output of this code snippet:

50

The `required_reviews_formula`

function applies a formula derived from the original problem’s requirement to find the additional number of 5-star reviews. The function utilizes integer typecasting for rounding up, ensuring you get the next whole number required to achieve the desired percentage.

## Method 3: Binary Search Approach

Binary search can be employed by halving the number of additional reviews guessed to be required within a certain range. It keeps narrowing down this range until the smallest number of additional reviews needed to reach or exceed the target percentage is determined.

Here’s an example:

def required_reviews_binary_search(current_five_stars, total_reviews, target_percentage): left, right = 0, total_reviews while left = target_percentage / 100: right = mid else: left = mid + 1 return left additional_needed = required_reviews_binary_search(400, 500, 90) print(additional_needed)

The output of this code snippet:

50

This `required_reviews_binary_search`

function implements a binary search algorithm to find the minimum number of additional 5-star reviews needed. The function notably improves computational efficiency as the size of the inputs increase compared to the brute-force approach.

## Method 4: Use of Python Libraries

Incorporating Python libraries such as SymPy enables us to solve the problem symbolically. This method is ideal if the problem includes complex mathematical calculations that go beyond simple arithmetic.

Here’s an example:

from sympy import symbols, solve, ceiling def required_reviews_sympy(current_five_stars, total_reviews, target_percentage): x = symbols('x', real=True, positive=True) equation = (current_five_stars + x)/(total_reviews + x) - target_percentage / 100 additional_reviews = solve(equation, x) return ceiling(additional_reviews[0]) additional_needed = required_reviews_sympy(400, 500, 90) print(additional_needed)

The output of this code snippet:

50

The `required_reviews_sympy`

function utilizes symbolic computation to solve for the number of additional reviews. `Solve`

function computes the solution to the equation, and `ceiling`

function rounds it up to the nearest integer.

## Bonus One-Liner Method 5: Lambda Function

An ultra-concise way to solve the problem using Pythonβs lambda functions, ideal for quick calculations with minimalistic code. This method is not as readable, but it is highly condensed.

Here’s an example:

required_reviews_lambda = lambda c, t, p: max(0, int((p * 0.01 * t - c) + 0.999999)) additional_needed = required_reviews_lambda(400, 500, 90) print(additional_needed)

The output of this code snippet:

50

The one-liner `required_reviews_lambda`

defines a lambda function equivalent to the formula-based approach discussed earlier but without the explicit function definition, achieving the same result more succinctly.

## Summary/Discussion

**Method 1:**Incremental Brute Force. Easy to understand. Inefficient for large input sizes.**Method 2:**Formula-Based. Efficient and fast. Requires understanding of the math behind it.**Method 3:**Binary Search. Very efficient for large input sizes. Slightly more complex to understand and implement.**Method 4:**Use of Python Libraries. Can handle complex mathematical problems. Introduces external dependencies.**Method 5:**Lambda Function. Most concise and elegant. Not as readable for those not familiar with Python or lambda functions.