**π‘ Problem Formulation:** The challenge is to determine if a particular ratio of cost to quantity can be achieved within specified ranges for each. For instance, given a cost range from $25 to $50 and a quantity range from 100 to 200 units, we want to know if a desired cost-to-quantity ratio, like 1:4, is possible. This article explores five methods to solve this in Python.

## Method 1: Brute Force Iteration

The Brute Force Iteration method involves iterating through all possible combinations of costs and quantities within the given ranges to check if the desired ratio can be achieved. This method is easy to implement but may be inefficient for large ranges as it checks each possible pair.

Here’s an example:

def find_ratio(cost_range, quantity_range, desired_ratio): for cost in range(cost_range[0], cost_range[1] + 1): for quantity in range(quantity_range[0], quantity_range[1] + 1): if cost / quantity == desired_ratio: return True return False # Example usage: is_possible = find_ratio((25, 50), (100, 200), 0.25)

Output: `True`

or `False`

This code snippet defines a function called `find_ratio`

that takes a cost range, a quantity range, and a desired ratio. It iterates through every possible cost and quantity within those ranges to check for the ratio. The function returns `True`

if the desired ratio is found and `False`

otherwise.

## Method 2: Ratio Boundary Checking

Ratio Boundary Checking simplifies the problem by checking if the desired ratio is between the minimum and maximum possible ratios that can be formed from the given ranges. This method is efficient as it avoids checking each combination and instead uses range analysis.

Here’s an example:

def check_ratio_bounds(cost_range, quantity_range, desired_ratio): min_ratio = cost_range[0] / quantity_range[1] max_ratio = cost_range[1] / quantity_range[0] return min_ratio <= desired_ratio <= max_ratio # Example usage: is_possible = check_ratio_bounds((25, 50), (100, 200), 0.25)

Output: `True`

or `False`

In the code snippet, the function `check_ratio_bounds`

is defined to calculate the minimum and maximum ratios possible within the given cost and quantity ranges. By comparing these with the desired ratio, the function quickly determines its possibility without iteration.

## Method 3: Mathematical Inference

Mathematical Inference uses algebraic manipulation to transform the problem into a solvable inequality. This method provides a precise mathematical approach to conclude whether the ratio is achievable.

Here’s an example:

def is_ratio_achievable(cost_range, quantity_range, desired_ratio): return desired_ratio * quantity_range[0] = cost_range[0] # Example usage: is_possible = is_ratio_achievable((25, 50), (100, 200), 0.25)

Output: `True`

or `False`

The function `is_ratio_achievable`

takes cost and quantity ranges along with the desired ratio, applying algebra to the problem. It checks if the desired ratio falls between the scaled cost range. This method is a quick and effective solution for the problem.

## Method 4: Optimized Search

The Optimized Search method employs binary search or other searching algorithms to find if the desired ratio is attainable. This is faster than brute force for large datasets since it narrows down the search space significantly.

Here’s an example:

# Assuming ranges are sorted and there is no zero in quantity ranges. # We will only implement a pseudo-binary search for demonstrative purposes. def optimized_search(cost_range, quantity_range, desired_ratio): # Binary search logic goes here # For simplicity, it's omitted. pass # This method would require a proper implementation of the binary search algorithm.

Output: Typically `True`

or `False`

after implementation.

This code snippet outlines a proposed function `optimized_search`

that would utilize a searching algorithm to find the desired ratio. It hints at a faster approach compared to brute force, though it requires an implementation of a suitable optimized search algorithm.

## Bonus One-Liner Method 5: Comprehension Check

This single-line approach uses Python’s list comprehensions to tersely express the brute force check. It’s concise and readable for small ranges, but not as efficient for larger ones.

Here’s an example:

is_possible = any(c/q == 0.25 for c in range(25, 51) for q in range(100, 201))

Output: `True`

or `False`

.

The one-liner code utilizes a list comprehension to iterate through the cost and quantity ranges, checking if any combination matches the desired ratio. It’s a condensed version of Method 1 for quick and simple cases.

## Summary/Discussion

**Method 1: Brute Force Iteration.**Easy to implement. Suitable for small ranges. Inefficient for large ranges due to time complexity.**Method 2: Ratio Boundary Checking.**More efficient by avoiding iteration through combinations. Quick results through comparative boundary checks.**Method 3: Mathematical Inference.**Direct approach through algebraic analysis. No iteration necessary. Provides a precise and fast solution.**Method 4: Optimized Search.**Best for large data sets. Requires advanced implementation of search algorithms. Highly efficient once set up.**Bonus One-Liner Method 5: Comprehension Check.**Quick and concise. Best for small and simple ranges. Not recommended for larger data due to inefficiency.