How Can I Effectively Design A Word Problem That Illustrates The Concept Of Inverse Proportionality In A Mixture Scenario, Where The Ratio Of Two Ingredients Changes In Such A Way That Increasing One Ingredient By A Certain Percentage Results In A Corresponding Decrease In The Other Ingredient, While Still Allowing Students To Calculate The Resulting Proportions Using Algebraic Equations?

Problem:

Tom is making a special cleaning solution where the strength of the solution depends on the product of the amounts of two key ingredients, A and B. The strength ${ S }$ is given by ${ S = A \times B }$. Initially, Tom uses 5 liters of ingredient A and 4 liters of ingredient B, resulting in a strength of 20 liter². Tom wants to increase the amount of ingredient A by 25% while keeping the strength the same. How many liters of ingredient B should Tom use now?

Solution:

1. Calculate the new amount of ingredient A: ${ \text{New amount of A} = 5 \text{ liters} \times 1.25 = 6.25 \text{ liters} }$

2. Set up the equation for the strength ${ S }$: ${ 20 = 6.25 \times B }$

3. Solve for ${ B }$: ${ B = \frac{20}{6.25} = 3.2 \text{ liters} }$

Answer: Tom should use 3.2 liters of ingredient B.

This problem illustrates inverse proportionality because as the amount of ingredient A increases, the amount of ingredient B must decrease to keep the product (and thus the strength) constant.