Consider the following students method of solving the proportion 8/5 = 20/x and determine whether the methods were correct: Richie: First I divided 20 by 8 and got 2 1/2. Then I multiplied 2 1/2 by 5 and got 12 1/2 for my answer. Martha: I think of it this way. Every time a person takes 8 steps, another person takes 5 steps. So 16 steps correspond to 10 steps, and 24 steps correspond to 15 steps. As 20 is halfway bwtween 10 and 15. I got 12 1/2 for the answer. Explain your reasoning.

I will be happy to critique your thinking.

To determine whether Richie and Martha's methods are correct for solving the proportion 8/5 = 20/x, we can compare their reasoning step by step.

Let's start with Richie's method:
1. Richie states that he divided 20 by 8 and got 2 1/2. This step is incorrect because he divided 20 by 8, which should have resulted in 2.5 (a decimal), not 2 1/2 (a mixed fraction).
2. Then Richie multiplied 2 1/2 by 5 and got 12 1/2 for his answer. Since his initial division was incorrect, the subsequent multiplication is also incorrect. The correct approach would have been to multiply 2.5 by 5, which equals 12.5.

Next, let's look at Martha's method:
1. Martha uses a different approach, considering the proportion as a ratio of steps. She explains that for every 8 steps taken by one person, another person takes 5 steps.
2. Martha then creates a proportion between the steps. According to her reasoning, if 16 steps correspond to 10 steps and 24 steps correspond to 15 steps, then as 20 is halfway between 10 and 15, it should correspond to a proportion of halfway between 16 and 24.
3. Martha concludes that the answer to 8/5 = 20/x should be 12 1/2 because 20 falls between 16 and 24.

Based on this analysis, we can conclude that Richie's method is incorrect because he made errors in both the division and multiplication steps. Martha's method, on the other hand, provides a logical and correct reasoning for solving the proportion. Thus, Martha's method is correct.