Consider the following students’ methods 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 by 5 and got 12 ½ 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 between 16 and 24, I reasoned
that the answer was halfway between 10 and 15. I got
12 ½ for the answer.
Explain your reasoning.
I feel that both methods were correct in determining the answer, they were both based on comparison, the first one cross multiplied where the second one used equivalent ratios ?
Is that correct ? am I wrong or missing something ?
OK except for your typo. 12 1/2 is correct, not 2 1/2
Yes, your understanding is correct. Both Richie and Martha used different approaches to solve the proportion, but both methods are valid and lead to the correct answer.
First, let's understand their methods in more detail:
1. Richie's method:
Richie started by dividing 20 by 8, which is correct. This step gives him the value of x when 20 is proportionate to 8, which is 2.5. Then he multiplied this result by 5, which is also correct because he is scaling up the proportion by the same factor. The final result he obtained, 12.5, is the value of x in the given proportion.
2. Martha's method:
Martha reasoned using the concept of equivalent ratios. She realized that for every 8 steps taken by one person, there are 5 steps taken by another person. So, she found the ratio of 16 steps to 10 steps and the ratio of 24 steps to 15 steps. Since 20 is halfway between 16 and 24, she reasoned that the answer should be halfway between 10 and 15. Therefore, she obtained the same result as Richie, 12.5, for the value of x in the given proportion.
In conclusion, both Richie and Martha correctly solved the proportion using different approaches. Richie used cross multiplication and scaling, while Martha used the concept of equivalent ratios and interpolation. Both methods are valid and can be used to determine the missing value in a proportion.