tkfull wrote:

As I was practicing I came across the following question:

At a certain school, the ratio of students to teachers is 3:5, and the ratio of teachers to administrators is 4:5. Which of the following could be the total number of students at the school? (Indicate all that apply):

A. 150

B. 180

C. 200

D. 240

E. 300

I got the right answer, but my method was somewhat convoluted. The explanation offered by the website seems much easier (especially because these types of ratio problems confuse me). This their explanation:

The Correct answers are B, D, and E. Whenever a question provides multiple ratios with a common element, manipulate the ratios so that the common element has the same value in both ratios. In this case, the common element is teachers, so we should manipulate the ratios so that the number of teachers is the least common multiple of 5 and 4: 20. In the first ratio: students/teachers = 3/5 = 12/20. In the second ratio: teachers/administrators = 4/5 = 20/25. The ratio of students to teachers to administrators is thus 12:20:25. The number of students must therefore be a multiple of 12. Any choice that is a multiple of 12 is a potential value for the number of students. The correct answer is B, D, and E.

The bold part of the explanation is the part that I don't fully buy because I can't come up with the logic for why that must be the case. Can anyone verify (and perhaps attempt to explain why) that this is a valid method and will always work for this type of question.

Much appreciated!

Because it has asked for total number of students at the school and in the school we also have teachers and administrators.

By combining the three ratios 12:20:25 i.e 57 (12+20+25)this is the minimum total member in the school including students, teachers and administrators in the school.

The value of students or teachers or administrators goes on increasing only in this ratios in the school.

Now if it ask for teacher then it will be the multiple of 20 similarly for administrators it will be multiple of 25

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