Bunuel wrote:

Tough and Tricky questions: Ratios.

On a certain field trip, the ratio of teachers to students was 3 : 7. What was the number of parents on the field trip?

(1) On the field trip, the ratio of the number of students to parents was 5 : 1.

(2) The number of parents on the trip was less than 20.

Kudos for a correct solution. Official Solution:On a certain field trip, the ratio of teachers to students was 3 : 7. What was the number of parents on the field trip?We need to determine the number of parents on a field trip. Since this problem involves three unknown quantities, we will assign each quantity a variable. The number of teachers, students, and parents will be \(t\), \(s\), and \(p\), respectively. The problem tells us that \(t : s\), the ratio of teachers to students, was 3 : 7.

Statement 1 tells us that \(s : p\), the ratio of the number of students to parents, was 5 : 1. Since this ratio and the ratio in the prompt contain \(s\), the number of students, then it is possible to create a ratio using all three variables by finding the least common multiple (LCM) of the two values for \(s\).

The LCM of 7 and 5 is 35; multiply both ratios so that their \(s\) term is 35. Thus, \(t : s = 3 : 7\) becomes \(t : s = 15 : 35\), and \(s : p = 5 : 1\) becomes \(s : p = 35 : 7\).

Combine the ratios to get \(t : s : p = 15 : 35 : 7\).

This ratio gives us the relative quantities of teachers, students, and parents. However, any numbers of students, teachers, and parents that satisfy this ratio could have gone on the field trip. The numbers of teachers, students, and parents on the trip could have been 15, 35, and 7, respectively; they could also have been 30, 70, and 14. In fact, any multiple of this ratio would satisfy this statement. Statement 1 is therefore NOT sufficient. Eliminate answer choices A and D. The correct answer choice must be B, C, or E.

Statement 2 tells us that the number of parents on the field trip was less than 20. This does not allow us to identify the number of parents on the field trip; it could be anywhere between 0 or 19 and still satisfy the statement. This statement is NOT sufficient to answer the question. Eliminate answer choice B. The correct answer choice is either C or E.

Taken together, the statements give us the relative amounts of all three quantities and place an upper bound on the number of parents: \(p \lt 20\). However, as we saw in looking at statement 1, the first two multiples of our ratio both call for less than 20 parents. The possible values of \(p\) are limited to 7 and 14, but there is not enough information to find a single value. Both statements together are NOT sufficient.

Answer: E.

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