Plural of “the [x] of [y]”

Question

What is the proper usage between 1) and 2)?
1) "Determine the values of the coefficients."
2) "Determine the value of the coefficients."
Each coefficient has one value in this context.

Similarly, which of the following forms is correct?
1) "Write the values of the coefficients in the book."
2) "Write the value of the coefficients in the book."

Is the rule consistent for all possible [x] of [y], excluding constructs such as "every" and word-specific exceptions such as "cattle"? I'm aware this question is very vague. Basically, I'm looking for clues like "the rule depends on whether you want to emphasise [x] over [y]", etc.

I read about the similar question 'Does a [x] of [y] take a singular or plural verb?', but I'm still in doubt after reviewing the discussion.

Thanks for any help!

Answer 1

Having taken many math and science courses in my life, I would say that there is certainly a difference between your choices and which one is correct depends on what you mean. Though I cannot provide you with formal gramatical reasons, my interpretation is as follows:

1) "Determine the values of the coefficients."

This phrase means that there are at least two coefficients and at least two values. This could arise in several ways. Either x = y = {a, b, ...} where x and y are coefficients and {a, b, ...} are multiple values, or x = a and y = b where a and b are different values, or a combination of these.

2) "Determine the value of the coefficients."

This phrase means that there are at least two coefficients but only one value. This is only the case when x = y = a where x and y are coefficients and a is a value.

The only exception to this that comes to mind is if you have a series of multiple questions. In that case, the instruction should either use the plural of both words or a phrasing that applies to each question individually with the implication that the instruction should be applied many times.

I will say that many teachers are not as careful in their wording and I have seen much more misleading grammar in science questions. I honestly doubt many students would be tripped up by this relatively minor point provided they understand the math involved. Since you asked, however, I would say that the first choice (with both words pluralized) is the more broadly correct and therefore the best choice for most situations.