To solve this question, we will first calculate the value of the 7th mean and the (m-1)th mean. Then, using the given ratio, we can find the value of m.

Let's start by calculating the value of the 7th mean. We know that there are m arithmetic means between 1 and 31. So, the common difference between the means is given by:

d = (31 - 1) / (m + 1) = 30 / (m + 1)

Now, we can find the value of the 7th mean using the formula for the nth term of an arithmetic progression:

a_n = a + (n-1)d

where a is the first term, n is the term number, and d is the common difference.

In this case, the first term (a) is 1, and the term number (n) is 7. Plugging in these values, we have:

a_7 = 1 + (7-1)d
= 1 + 6d
= 1 + 6 * (30 / (m + 1))
= 1 + 180 / (m + 1)
= (m + 1 + 180) / (m + 1)

Similarly, we can find the value of the (m-1)th mean:

a_(m-1) = 1 + (m-1)d
= 1 + (m-1) * (30 / (m + 1))
= 1 + (30m - 30) / (m + 1)
= (m + 1 + 30m - 30) / (m + 1)
= (31m - 29) / (m + 1)

Given that the ratio of the 7th mean to the (m-1)th mean is 5:9, we can set up the following equation:

(a_7) / (a_(m-1)) = 5/9

Substituting the values we found earlier:

[(m + 1 + 180) / (m + 1)] / [(31m - 29) / (m + 1)] = 5/9

Simplifying the equation:

[(m + 1 + 180) / (31m - 29)] * (m + 1) = 5/9

Cross-multiplying:

[(m + 1 + 180) * 9] = 5 * (31m - 29)

Expanding:

9m + 189 + 1620 = 155m - 145

Combining like terms:

9m - 155m = -145 - 189 - 1620

-146m = -1954

Solving for m:

m = 1954 / 146
m ≈ 13.4

Since m represents the number of arithmetic means, it must be a positive integer. The nearest positive integer to 13.4 is 14, which is the correct answer (option A).