Final answer:
The equation given suggests m and n are multiples of 5. Factoring the equation and substituting m = 5k and n = k (where k is a natural number) indicates m ÷ n = 5k ÷ k which simplifies to 5. Hence, m ÷ n = 5.
Step-by-step explanation:
The student is asking to find the value of m ÷ n from the equation m² - 3m = 25n² - 15n, where m and n are natural numbers. First, we can isolate the terms involving m and n respectively, by placing all terms involving m on one side of the equation and all terms involving n on the other side:
m² - 3m + 0 = 25n² - 15n
We notice that the equation can be factored to take the form:
m(m - 3) = 25n(n - 3)
For this equation to hold true for natural number values of m and n, it is necessary that m and n are multiples of 5, and in particular, we can see that m = 5k and n = k where k is a natural number. This gives us:
(5k)^2 - 3(5k) = 25(k^2) - 15k
Simplifying both sides leads us to:
25k^2 - 15k = 25k^2 - 15k
Since the terms are now identical, the equation is true for all values of k, and thus for all corresponding values of m and n. We then calculate the ratio m ÷ n = (5k) ÷ k, which simplifies to:
5
So, the answer to the student's question is A) m ÷ n = 5.