Let's denote the initial number of students in the dormitory as "N," and the rate at which food is consumed by each student per day as "R" (measured in student-days per day). We're given that initially, there was enough food to feed 300 students for 15 days, so:

Initial food supply = N * R * 15

Now, due to the admission of some new students, the food only lasted for 10 days, so:

Remaining food supply = (N + X) * R * 10, where "X" represents the number of new students admitted.

Since the amount of food is the same in both cases, we can set the two expressions equal to each other:

N * R * 15 = (N + X) * R * 10

Now, we can cancel out the common factor of "R" on both sides of the equation:

N * 15 = (N + X) * 10

Now, we can solve for "X," the number of new students admitted:

15N = 10(N + X)

15N = 10N + 10X

Subtract 10N from both sides:

5N = 10X

Divide both sides by 5 to solve for "X":

X = (5N) / 10

X = (1/2) * N

Now, we're given that X should be an integer because we can't admit a fraction of a student. Therefore, (1/2) * N must be a whole number.

To find out how many new students were admitted, we need to find a value of "N" where (1/2) * N is a whole number. Since we're looking for the number of new students, we want to find a value of "N" such that (1/2) * N is an integer.

If we let "N" be a multiple of 2, then (1/2) * N will be a whole number. Let's try "N = 300" because it's given that initially, there was enough food for 300 students:

X = (1/2) * 300 X = 150

So, if "N" was initially 300 students, then 150 new students were admitted to the dormitory.