Step-by-step explanation:

If two numbers are directly proportional to 7 and 4, we can say that:

x/y = k (where k is a constant of proportionality)

We can find the value of k by using the information that the sum of the two numbers is 55:

x + y = 55

We can use the first equation to write:

x = ky

We can substitute the value of x in the second equation to get:

ky + y = 55

We can then solve for y in terms of k:

y(k + 1) = 55

y = 55 / (k + 1)

We can then use this value of y to find the value of x by using the first equation:

x = ky = k(55 / (k + 1))

Now we can use the information that x is directly proportional to 7 and y is directly proportional to 4 to find the value of k:

x = 7k y = 4k

So we can say that x = 7k and y = 4k.

We can solve for k by taking the value of x and y and divide by 7 and 4 respectively.

k = x/7 = y/4

Now we have the value of k and we can solve for x and y.

x = 7k = 7(55 / (k + 1)) = 385 / (k + 1)

y = 4k = 4(55 / (k + 1)) = 220 / (k + 1)

So the two numbers are 385 and 220.