If one number is greater than another by 23, and the difference of four times the smaller and half the larger is 90, what are the two numbers?

To solve this problem, we start by defining the two numbers. Let the smaller number be (x) and the larger number be (y). According to the information given, we can write two equations based on the problem's statements:

1. The first statement tells us that one number is greater than the other by 23, which we can express as:

[

y = x + 23

]

2. The second statement describes the relationship between four times the smaller number and half of the larger number:

[

4x - \frac{y}{2} = 90

]

Now, substitute the expression for (y) from the first equation into the second equation:

[

4x - \frac{x + 23}{2} = 90

]

Next, multiply through by 2 to eliminate the fraction:

[

8x - (x + 23) = 180

]

Distributing the negative sign gives:

[

8x - x - 23 = 180

]

[

7x - 23 = 180

]

Now, add 23 to both sides:

[

7x = 203