The sum of two numbers is 184. If one-third of the one exceeds one - seventh of the other by 8, find the smaller number?

Let's denote the two numbers as x and y.

From the problem, we know that:

1. x + y = 184 (because the sum of the two numbers is 184)

2. (1/3)x - (1/7)y = 8 (because one-third of the one exceeds one-seventh of the other by 8)

We can solve these two equations to find the values of x and y.

First, let's multiply the second equation by 21 to get rid of the fractions:

7x - 3y = 168

Now we have two new equations:

1. x + y = 184

2. 7x - 3y = 168

We can multiply the first equation by 3 and the second equation by 1:

3x + 3y = 552

7x - 3y = 168

Now, add these two equations together:

10x = 720

Divide by 10:

x = 72

Substitute x = 72 into the first equation:

72 + y = 184

Subtract 72 from both sides:

y = 184 - 72 = 112

So, the two numbers are 72 and 112. The smaller number is 72.