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

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

Solution:
Let the two numbers be x and y.

Given:
x + y = 184
x = 184 - y -------(1)

Also, one-third of one number exceeds one-seventh of the other by 8, so:
x/3 - y/7 = 8
Multiply by 21:
7x - 3y = 168 ------(2)

From (1) and (2), we get
7(184 - y) - 3y = 168
⇒ 1288 - 7y - 3y = 168
⇒ 1288 - 10y = 168
⇒ - 10y = 168 - 1288
⇒ - 10y = - 1120
∴ y = 112

Now find x:
x = 184 - 112 = 72

Smaller number = 72