The difference of two numbers is 0.77. if the larger number is increased 5 times, then the difference becomes 77. find the numbers.

For this case, the first thing we must do is define variables.
 We have then:
 x = larger number
 y = smaller number
 By writing the system of equations we have:
 x-y = 0.77
 5x-y = 77
 Solving the system we have:
 x = 19.0575
 y = 18.2875
 
 The numbers are:
 x = 19.0575
 y = 18.2875