When defining a member function of a class template outside the class definition, the rules for template parameter repetition depend on the context:

1. Before :: (Qualified Name)

You must repeat the template parameters because you're specifying the full type of the class template:

template<std::signed_integral T>
Ratio<T>& Ratio<T>::operator+=(...) { ... }
//      ^^^^^^^^ Required: `Ratio<T>`

2. Return Type

You must repeat the template parameters because the return type is outside the scope of the class:

template<std::signed_integral T>
Ratio<T>& Ratio<T>::operator+=(...) { ... }
// ^^^^^^^ Required: `Ratio<T>&`

3. Function Parameters

You do not need to repeat them because:

• The Ratio<T>:: prefix already establishes the scope.

• The compiler can deduce Ratio inside the parameter list as Ratio<T> (due to injected-class-name behavior).

template<std::signed_integral T>
Ratio<T>& Ratio<T>::operator+=(const Ratio& R) { ... }
//                              ^^^^^^ No `<T>` needed here

And why this happen ?

• Inside the class, the template parameter is "injected," so Ratio alone means Ratio<T>:

template<std::signed_integral T>
struct Ratio {
    Ratio& operator+=(const Ratio& R); // `Ratio` = `Ratio<T>`
};

• Outside the class, you must explicitly qualify the name before :: and in the return type. However, after ::, the compiler knows Ratio refers to Ratio<T>.