The linear density of a string decreases by 1/4 when it is stretched. what happens to the speed of a wave on the string?

The speed doubles.

Explanation:

The speed of a wave in a string is:

v = √(T / ρ)

where T is the tension and ρ is the linear density (mass per length).

If T is held constant, and ρ decreases by 1/4, then:

v = √(T / (ρ/4))

v = √(4T / ρ)

v = 2√(T / ρ)

The speed doubles.