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.