Ared, 0.1 kg marble hits a blue, 70 g marble. after the collision the red marble is moving at 10 m/s and the blue marble is moving at 15 m/s. what was the red marbles original speed before the collision

By law of conservation of momentum:
Summation of momentum before collision = Summation of momentum after collision

Momentum = mass*velocity

If v and u represents initial and final velocities and assuming that the bred marble was at rest before collision;
m1u1+m2u2 = m1v1+m2v2
But u2 = 0
Then,
m1u1 = m1v1+m2v2
u1 = (m1v1+m2v2)/m1

Where;
u1 = Initial velocity of red marble before collision
m1 = mass of blue marble = 0.1 kg
v1 = Final velocity of red marble = 10 m/s
v2 = Final velocity of blue marble = 15 m/s
m2 = mass of blue marble = 70 g = 0.07 kg

Substituting;
u1 = (0.1*10+0.07*15)/0.1 = 20.5 m/s

Therefore, red marble original velocity before collision was 20.5 m/s.