To find the speeds of both skiers, we can set up a system of equations based on the given information.
Let's call the speed of the slower skier "x" km/hour. Then, the speed of the faster skier would be "x + 2" km/hour.
We know that the skier who traveled 20 km faster also took 20 minutes less time. To convert this time difference to hours, we divide by 60:
20 minutes = 20/60 = 1/3 hours
Now, using the formula Time = Distance / Speed, we can write the following equation for the slower skier:
(20 km) / x = (20 km) / (x + 2) + 1/3
To solve this equation, we can first eliminate the denominators by multiplying both sides by x(x + 2):
20(x + 2) = 20x + x(x + 2)/3
Now, distribute and simplify:
20x + 40 = 20x + (x^2 + 2x)/3
Next, multiply through by 3 to clear the fractions:
60x + 120 = 60x + x^2 + 2x
Combine like terms and rearrange the equation:
0 = x^2 + 2x - 120
Now, we can factor the quadratic equation:
0 = (x + 12)(x - 10)
Setting each factor to zero, we get:
x + 12 = 0 --> x = -12 (extraneous solution, as speed cannot be negative)
x - 10 = 0 --> x = 10
Therefore, the speed of the slower skier is 10 km/hour, and the speed of the faster skier is 10 + 2 = 12 km/hour.
So, the speeds of the skiers are 10 km/hour and 12 km/hour, respectively.