One skier covered the distance of 20 km 20 minutes faster than the other skier. Find the speed of both skiers if one of them is travelling 2 km/hour faster than the other one.
speed of slower --- x km/h
speed of faster ---- x+2 km/h
20/x - 20/(x+2) = 20/60 = 1/3
times 3x(x+2) , the LCD
60(x+2) - 60x = x(x+2)
x^2 + 2x - 120 = 0
(x+12)(x-10) = 0
x = 10 or x = -12, the ladder is silly
The slower one went 10 km/h, and the faster went 12 km/h
check:
20/10 - 20/12
= 1/3 hr or 20 minutes.
My answer is correct
Let's say the speed of the slower skier is x km/hour.
According to the problem, the speed of the faster skier is (x + 2) km/hour.
Distance = Speed x Time
For the slower skier:
Distance = 20 km
Time = t (in hours)
For the faster skier:
Distance = 20 km
Time = t - 20/60 = t - 1/3 (converted 20 minutes to hours)
Now we can calculate the time for each skier:
Time for the slower skier = Distance/Speed = 20/x
Time for the faster skier = Distance/Speed = 20/(x + 2)
Since the faster skier is 20 minutes (1/3 hours) faster than the slower skier, we can set up the equation:
20/x = 20/(x + 2) + 1/3
To get rid of the fraction, let's multiply both sides of the equation by 3x(x + 2):
3(20)(x + 2) = 20(3x) + x(x + 2)
Now, solve for x:
60(x + 2) = 60x + x^2 + 2x
60x + 120 = 60x + x^2 + 2x
Rearranging the equation:
x^2 + 2x - 120 = 0
Now, let's solve this quadratic equation by factoring or using the quadratic formula:
(x - 10)(x + 12) = 0
Setting each factor equal to zero to solve for x:
x - 10 = 0 or x + 12 = 0
x = 10 or x = -12
Since speed cannot be negative, we disregard x = -12.
Therefore, the speed of the slower skier is 10 km/hour, and the speed of the faster skier is (10 + 2) = 12 km/hour.
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.