A motorist averages 30 kph on ordinary loads and 12 kph on roads under repair. His average speed for a distance of 50 km is 24 km/h. What length of the road is under repair?
avgspeed=distance/time=50km/time=24km/hr
time=50/24 hrs
time rough+timesmooth= 50/24
sum of distances equals 50km or
30*timerough+12*(50/24-timerough)=50
timerough(30 -12)=50(1-1/2)
time rough= 25/18 hrs
distance rough= 25/18 * 12=50/3 km
check all that.
Let the distance on good road be x km
then distance on rough road is 50-x km
time on good road = x/30
time on rough road = (50-x)/12
total time = x/30 + (50-x)/12
= (2x + 250 - 5x)/60
= (250 - 3x)/60
but total time = 50/24
so (250-3x)/60 = 50/24
3000 = 6000 - 72x
72x = 3000
x = 3000/72 = 125/3 or 41 .67 km
so 42.67 km were good road and
50 - 41.7 or 8.3 km were under repair.
check: total time = 50/24 = 2.083 hrs
time on good road = 41.67/30 = 1.389
time on rought road = 8.3/12 = .692
total time = 2.081 , close enough allowing for round-off
all checks out
Let's assume that the length of the road under repair is x km.
The distance traveled on ordinary roads would be (50 - x) km.
To calculate the time taken for each segment, we divide the distance by the speed.
Time taken on ordinary roads: (50 - x) km / 30 km/h = (50 - x)/30 hours
Time taken on roads under repair: x km / 12 km/h = x/12 hours
The average speed is calculated by dividing the total distance by the total time taken:
Average speed = Total distance / Total time
24 km/h = 50 km / ( (50 - x)/30 hours + x/12 hours )
To simplify the equation, we can multiply both sides by the least common multiple (LCM) of 30 and 12, which is 60:
24 * 60 = 50 * 60 / (2*(50 - x) + 5*x)
1440 = 3000 / (100 - 3x)
1440(100 - 3x) = 3000
144000 - 4320x = 3000
144000 - 3000 = 4320x
141000 = 4320x
x = 141000 / 4320
x ≈ 32.64 km
Therefore, the length of the road under repair is approximately 32.64 km.
To solve this problem, let's assume that the length of the road under repair is 'x' km.
We know that the motorist averages 30 kph on ordinary roads, so the time taken to cover the distance on ordinary roads is given by:
Time taken on ordinary roads = Distance / Speed
= (50 - x) / 30
Similarly, the motorist averages 12 kph on roads under repair, so the time taken to cover the distance on roads under repair is given by:
Time taken on roads under repair = x / 12
Now, we can use the average speed formula to represent the total time taken for the entire distance of 50 km:
Average speed = Total distance / Total time
Given that the average speed for the entire distance is 24 km/h, we can write:
24 = 50 / ( (50 - x) / 30 + x / 12)
To solve for 'x', we can simplify this equation:
24 = 50 / ( (50 - x) / 30 + x / 12)
24 = 50 / ( 5/6 - x/30 + x / 12)
24 = 50 / ( (25 - x)/30)
24 * (25 - x)/30 = 50
24 * (25 - x) = 50 * 30
600 - 24x = 1500
-24x = 900
x = 900 / -24
x ≈ -37.5
Since the length of the road can't be negative, we disregard this negative value. Therefore, the length of the road under repair is approximately 37.5 km.