Mr. Njinga cycled a distance of 42km from his village. On his return journey he increased his speed by 2km/h and took half an hour less. Calculate the average speed on the journey from his village.
speed on first part ---- x km/h
speed on return trip -- x+2 km/h
time on first trip = 42/x
time on 2nd trip = 42/(x+2)
42/x - 42/(x+2) =1/2
multiply each term by 2x(x+2)
84(x+2) - 84x = x(x+2)
84x + 168 - 84x = x^2 + 2x
x^2 + 2x - 168 = 0
(x-12)(x+14)=0
x = 12 or x =-14, let's reject the x = -14
x = 12 km/h
find each individual times by subbing this back into my time definitions.
average speed = 84/(sum of the above times)
= ....
Thank you very much
To calculate the average speed, we need to find the total time spent on the journey and the total distance traveled.
Let's start by finding the total time spent on the journey.
Let 'x' be Mr. Njinga's speed on his initial journey.
On the initial journey:
Distance = 42 km
Speed = x km/h
Time = Distance / Speed = 42 km / x km/h = 42/x hours
On the return journey:
Distance = 42 km
Speed = (x + 2) km/h (as he increased his speed by 2 km/h)
Time = Distance / Speed = 42 km / (x + 2) km/h = 42/(x + 2) hours
According to the problem, the return journey took half an hour less.
So, the new time taken for the return journey = 42/(x + 2) - 1/2 hours = (84 - (x + 2))/(2(x + 2)) hours.
Now, let's find the total time spent:
Total Time = Time spent on the initial journey + Time spent on the return journey
= 42/x + (84 - (x + 2))/(2(x + 2)) hours
= 42/x + (84 - x - 2)/(2(x + 2)) hours
= 42/x + (82 - x)/(2(x + 2)) hours
Now, let's find the average speed:
Average Speed = Total Distance / Total Time
Total Distance = 42 km (on the initial journey) + 42 km (on the return journey) = 84 km
Average Speed = 84 km / [42/x + (82 - x)/(2(x + 2)) hours]
Simplifying the expression:
Average Speed = 84 km / [(42x + 2(82 - x))/(x(x + 2))] = 84(x(x + 2))/(42x + 2(82 - x))
= 84(x(x + 2))/(42x + 164 - 2x)
= 84(x^2 + 2x)/(40x + 164)
= 2(x^2 + 2x)/(10x + 41)
So, the average speed on the journey from his village is 2(x^2 + 2x)/(10x + 41) km/h.
To calculate the average speed on the journey from his village, we need to find the total time taken for the entire journey and divide it by the total distance traveled.
Let's assume Mr. Njinga's initial speed while traveling from his village is 'x' km/h. On his return journey, his speed increases by 2km/h, so his speed becomes 'x + 2' km/h.
Now, let's break down the problem step by step:
Step 1: Calculate the time taken for the onward journey.
Time = Distance / Speed
Given that the distance is 42 km and the speed is 'x' km/h, the time taken for the outward journey is:
Time taken for the onward journey = 42 km / x km/h
Step 2: Calculate the time taken for the return journey.
Since Mr. Njinga increased his speed by 2 km/h, his speed for the return journey becomes 'x + 2' km/h. Additionally, it is mentioned that he took half an hour (0.5 hours) less on his return journey.
Time = Distance / Speed
Given that the distance is 42 km and the speed is 'x + 2' km/h, the time taken for the return journey is:
Time taken for the return journey = 42 km / (x + 2) km/h - 0.5 hours
Step 3: Calculate the average speed.
The average speed is calculated by dividing the total distance by the total time.
The total distance covered in the journey from the village is 42 km + 42 km = 84 km. (Since it is mentioned that he returned from the same point)
The total time taken for the entire journey is the sum of the time taken for the onward journey and the time taken for the return journey:
Total time taken = Time taken for the onward journey + Time taken for the return journey
Now we can substitute the values we calculated earlier:
Total time taken = 42 km / x km/h + (42 km / (x + 2) km/h - 0.5 hours)
Finally, the average speed is calculated by dividing the total distance by the total time taken:
Average speed = Total distance / Total time taken
Average speed = 84 km / Total time taken
So, to get the value of the average speed, substitute the value of Total time taken in the above expression and calculate the result.