CONSTANT SPEED
About "Constant speed"
Constant speed is also called as uniform rate which involves something travelling at fixed and steady pace or else moving at some average speed.
For example, A car travels 3 hours. It travels 30 miles in the first hour, 45 miles in the second hour and 75 miles in the third hour.
Speed in the first hour = 30 miles / hour
Speed in the second hour = 45 miles / hour
Speed in the third hour = 75 miles / hour
We have three different speeds in the three hour journey.
If we want to find the constant speed for the whole journey of three hours, we have to find the ratio between the total distance covered and total time taken.
That is, constant speed = (30 + 45 + 75) / 3
= 150 / 3
= 50 miles / hour
Based on the above example, the formula is to find the constant speed is given below.
If a person travels from A to B at some speed, say "x" miles per hour. He comes back from B to A at different speed, say "y" miles per hour. Both the ways, he covers the same distance, but at different speeds.
Then, the formula is to find the constant speed for the whole journey is given below.
Constant Speed - Examples
To have better understanding on "Constant-speed", let us look at some examples.
Example 1 :
David drove for 3 hours at a rate of 50 miles per hour, for 2 hours at 60 miles per hour and for 4 hours at a rate of 70 miles per hour. What was his constant-speed for the whole journey ?
Solution :
Step 1 :
Formula for constant-speed = Total distance / Total time taken.
And also, for for distance = Rate x Time
Step 2 :
Distance covered in the first 3 hours = 50 x 3 = 150 miles
Distance covered in the next 2 hours = 60 x 2 = 120 miles
Distance covered in the last 4 hours = 70 x 5 = 350 miles
Step 3 :
Then, total distance = 150 + 120 + 350 = 620 miles
Total time = 3 + 2 + 5 = 10 hours
Step 4 :
So, constant-speed = 620 / 10 = 62 miles per hour
Hence, the constant speed for the whole journey is 62 miles per hour.
Let us look at the next problem on "constant-speed"
Example 2 :
A person travels from Newyork to Washington at the rate of 45 miles per hour and comes backs to the Newyork at the rate of 55 miles per hour. What is his constant-speed for the whole journey ?
Solution :
Step 1 :
Here, both the ways, he covers the same distance.
Then, formula for constant speed = 2xy / (x+y)
Step 2 :
x ----> Rate at which he travels from Newyork to Washington
x = 45
y ----> Rate at which he travels from Newyork to Washington
y = 55
Step 3 :
So, constant-speed = 2(45)(55) / (45+55)
constant-speed = 4950 / 100
constant-speed = 49.5 miles per hour
Hence, the constant speed for the whole journey is 45 miles per hour.
Let us look at the next problem on "constant-speed"
Example 3 :
Jose travels from the place A to place B at a certain speed. When he comes back from place B to place A, his speed is 60 miles per hour.If the constant-speed for the whole journey is 72 miles per hour, find his speed when he travels from the place A to B.
Solution :
Step 1 :
Let "a" be the speed from place A to B.
Speed from place B to A = 60 miles/hour
Step 2 :
Here, both the ways, he covers the same distance.
Then, formula for constant-speed = 2xy / (x+y)
Step 3 :
x ----> Speed from place A to B
x = a
y ----> Speed from place B to A
y = 60
Step 4 :
Given : Constant-speed = 72 miles/hour
2(a)(60) / (a+60) = 72
120a = 72(a+60)
120a = 72a + 4320
48a = 4320
a = 90
Hence, the speed from place A to B is 90 miles per hour.
Let us look at the next problem on "constant-speed"
Example 4 :
A man takes 10 hours to go to a place and come back by walking both the ways. He could have gained 2 hours by riding both the ways. The distance covered in the whole journey is 18 miles. Find the constant-speed for the whole journey if he goes by walking and comes back by riding.
Solution :
Step 1 :
Walking + Walking = 10 hours ---------> walking = 5 hours
Riding + Riding = 8 hours (Because 2 hours gained)
Then, Riding = 4 hours
Walking + Riding --------> ( 5 + 4 ) = 9 hours
Step 2 :
Total time taken = 9 hours
Total distance covered = 18 miles
Step 3 :
So, constant-speed = Total distance / Total time
= 18 / 9
= 2 miles per hour
Hence, the required constant speed is 2 miles per hour.
Let us look at the next problem on "constant-speed"
Example 5 :
David travels from the place A to place B at a certain speed. When he comes back from place B to place A, he increases his speed 2 times. If the constant-speed for the whole journey is 80 miles per hour, find his speed when he travels from the place A to B.
Solution :
Step 1 :
Let "a" be the speed from place A to B.
Then, speed from place B to A = 2a
Step 2 :
Here, both the ways, he covers the same distance.
Then, formula for constant speed = 2xy / (x+y)
Step 3 :
x ----> Speed from place A to B
x = a
y ----> Speed from place B to A
y = 2a
Step 4 :
Given : Constant speed = 80 miles/hour
2(a)(2a) / (a+2a) = 80
4a² / 3a = 80
4a / 3 = 80
a = 60
Hence, the speed from place A to B is 60 miles per hour.
Let us look at the next problem on "constant-speed"
Example 6 :
Lily takes 3 hours to travel from place A to place B at the rate of 60 miles per hour. She takes 2 hours to travel from place B to C with 50% increased speed. Find the constant-speed from place A to C.
Solution :
Step 1 :
Speed ( from A to B ) = 60 miles/hour
Speed ( from B to C ) = 90 miles/hour (50% increased)
Step 2 :
Formula for distance = Rate x Time
Distance (A to B) = 60 x 3 = 180 miles
Distance (B to C) = 90 x 2 = 180 miles
Total distance traveled = 360 miles
Total time taken = 3 + 2 = 5 hours
Step 3 :
Formula for constant speed = Total distance / Total time
= 360 / 5
= 72
Hence, the constant speed from place A to B is 72 miles/hour.
Let us look at the next problem on "constant-speed"
Example 7 :
A person takes 5 hours to travel from place A to place B at the rate of 40 miles per hour. He comes back from place B to place A with 25% increased speed. Find the constant-speed for the whole journey.
Solution :
Step 1 :
Speed ( from A to B ) = 40 miles/hour
Speed ( from B to A ) = 50 miles/hour (25% increased)
Step 2 :
Formula for distance = Rate x Time
Distance (A to B) = 40 x 5 = 200 miles
Total distance = 200 + 200 = 400 miles
Time (A to B) = 5 hours
Time (B to A) = Distance / Speed = 200 / 50 = 4 hours
Total time = 5 + 4 = 9 hours
Step 3 :
Formula for constant speed = Total distance / Total time
= 400 / 9
= 44.44
Hence, the constant speed for the whole journey is 44.44 miles/hour.
Let us look at the next problem on "constant-speed"
Example 8 :
Distance from A to B = 200 miles,
Distance from B to C = 300 miles,
Distance from C to D = 540 miles
The speed from B to C is 50% more than A to B. The speed from C to D is 50% more than B to C. If the speed from A to B is 40 miles per hour, find the constant-speed from A to D.
Solution :
Step 1 :
Speed ( from A to B ) = 40 miles/hour
Speed ( from B to C ) = 60 miles/hour (50% more)
Speed ( from C to D ) = 90 miles/hour (50% more)
Step 2 :
Formula for Time = Distance / Speed
Time (A to B) = 200 / 40 = 5 hours
Time (B to C) = 300 / 60 = 5 hours
Time (C to D) = 540 / 90 = 6 hours
Total time (from A to D) = 5 + 5 + 6 = 16 hours
Total distance (from A to D) = 200 + 300 + 540 = 1040 miles
Step 3 :
Formula for constant speed = Total distance / Total time
= 1040 / 16
= 65
Hence, the constant speed from A to D is 65 miles per hour.
Let us look at the next problem on "constant-speed"
Example 9 :
Speed ( A to B ) = 20 miles/hour,
Speed (B to C ) = 15 miles/hour,
Speed (C to D ) = 30 miles/hour
If the distances from A to B, B to C and C to D are equal and it takes 3 hours to travel from A to B, find the constant-speed from A to D
Solution :
Step 1 :
Formula for distance = Rate x Time
Distance from A to B = 20 x 3 = 60 miles
Given : Distance from A to B, B to C and C to D are equal.
Total distance (A to D) = 60 + 60 + 60 = 180 miles
Step 2 :
Formula for Time = Distance / Speed
Time (A to B) = 60 / 20 = 3 hours
Time (B to C) = 60 / 15 = 4 hours
Time (C to D) = 60 / 30 = 2 hours
Total time (from A to D) = 3 + 4 + 2 = 9 hours
Step 3 :
Formula for constant speed = Total distance / Total time
= 180 / 9
= 20
Hence, the constant speed from A to D is 20 miles per hour.
Let us look at the next problem on "constant-speed"
Example 10 :
Time ( A to B ) = 3 hours,
Time (B to C ) = 5 hours,
Time (C to D ) = 6 hours
If the distances from A to B, B to C and C to D are equal and the speed from A to B is 70 miles per hour, find the constant-speed from A to D
Solution :
Step 1 :
Formula for distance = Rate x Time
Distance from A to B = 70 x 3 = 210 miles
Given : Distance from A to B, B to C and C to D are equal.
Total distance (A to D) = 210 + 210 + 210 = 630 miles
Total time (A to D) = 3 + 5 + 6 = 14
Step 2 :
Formula for constant speed = Total distance / Total time
= 630 / 14
= 45
Hence, the constant speed from A to D is 45 miles per hour.
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