# 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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