Average speed problems are based on uniform rate which involves something travelling at fixed and steady pace.

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 average 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 average 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 average speed for the whole journey is given below.

To have better understanding on "Average speed problems", 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 average speed for the whole journey ?

**Solution :**

**Step 1 :**

Formula for average 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, average 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 "Average speed problems"

**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 average 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, average speed = 2(45)(55) / (45+55)

Average speed = 4950 / 100

Average speed = 49.5 miles per hour

**Hence, the average speed for the whole journey is 45 miles per hour. **

Let us look at the next problem on "Average speed problems"

**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 average 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 :** Average speed = 72 miles/hour

2(a)(60) / (a+60) = 72

120a = 72(a+60)

120a = 72a + 4320

48a = 4320

a = 90

**Hence, the average speed from place A to B is 90 miles per hour. **

Let us look at the next problem on "Average speed problems"

**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 average 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, average speed = Total distance / Total time

= 18 / 9

= 2 miles per hour

**Hence, the required average speed is 2 miles per hour. **

Let us look at the next problem on "Average speed problems"

**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 : Average 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 "Average speed problems"

**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 average 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 average speed = Total distance / Total time **

= 360 / 5

= 72

**Hence, the average speed from place A to B is 72 miles/hour. **

Let us look at the next problem on "Average speed problems"

**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 average 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 average speed = Total distance / Total time **

= 400 / 9

= 44.44

**Hence, the average speed for the whole journey is 44.44 miles/hour. **

Let us look at the next problem on "Average speed problems"

**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 average 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 average speed = Total distance / Total time **

= 1040 / 16

= 65

**Hence, the average speed from A to D is 65 miles per hour. **

Let us look at the next problem on "Average speed problems"

**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 average 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 average speed = Total distance / Total time **

= 180 / 9

= 20

**Hence, the average speed from A to D is 20 miles per hour. **

Let us look at the next problem on "Average speed problems"

**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 average 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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