**Average Speed Problems : **

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.

**Example 1 : **

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 ?

**Answer :**

**Step 1 :**

Here, both the ways, he covers the same distance.

Then, the formula to find average speed is

= 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, the average speed is

= (2 ⋅ 45 ⋅ 55) / (45 + 55)

= 4950 / 100

= 49.5

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

**Example 2 :**

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.

**Answer :**

**Step 1 :**

**Given :** A man takes 10 hours to go to a place and come back by walking both the ways.

That is,

Walking + Walking = 10 hours

2 ⋅ Walking = 10 hours

Walking = 5 hours

**Given :** He could have gained 2 hours by riding both the ways.

That is,

Riding + Riding = 8 hours

2 ⋅ Riding = 8 hours

Riding = 4 hours

**Step 2 :**

If he goes by walking and comes back by riding, time taken by him :

Walking + Riding = 5 + 4

Walking + Riding = 9 hours

**Step 3 :**

Total time taken = 9 hours

Total distance covered = 18 miles

**Step 4 :**

So, the average speed is

= Total distance / Total time

= 18 / 9

= 2

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

**Example 3 :**

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.

**Answer :**

**Step 1 :**

Speed ( from A to B ) = 60 miles/hour

Speed ( from B to C ) = 90 miles/hour (50% increased)

**Step 2 :**

Formula to find distance is

= Rate ⋅ Time

Distance from A to B is

= 60 ⋅ 3

= 180 miles

Distance from B to C

= 90 ⋅ 2

= 180 miles

Total distance traveled from A to B is

= 180 + 180

= 360 miles

Total time taken from A to B is

= 3 + 2

= 5 hours

**Step 3 :**

Formula to find average speed is

= Total distance / Total time

= 360 / 5

= 72

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

**Example 4 :**

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.

**Answer :**

**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 to find time is

= Distance / Time

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 taken from A to D is

= 5 + 5 + 6

= 16 hours

Total distance from A to D is

= 200 + 300 + 540

= 1040 miles

**Step 3 :**

Formula to find average speed is

= Total distance / Total time

= 1040 / 16

= 65

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

**Example 5 :**

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

**Answer :**

**Step 1 :**

Formula to find distance is

= Rate ⋅ Time

Distance from A to B is

= 70 ⋅ 3

= 210 miles

**Given :** Distance from A to B, B to C and C to D are equal.

Total distance from A to D is

= 210 + 210 + 210

= 630 miles

Total time taken A to D is

= 3 + 5 + 6

= 14 hours

**Step 2 :**

Formula to find average speed is

= Total distance / Total time

= 630 / 14

= 45

Hence, the average speed from A to D is 45 miles per hour.

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