**Problem 1 :**

David drove for 3 hours at a rate of 50 miles per hour, for 2 hours at 60 miles per hour and for 5 hours at a rate of 70 miles per hour. What was his average speed for the whole journey ?

**Answer :**

**Step 1 :**

Formula for average speed is

= Total distance / Total time taken.

And also, formula for the distance is

= Rate ⋅ Time

**Step 2 :**

Distance covered in the first 3 hours is

= 50 ⋅ 3

= 150 miles

Distance covered in the next 2 hours is

= 60 ⋅ 2

= 120 miles

Distance covered in the last 5 hours is

= 70 ⋅ 5

= 350 miles

**Step 3 :**

Then, total distance is

= 150 + 120 + 350

= 620 miles

Total time is

= 3 + 2 + 5

= 10 hours

**Step 4 :**

So, the average speed is

= 620 / 10

= 62

So, the average speed for the whole journey is 62 miles per hour.

**Problem 2 :**

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.

**Answer :**

**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 to find average speed is

= 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 is 72 miles/hour.

(2 ⋅ a ⋅ 60) / (a + 60) = 72

120a = 72(a + 60)

120a = 72a + 4320

48a = 4320

a = 90

So, the speed from place A to B is 90 miles per hour.

**Problem 3 :**

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.

**Answer :**

**Step 1 :**

Let "a" be the speed from place A to B.

Then, speed from place B to A = 2a

**Step 2 :**

The distance traveled in both the ways (A to B and B to A) is same.

So, the formula to find average speed is

= 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

So, the speed from place A to B is 60 miles per hour.

**Problem 4 :**

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.

**Answer :**

**Step 1 :**

Speed ( from A to B ) = 40 miles/hour

Speed ( from B to A ) = 50 miles/hour (25% increased)

**Step 2 :**

The distance traveled in both the ways (A to B and B to A) is same.

So, the formula to find average distance is

= 2xy / (x + y)

**Step 3 :**

x ----> Speed from place A to B

x = 40

y ----> Speed from place B to A

y = 50

**Step 4 :**

Average speed = (2 ⋅ 40 ⋅ 50) / (40 + 50)

Average speed = 44.44

So, the average speed for the whole journey is about 44.44 miles/hour.

**Problem 5 :**

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.

**Answer :**

**Step 1 :**

Formula to find distance is

= Rate ⋅ Time

Distance from A to B is

= 20 ⋅ 3

= 60 miles

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

Total distance from A to D is

= 60 + 60 + 60

= 180 miles

**Step 2 :**

Formula to find time is

= 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 taken from A to D is

= 3 + 4 + 2

= 9 hours

**Step 3 :**

Formula to find average speed is

= Total distance / Total time

= 180 / 9

= 20

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

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