**Constant rates of change worksheet : **

Worksheet on constant rates of change is much useful to the students who would like to practice solving problems on "Average rates".

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 ?

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 ?

3. 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.

4. 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.

5. Kemka is preparing 20 cups of apple juice in the first 4 minutes and 60 cups in the next 12 minutes. How many cups does she prepare per minute ?

**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 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 example on "Constant rates of change worksheet"

**Problem 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 example on "Constant rates of change worksheet"

**Problem 3 : **

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 example on "Constant rates of change worksheet"

**Problem 4 : **

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 example on "Constant rates of change worksheet"

**Problem 5 : **

Kemka is preparing 20 cups of apple juice in the first 4 minutes and 60 cups in the next 12 minutes. How many cups does she prepare per minute ?

**Solution :**

**Step 1 :**

Total no. of cups of apple juice prepared is

= 20 + 60

= 80 cups

**Step 2 :**

Total time taken is

= 4 + 12

= 16 minutes

**Step 3 :**

Number of cups of apple juice prepared per minute is

= Total number of cups / Total time taken

= 80 / 16

= 5

Hence, Kemka prepares 5 cups of apple juice per minutes.

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