Rate of change is generally expressed as a ratio between a change in one variable relative to a corresponding change in another variable.

Graphically, the rate of change is represented by the slope of a line.

That is,

Change in y-value / change in x-value

**Example 1 :**

What is the rate-of-change between 6 and 8 ?

**Solution : **

**Rate-of-change = Change in y-value / change in x-value**

**Change in y - value from 6 and 8 = 42 - 52 = -10**

**Change in x - value from 6 and 8 = 6 - 8 = -2**

** = -10/(-2) **

** = 5**

**Example 2 :**

A phone salesperson is paid a minimum weekly salary and a commission for each phone sold, as shown below. Find the rate of change between 20 and 40.

**No. of phones sold**

10

20

30

40

**Weekly income ($)**

$480

$630

$780

$930

**Solution : **

**Rate-of-change = Change in y-value / change in x-value**

**Change in y - value from 20 and 40 = 630 - 930 = -300**

**Change in x - value from 20 and 40 = 20 - 40 = -20**

** = -300/(-20) **

** = 15**

**Example 3 :**

**The graph shows the distance Nathan bicycled over time. What is Nathan’s rate of change between 1 hour to 4 hours ?**

**Solution : **

**Rate-of-change = Change in y-value / change in x-value**

**Change in y - value from 1 hour to 4 hours = 15-45 = -30**

**Change in x - value from 1 hour to 4 hours = 20 - 40 = -20**

** = -300/(-20) **

** = 15**

Constant rate is also called as uniform rate which involves something travelling at fixed and steady pace or else moving at some average speed.

**Example 4 : **

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

Formula 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

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

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