RATE OF CHANGE

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.

Change in y-value/change in x-value

Determining Rate of Change and Initial 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.

Rate of change :

= -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.

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

Rate of change :

= -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

Rate of change :

= -300/(-20)

= 15

Constant Rate of Change

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 :

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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