# 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.

That is,

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

=  -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 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 :

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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