# Sec 2.4 Change in Quantity and Subtracting Real Numbers

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```					                Sec 2.4 Change in Quantity and Subtracting Real Numbers

Change in Quantity:
If you had 15gals of gas in your tank and at the end of your trip you had only 2gals left, then
the gas quantity decreased from 15 to 2. This change is called the change in quantity. In our
case the quantity of gasoline decreased from 15 to 2 gallons. The actual change in quantity
could be calculated by subtracting the starting quantity from the ending: 2 15  13

The change in quantity is the ending amount minus the beginning amount.

Change in Quantity = Ending Amount – Beginning amount

If the quantity increased the change is positive, if it decreased, the change is negative.

Subtraction of Real Numbers.
Subtracting some amount of money has the same effect as incurring debt or spending. That
means that subtracting is the same as adding negative numbers, which suggests that in
general subtracting a number is the same as adding the opposite of that number:
a  b  a   b 
Example:        Subtract: 5  2
5  2  3 , which is the same as 5   2  3 , so 5  2  5   2

Subtracting a Negative Number:

Example:
A helicopter starts from the bottom of the canyon 150 ft below the sea level and rises to the
altitude of 423 ft above the sea level. Find the change in altitude.
Solution: Since the altitude changed from 150 ft (below the sea level altitude) to 423 ft
(above the sea level), the change in altitude is the ending altitude minus the starting altitude:
423   150 the difference should be larger than 423 since the helicopter started from the
altitude lower than the sea level. If we recall that a  b  a   b  , we get
423   150  423   opposite of 150  423  150  573 ft

An opposite of a number is denoted by a negative sign, so the opposite of 5 is 5 . To denote
the opposite of 5 we write it   5 . Since we know that the opposite of 5 is 5 we can
observe that
  a   a
The opposite of an opposite of a number is the number itself.

Example:        Find an opposite of a number:
a) 7                Opposite 7
b) 5               Opposite 5

Subtractive a negative number:
4   5  4  5  9                    8   1  8  1  9
Changes of Increasing and Decreasing Quantities.
 An increasing quantity has a positive change.
 A decreasing quantity has a negative change.
Example:
The wolf population in the Greater Yellowstone area for various years is shown in the table.

Year      Population      Change in population
from each year to next
1996            40                   -
1997            86                  46
1998           112                  26
1999           118                   6
2000           177                  59
2001           218                  31

a. From what year to the next the changes in population were the greatest?
What was that change?
The change was the largest from year 1999 to 2000, The change was 59.

b. From what year to the next the changes in population were the least?
What was that change?
The change was least from year 1998 to 1999, The change was 6.

Group Exploration:
A set of points is given in the table (by x- and y- coordinates).
x               y          Change in y (current
value minus previous)
0               3                    -
1               5
2               7
3               9
4              11
5              13

a. Plot the points from the table. Do they lie on one line?

b. Complete the third column in the table. Describe any patterns.

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