Transcript The following lesson is one lecture in a series of Chemistry Programs developed by Professor Larry Byrd Department of Chemistry Western Kentucky University.

The following lesson is
one lecture in a series of
Chemistry Programs
developed by
Professor Larry Byrd
Department of Chemistry
Western Kentucky University
Excellent Assistance
has been provided by:
Dr. Robert Wyatt
Ms. Elizabeth Romero
Ms. Kathy Barnes
Mathematical Concepts
(Part 1)
MATHEMATICAL CONCEPTS
[ Updated January 10, 2006 ]
Often in the sciences we will be involved in using
many mathematical concepts.
In this section, we will review all the fundamental
mathematical processes that you will use in a basic
chemistry course.
I. Fractions:
Fractions are commonly used to indicate the division of
one number by another and they are written as:
1 or
10
1
10
4 or
12
4
12
When we read the first above fraction, we may say that in
this fraction one is divided by ten or equally correct, we
could say that it is one part per 10 parts.
Notice that the term per means division.
The other fraction could be read as four is divided by
twelve or as four parts per twelve parts.
The number above the line is the numerator, which is
divided by the number below the line, called the
denominator. A whole number can also be considered a
fraction in which the denominator equals one.
3 = 3
1
180 = 180
1
If the numerator and denominator of a fraction are both
multiplied by the same number, a fraction equal to the
first is formed.
The fraction 2/3 is equivalent to 4/6 and is also
equivalent to 10/15:

Numerator 

Denominator 

Numerator 

Denominator 
2 times 2
3 times 2
=
2 times 5
3 times 5
=
4
6
10
15
Thus, 2 = 4
3
6
Thus, 2 = 10
3
Observe that the term times means to multiply!
15
II. Fractions as ratios
II.
A fraction is nothing more than a representation of how
many parts we have out of a total amount.
If a chemistry class is composed of 20 students and 11 of
those are girls, we can say that 11/20 represents the ratio of
the girls in the class.
Notice, that fractions are simply “ratios”!
11 girls in a class of 20 students is the same as
11 members per 20 members
or as a fraction :
11 members
20 members
III. Adding and Subtracting
When adding and subtracting fractions they must have the same
common denominator.
If they do not have a common denominator, we must first
convert them so each has the same common denominator.
Addition of fractions is easy to do if both fractions have the same
denominator:
Simply add the numerators and place that value over the common
denominator.
2 + 1 = 21 = 3
5
5
5
5
If the denominators are not the same,
then one or both of the fractions must
be changed to develop a common
denominator.
Example:
If we need to add 11/20 + 1/10, we
must first find a common denominator.
11/20 +
1/10
We must always pick the lowest number that both 20 and 10
will divide into a whole number of times.
For this problem, 20 is the lowest common denominator.
11 + 1 = 11 + ?
10
20
20
20
1 = ?
10
20
We can easily see that 1/10 is the same as 2/20.
Now we can do the math:
11 + 2 = 11  2 = 13
20
20
20
20
Notice, that the new numerator is a sum of the
NUMERATORS of the fractions with the same
denominator.
If we want to subtract 1/ 2 from 3/4, we first write it as
3 - 1
4
2
We see that the LOWEST COMMON
Denominator is “4 “
1
2
= ?
4
Thus, 1 = 2
2
4
3
4
- 2 = 3  2 = 1
4
4
4
If we want to subtract 1/16 from 2/3, we need to find a
common denominator that both 16 and 3 will divide into a
whole number of times.
In this it is found by simply multiplying the two given
denominators.
(16) (3) = 48 is the lowest common denominator.
Thus,
and
2/3
1/16
becomes
is
3/48.
32/48
Now, we can do the needed subtraction:
2 - 1
3
16
=
? - ?
48
48
Now we can do the math:
32 - 3 = 32  3 = 29
48
48
48
48
=
32 - 3
48
48
Other Examples: **Remember to always reduce
fractions to their lowest form.
(1)
8 + 1 =
9
2




2 8  9 1
    =



2  9  9  2
16  9 = 16  9 = 25 = 1 7
18
18 18
18
18
Other Examples: **Remember to always reduce
fractions to their lowest form.
(2) 2 + 3 =
5
2 + 3 =
5
1
The number 3 may also be written as 13
2 + ?
5
5
3 = ?
5
1
2 + 15 = 2  15 = 17 ** =
5
5
5
5
32
5
Thus, 3 = 15
5
1
IV. Common Denominators
Notice that when we converted 1/10 to 20's that we
found:
1 = 2
10
20
Then 1/10 is the same as 2/20 or in other words 2/20 will
reduce to 1/10.
If we use the Mathematical Rule known as the
MEANS and EXTREMES RULE, we can always be
sure our work is correct:
The rule states that if we have a fraction equal to
another fraction, then the numerator(A) of the left
fraction times the denominator(D) of the right fraction
will always be equal to the numerator(C) of the right
fraction times the denominator(B) of the left fraction .
A = C
D
B
If we have the fractions
Then :
(A)(D) =
(C)(B)
For example, if we want to convert
would do it as follows:
1 = ?
10
20
Thus, 1 = C
10
20
1/10 into 20’s we
Thus, 1 = C
10
20
Start with the unknown (C) and multiply it by (10) and
it must equal (1) times (20 ) :
Step #1
(C) (10)
=
(1) (20)
Step #2
10 C
=
20
Step # 3
10 C
10
Step #4
C
Step # 5
Thus,
=
20
10
Divide both sides by 10!
=
2
1
2
10 = 20
Another example of usage of the Means and Extremes
Rule to find new fractions:
2
17
=
?
204
2
17
=
C
204
(C) (17)
= (2) (204)
17C
= 408
17 C
17
=
C
408
17
= 24
Thus, 2
17
=
24
204
You can always CHECK your work by multiplying
across the equal sign to make sure your new fraction
[one with the new denominator]
is equal to the original fraction.
In our example , ( 2 ) times ( 204 ) must be equal
to ( 24 ) ( 17 ) or we have made an error!
Notice, we got it right!!!
2 = 24 ,
17
204
*** The proof is ( 2 ) ( 204 ) = 408
and ( 24 ) ( 17 ) also equals 408 !!!
*** Always do this above check to make sure
your new fraction is correct.
Practice Test #1
Questions
1. What are fractions?
2. For the fraction 1/5 and 2/7, what is the lowest
possible common denominator?
3.
24
17
4.
6
A
=
48
B
Find the value for B!!!
=
120
25
Find the value for A!!!
Practice Test #1
Answers
1. What are fractions?
They are just a ratio of some value (numerator) per some
other value (denominator)
2. For the fraction 1/5 and 2/7, what is the lowest possible
common denominator?
(5) (7)
3.
24 =
17
(24) (B)
4.
=
35
Thus, 35 is the lowest common denominator.
48 Find the value for B!!!
B
= (48) (17)
Thus, B is equal to
34 !!!
6 = 120 Find the value for A!!!
A
25
( 120 )( A ) = ( 6 )( 25 )
Thus, A is equal to
11
4
!!!