[RealPlayer Video] 3 minutes - Comparison of Fractions Size or Magnitude, and more examples of the use of common denominators in addition and subtraction.
Page Contents:
- Comparison of Fractions
- Comparison of Mixed Numbers
- Algebraic Viewpoint
Comparison of Fractions
How raising fractions over a common denominator leads to direct comparison (and justifies cross multiplication rule methods for comparison
After reading the following, say which is greater: four ninths or five elevenths, and find the difference of the larger minus the smaller.
Example 1. The question which is greater
|
5 |
or |
3 |
is often answered by comparing 5×4 = 20 with 6×3 =18. Let use look at this in more detail. The least common denominator is 12. The following diagram show both fractions in terms of twenty-fourths: Here 24 = 2× 12
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| 5 6 |
1 |
||||||||||||||||||||||
|
3 |
1 |
||||||||||||||||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
Here by putting both fractions over the common denominator 4×6= 24, we see that
|
5 |
= |
20 24 |
is more than | 18 24 |
= |
3 4 |
Therefore
|
5 |
> |
3 |
and we can calculate how much more:
|
5 |
- |
3 4 |
= |
20 24 |
- |
18 24 |
= |
2 24 |
= |
1 12 |
By putting both fractions over the common denominator, the original comparison can be decided by comparing the over 24 = 4×6 numerators
(i) 20 = 5×4 = (first numerator)×(second denominator)
with
(ii) 18 =6×3 = (first denominator)×(second denominator). These
These over 6×4 = 24 numerators indicate how many (6×4)ths there are in the original fractions.
Example 2: The question which is greater
|
9 |
or |
11 |
This can be answered by seeing how (13×17)ths there are in each fraction. We see that
|
9 |
= |
9×17 13×17 |
= | 153 13×17 |
while | 11 17 |
= |
13×11 13×17 |
= | 143 13×17 |
So the first fraction is greater. It provides 153- 143 = 10 more (13×17)ths than the second.
For those of you who insist on knowing, 13×17 =221, a number whose existence we need, but whose value is not required.
Mixed Numbers Comparison
The comparison there begins with comparison of the whole number parts. If those parts are equal, comparison then proceeds with the (proper) fraction parts. The cross-multiplication or common denominator method then applies. Note: there is no need to convert the mixed number into an improper fraction.
A few examples should go here (a to-do).
In process, the question of comparing numbers and fraction given in decimal form is similar.
- Which is larger 587 or 622? Comparison of the larger part of each number, the hundred digits gives the answer.
- Which is larger 3.46 or 2.61? Comparison of the whole number part, the digits in the ones place of each number provides the answer.
- Which is larger 453.145 or 52.93? Comparison of the whole number parts of each decimal gives the answer.
Teachers: The comparison of decimals is part lexicographic. It is fully lexicographic when the decimals are padded with zeroes on the left to make them have the same number of decimal places.
Another thought:
In comparing decimals 4352 and 4348, we observe the first difference occurs in the ten digit place. Here 5 tens or 50 is greater than 48. Thus the first number 4352 > 4350 > 4348. Like wise when we compare mixed numbers like
| 348 | 7 15 |
and | 343 | 56 57 |
the first number has the greater whole number part. So it is larger than the second regardless of the relative size of the proper fraction parts in each.
However in the comparison of
| 54 | 8 15 |
and | 54 | 7 10 |
the whole number parts are equal. Thus which is larger depends on the comparison of the proper fraction parts
| 8 15 |
= | 24 30 |
||
| 7 10 |
= | 21 30 |
Raising terms over a common denominator, the least for instance, or alternatively the product 150, expresses both fractions as multiplies of a common unit numerator fraction, over which the comparison is easy.
Remark: The comparison could be made by converting both mixed numbers into decimal using a calculator, but that would not develop or strengthen fraction skills and sense.
Algebraic Viewpoint/Description
for reading as part of algebra skill development - optional reading for now
Algebraic Shorthand Description of Ideas- the General Case: The key to comparing fractions is to put them over a common denominator.
Read (?) the following with the letters or literals (a,b,c,d) = (8, 5, 7, 11) in the first instance. Then let the literals or letters a, b, c and d be any whole number you like.
To compare
|
a |
with |
c |
we put them over a common denominator b×d. (Here we assume b and d are non-negative - why?)
|
a |
= |
a×d b×d |
needs to be compared with | b×c b×d |
= |
c d |
Now we need to compare numerators a×d and b×c.
There are three possibilities:
| (i) if a×d > b×c then |
a |
> |
c |
| (ii) if a×d = b×c then |
a |
= |
c |
| (i) if a×d < b×c then |
a |
< |
c |
The foregoing explains and justifies the so called cross-multiplication rule for the comparison of fractions with non-negative denominators. The product of the denominators b×d gives a common denominator, most likely not the least common, but it will do. The use of the least common denominator is optional in the case of comparison.
Another Method for Comparison of Fractions
[Play Video] 3 minutes - Comparison of Fractions Size or Magnitude, and (?) more examples of the use of common denominators in addition and subtraction.
To compare two fractions with unlike denominators
| A B | and | C D |
express both over a common denominator M. Then
| A B |
= | A(M B)M |
and | C D |
= | C(MD) M |
Then compare the numerators.
The case M = BD gives
| A B |
= | AD M |
and | C D |
= | CB M |
The first fraction A/B is then
- less than the second fraction C/D if AD < BD.
- greater than the second fraction C/D if AD > BD, and
- has the same value as the second fraction C/D if AD = BD
In case III, we may also say the two fractions are equivalent.
In general, when
| A B |
= | A(MB) M |
and | C D |
= | C(MD) M |
The first fraction A/B is then
- less than the second fraction C/D
if A(M
B)
< C(MD) - greater than the second fraction C/D
if A(MB)
> C(MD),
and - has the same value as the second fraction C/D
if A(MB)
= C(MD)
In case III, we may also say the two fractions are equivalent.
|
Four Topics
|
|
||||||||||||