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Welcome to our Math lesson on Adding or Subtracting the Numerator or Denominator of a Fraction by the Same Number, this is the fifth lesson of our suite of math lessons covering the topic of Comparing Fractions. Ordering Fractions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Adding or Subtracting the Numerator or Denominator of a Fraction by the Same Number
In this part of the tutorial, we will explain what happens to the original fraction (increases or decreases) when the same number is added to or subtracted by both numerator and denominator. Let's begin with addition by the same number. When we add the same number to both components of a fraction, it basically gets closer to 1. Hence, when the same number is added to both numerator and denominator of a proper fraction, its value increases, when this occurs to an improper fraction, its value decreases. In symbols we have
a/b (a < b) ⟹ (a + c)/(b + c) > a/b
a/b (a > b) ⟹ (a + c)/(b + c) < a/b
Example 4
Which fraction is the greatest in each pair below?
21/17 or 23/19
32/41 or 35/44
Solution 4
Both 2
1/17 and
23/19 are improper fractions where the second fraction is obtained by adding 2 to both numerator and denominator of the first fraction, i.e.
23/19 = (21 + 2)/(17 + 2)
Therefore, based on the corresponding rule explained earlier, we have
23/19 < 21/17
Both
32/41 and
35/44 are proper fractions where the second fraction is obtained by adding 3 to both numerator and denominator of the first fraction, i.e.
35/44 = (32 + 3)/(41 + 3)
Therefore, based on the corresponding rule explained earlier, we have
32/41 < 35/44
Since subtraction is the opposite operation of addition, we obtain the reverse rules of those explained above when subtracting the numerator and denominator of a fraction by the same number. Thus, if we subtract the numerator and denominator of a proper fraction by the same number, we obtain a smaller fraction when we do this to an improper fraction, we obtain a bigger fraction. In symbols, we have
a/b (a < b) ⟹ (a - c)/(b - c) < a/b
a/b (a > b) ⟹ (a - c)/(b - c) > a/b
Example 5
Compare the following pairs of fractions:
15/23 and 11/19
31/25 and 29/23
Solution 5
The second fraction (
11/19) is obtained by subtracting the numerator and denominator of the first fraction (
15/23) by 4, that is
11/19 = (15 - 4)/(23 - 4)
Thus, since both fractions are proper, we obtain (based on the aforementioned rule):
15/23 > 11/19
The second fraction (
29/23) is obtained by subtracting the numerator and denominator of the first fraction (
31/25) by 2, that is
29/23 = (31 - 2)/(25 - 2)
Thus, since both fractions are improper, we obtain (based on the aforementioned rule):
31/25 < 29/23
It is impossible to have any similar rule as above when the numerator and denominator increase or decrease by a different number as the comparison depends on the numbers used in every specific situation.
More Comparing Fractions. Ordering Fractions Lessons and Learning Resources
Fractions Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 3.3 | Comparing Fractions. Ordering Fractions | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 3.3.1 | Comparing Fractions with the Same Denominator | | |
| 3.3.2 | Comparing Fractions with the Same Numerator | | |
| 3.3.3 | Comparing Fractions with Different Numerators and Denominators | | |
| 3.3.4 | Comparing Improper Fractions and Mixed Numbers | | |
| 3.3.5 | Adding or Subtracting the Numerator or Denominator of a Fraction by the Same Number | | |
| 3.3.6 | Ordering Fractions | | |
| 3.3.7 | Ordering Fractions in the Number Line | | |
| 3.3.8 | Applications of Fractions Comparison in Practice | | |
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