Which of the following is a reducible fraction?
105/112
Understanding Reducible Fractions
A fraction is called a reducible fraction if its numerator and denominator share a common factor other than 1. In other words, a reducible fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is a number greater than 1. If the only common factor is 1, the fraction is called an irreducible fraction or a fraction in its simplest form.
Analyzing the Given Fractions to Find Reducible One
We need to examine each option and determine if the numerator and the denominator have a common factor greater than 1. Let's check each fraction:
Option 1: \( \frac{91}{15} \)
Let's find the factors of 91 and 15.
- Factors of 91: 1, 7, 13, 91
- Factors of 15: 1, 3, 5, 15
The only common factor is 1. Therefore, the greatest common divisor (GCD) of 91 and 15 is 1. This fraction is irreducible.
Option 2: \( \frac{79}{26} \)
Let's find the factors of 79 and 26.
- Factors of 79: 1, 79 (79 is a prime number)
- Factors of 26: 1, 2, 13, 26
The only common factor is 1. Therefore, the GCD of 79 and 26 is 1. This fraction is irreducible.
Option 3: \( \frac{105}{112} \)
Let's find the factors of 105 and 112.
- Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105
- Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112
Common factors are 1 and 7. The greatest common divisor (GCD) of 105 and 112 is 7. Since the GCD (7) is greater than 1, this fraction is reducible. We can simplify it by dividing both numerator and denominator by 7:
\( \frac{105 \div 7}{112 \div 7} = \frac{15}{16} \)
The fraction \( \frac{105}{112} \) is a reducible fraction because its numerator and denominator share a common factor of 7.
Option 4: \( \frac{41}{17} \)
Let's find the factors of 41 and 17.
- Factors of 41: 1, 41 (41 is a prime number)
- Factors of 17: 1, 17 (17 is a prime number)
The only common factor is 1. Therefore, the GCD of 41 and 17 is 1. This fraction is irreducible.
Conclusion
Based on our analysis, only the fraction \( \frac{105}{112} \) has a common factor greater than 1 between its numerator and denominator (the common factor is 7). Therefore, \( \frac{105}{112} \) is a reducible fraction.
| Fraction | Numerator | Denominator | Factors of Numerator | Factors of Denominator | Common Factors | GCD | Reducible/Irreducible |
|---|---|---|---|---|---|---|---|
| \( \frac{91}{15} \) | 91 | 15 | 1, 7, 13, 91 | 1, 3, 5, 15 | 1 | 1 | Irreducible |
| \( \frac{79}{26} \) | 79 | 26 | 1, 79 | 1, 2, 13, 26 | 1 | 1 | Irreducible |
| \( \frac{105}{112} \) | 105 | 112 | 1, 3, 5, 7, 15, 21, 35, 105 | 1, 2, 4, 7, 8, 14, 16, 28, 56, 112 | 1, 7 | 7 | Reducible |
| \( \frac{41}{17} \) | 41 | 17 | 1, 41 | 1, 17 | 1 | 1 | Irreducible |
Revision Table: Key Fraction Concepts
| Concept | Definition | Example |
|---|---|---|
| Fraction | Represents a part of a whole or a ratio of two numbers. Written as \( \frac{\text{Numerator}}{\text{Denominator}} \). | \( \frac{3}{4} \) |
| Numerator | The top number in a fraction, indicating the number of parts being considered. | In \( \frac{3}{4} \), the numerator is 3. |
| Denominator | The bottom number in a fraction, indicating the total number of equal parts the whole is divided into. | In \( \frac{3}{4} \), the denominator is 4. |
| Reducible Fraction | A fraction where the numerator and denominator have a common factor greater than 1. Can be simplified. | \( \frac{6}{8} \) (common factor is 2) |
| Irreducible Fraction | A fraction where the only common factor of the numerator and denominator is 1. Cannot be simplified further. Also called simplest form. | \( \frac{3}{4} \) (GCD of 3 and 4 is 1) |
| Greatest Common Divisor (GCD) | The largest positive integer that divides two or more integers without leaving a remainder. | GCD(12, 18) is 6. |
Additional Information on Fraction Simplification
Simplifying a fraction (reducing it to its lowest terms) is done by dividing both the numerator and the denominator by their Greatest Common Divisor (GCD). This process makes the fraction easier to understand and work with.
Steps to simplify a reducible fraction:
- Find the GCD of the numerator and the denominator. You can do this by listing factors or using prime factorization.
- Divide both the numerator and the denominator by the GCD.
- The resulting fraction is the simplified form, which is an irreducible fraction.
For example, to simplify \( \frac{105}{112} \):
- Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105. Prime factorization: \( 3 \times 5 \times 7 \).
- Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112. Prime factorization: \( 2^4 \times 7 \).
- The common prime factor is 7. So, GCD(105, 112) = 7.
- Divide numerator and denominator by 7: \( \frac{105 \div 7}{112 \div 7} = \frac{15}{16} \).
The simplified form of \( \frac{105}{112} \) is \( \frac{15}{16} \), which is an irreducible fraction.
Similar Questions
Which of the following numbers will have an irrational square root?
Which of the numbers given below is NOT rational?
Which of the following is a rational number?
Which of the following numbers is irrational?
The square root of which of the following numbers is irrational?
Which of the following numbers will have an irrational square root?
Important Questions from Rational or Irrational Numbers
The product of \(\sqrt{2}\) and \(\sqrt{3}\) is:
A terminating decimal is always:
The decimal expansion of \(\frac{27}{25}\) will terminate after:
Which of the following is a rational number between \(\sqrt{5}\) and \(\sqrt{7}\) ?
- \((\sqrt2 -\sqrt3)^2\) is:
