Finding the HCF of Fractions Explained

To find the Highest Common Factor (HCF) of fractions, we use a specific formula. The HCF of a set of fractions is calculated by taking the HCF of their numerators and dividing it by the Least Common Multiple (LCM) of their denominators.

The formula is:

$$\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}$$

Let's apply this formula to the given fractions: \(\frac{12}{5}\) , \(\frac{14}{15}\) , and \(\frac{16}{17}\).

Step 1: Identify Numerators and Denominators

The numerators are the top numbers of the fractions, and the denominators are the bottom numbers.

• Numerators: 12, 14, 16
• Denominators: 5, 15, 17

Step 2: Calculate the HCF of the Numerators

We need to find the HCF of 12, 14, and 16. The HCF is the largest number that divides all the given numbers without leaving a remainder. We can find this by looking at their prime factorizations.

• Prime factorization of 12: \(12 = 2 \times 2 \times 3 = 2^2 \times 3^1\)
• Prime factorization of 14: \(14 = 2 \times 7 = 2^1 \times 7^1\)
• Prime factorization of 16: \(16 = 2 \times 2 \times 2 \times 2 = 2^4\)

To find the HCF, we take the common prime factors raised to the lowest power they appear in any of the factorizations. The only common prime factor is 2. The lowest power of 2 is \(2^1\).

So, HCF(12, 14, 16) = \(2^1 = 2\).

Step 3: Calculate the LCM of the Denominators

Next, we need to find the LCM of 5, 15, and 17. The LCM is the smallest number that is a multiple of all the given numbers. We can find this using prime factorizations.

• Prime factorization of 5: \(5 = 5^1\)
• Prime factorization of 15: \(15 = 3 \times 5 = 3^1 \times 5^1\)
• Prime factorization of 17: \(17 = 17^1\) (17 is a prime number)

To find the LCM, we take all the prime factors that appear in any of the factorizations, raised to the highest power they appear. The prime factors are 3, 5, and 17. The highest power of 3 is \(3^1\), the highest power of 5 is \(5^1\), and the highest power of 17 is \(17^1\).

So, LCM(5, 15, 17) = \(3^1 \times 5^1 \times 17^1 = 3 \times 5 \times 17 = 15 \times 17\).

Calculation: \(15 \times 17\)

So, LCM(5, 15, 17) = 255.

Step 4: Apply the HCF of Fractions Formula

Now we use the formula:

$$\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}} = \frac{\text{HCF}(12, 14, 16)}{\text{LCM}(5, 15, 17)}$$

Substitute the values we found:

$$\text{HCF of fractions} = \frac{2}{255}$$

Final HCF Calculation Result

The HCF of \(\frac{12}{5}\), \(\frac{14}{15}\), and \(\frac{16}{17}\) is \(\frac{2}{255}\).

Revision Table: HCF and LCM for Fractions

Additional Information on HCF and LCM

Understanding HCF and LCM is fundamental when working with fractions. The HCF helps in simplifying fractions, while the LCM is crucial for adding or subtracting fractions by finding a common denominator, and also for finding the LCM of fractions.

For integers:

• To find HCF: List factors or use prime factorization. The product of common prime factors with the lowest power is the HCF.
• To find LCM: List multiples or use prime factorization. The product of all prime factors with the highest power is the LCM.

In this problem, the denominators 5, 15, and 17 are relatively simple. 5 and 17 are prime numbers. 15 is \(3 \times 5\). Since 17 is not a factor of 5 or 15, and 3 is not a factor of 5 or 17, the LCM is simply the product of the distinct prime factors raised to their highest powers, which in this case are just the numbers themselves multiplied together after considering common factors (which is just 5 and 15 sharing a factor of 5, and 17 being unique). So LCM(5, 15, 17) = LCM(5, 3x5, 17). The distinct prime factors are 3, 5, 17. Highest powers are \(3^1, 5^1, 17^1\). LCM = \(3 \times 5 \times 17 = 255\).