Reduce the following fractions into their lowest terms by prime factorisation method :
(i) `48/60" (ii) "510/210`

Answer

To reduce the fractions \( \frac{48}{60} \) and \( \frac{510}{210} \) into their lowest terms using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of 48
- Start dividing 48 by the smallest prime number, which is 2.
- \( 48 \div 2 = 24 \)
- \( 24 \div 2 = 12 \)
- \( 12 \div 2 = 6 \)
- \( 6 \div 2 = 3 \)
- 3 is a prime number, so we stop here.
- The prime factorization of 48 is:
\[ 48 = 2^4 \times 3^1 \]

### Step 2: Prime Factorization of 60
- Now, we will factor 60 in a similar way.
- \( 60 \div 2 = 30 \)
- \( 30 \div 2 = 15 \)
- \( 15 \div 3 = 5 \)
- 5 is a prime number, so we stop here.
- The prime factorization of 60 is:
\[ 60 = 2^2 \times 3^1 \times 5^1 \]

### Step 3: Write the Fraction with Prime Factors
- Now we can write the fraction \( \frac{48}{60} \) using its prime factors:
\[ \frac{48}{60} = \frac{2^4 \times 3^1}{2^2 \times 3^1 \times 5^1} \]

### Step 4: Cancel Common Factors
- We can cancel the common factors in the numerator and denominator:
- Cancel \( 2^2 \) from both:
\[ \frac{2^{4-2} \times 3^{1-1}}{5^1} = \frac{2^2}{5^1} \]
- This simplifies to:
\[ \frac{4}{5} \]

### Step 5: Prime Factorization of 510
- Now, we will factor 510.
- \( 510 \div 2 = 255 \)
- \( 255 \div 3 = 85 \)
- \( 85 \div 5 = 17 \)
- 17 is a prime number, so we stop here.
- The prime factorization of 510 is:
\[ 510 = 2^1 \times 3^1 \times 5^1 \times 17^1 \]

### Step 6: Prime Factorization of 210
- Now, we will factor 210.
- \( 210 \div 2 = 105 \)
- \( 105 \div 3 = 35 \)
- \( 35 \div 5 = 7 \)
- 7 is a prime number, so we stop here.
- The prime factorization of 210 is:
\[ 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \]

### Step 7: Write the Fraction with Prime Factors
- Now we can write the fraction \( \frac{510}{210} \) using its prime factors:
\[ \frac{510}{210} = \frac{2^1 \times 3^1 \times 5^1 \times 17^1}{2^1 \times 3^1 \times 5^1 \times 7^1} \]

### Step 8: Cancel Common Factors
- We can cancel the common factors in the numerator and denominator:
- Cancel \( 2^1 \), \( 3^1 \), and \( 5^1 \):
\[ \frac{17^1}{7^1} = \frac{17}{7} \]

### Final Answers
- The reduced form of \( \frac{48}{60} \) is \( \frac{4}{5} \).
- The reduced form of \( \frac{510}{210} \) is \( \frac{17}{7} \).