Compare the fractions
(i) `3/5, 5/8`
(ii)`9/16, 13/24`

Answer

To compare the fractions \( \frac{3}{5} \) and \( \frac{5}{8} \), and \( \frac{9}{16} \) and \( \frac{13}{24} \), we will follow these steps:

### Step 1: Find the LCM of the denominators
For the first part, we need to find the LCM of 5 and 8.

- **Factors of 5**: \( 5 \) (it is a prime number)
- **Factors of 8**: \( 2 \times 2 \times 2 = 8 \)

The LCM of 5 and 8 is \( 40 \).

### Step 2: Convert the fractions to have the same denominator
Now we convert both fractions to have the denominator of 40.

- For \( \frac{3}{5} \):
\[ \frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40} \]

- For \( \frac{5}{8} \):
\[ \frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40} \]

### Step 3: Compare the numerators
Now we compare \( 24 \) and \( 25 \):
- Since \( 24 < 25 \), we conclude that:
\[ \frac{3}{5} < \frac{5}{8} \]

### Step 4: Repeat for the second set of fractions
Now we will compare \( \frac{9}{16} \) and \( \frac{13}{24} \).

### Step 5: Find the LCM of the denominators
The LCM of 16 and 24 is found as follows:
- **Factors of 16**: \( 2^4 \)
- **Factors of 24**: \( 2^3 \times 3 \)

The LCM is \( 48 \).

### Step 6: Convert the fractions to have the same denominator
Convert both fractions to have the denominator of 48.

- For \( \frac{9}{16} \):
\[ \frac{9}{16} = \frac{9 \times 3}{16 \times 3} = \frac{27}{48} \]

- For \( \frac{13}{24} \):
\[ \frac{13}{24} = \frac{13 \times 2}{24 \times 2} = \frac{26}{48} \]

### Step 7: Compare the numerators
Now we compare \( 27 \) and \( 26 \):
- Since \( 27 > 26 \), we conclude that:
\[ \frac{9}{16} > \frac{13}{24} \]

### Final Conclusion
1. \( \frac{3}{5} < \frac{5}{8} \)
2. \( \frac{9}{16} > \frac{13}{24} \)

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