Which of the following is true ?

Options

A. `(9)/(16)lt(13)/(24)`
B. `(9)/(16)=(13)/(24)`
C. `(9)/(16)gt(13)/(24)`
D. `(9)/(16)le(13)/(24)`

Answer

To determine the relationship between the two fractions \( \frac{9}{16} \) and \( \frac{13}{24} \), we will follow these steps:

### Step 1: Identify the fractions
We have two fractions:
- \( \frac{9}{16} \)
- \( \frac{13}{24} \)

### Step 2: Find a common denominator
The denominators of the fractions are 16 and 24. We need to find the Least Common Multiple (LCM) of these two numbers to compare them easily.

**Calculation of LCM:**
- The prime factorization of 16 is \( 2^4 \).
- The prime factorization of 24 is \( 2^3 \times 3^1 \).
- The LCM is obtained by taking the highest power of each prime factor:
- For 2, the highest power is \( 2^4 \).
- For 3, the highest power is \( 3^1 \).

Thus, the LCM of 16 and 24 is:
\[ LCM = 2^4 \times 3^1 = 16 \times 3 = 48 \]

### Step 3: Convert both fractions to have the same denominator
Now we will 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 4: Compare the two fractions
Now we can compare the two fractions:
- \( \frac{27}{48} \) and \( \frac{26}{48} \)

Since both fractions have the same denominator, we can compare their numerators directly:
- \( 27 > 26 \)

### Conclusion
Thus, we can conclude that:
\[ \frac{9}{16} > \frac{13}{24} \]

### Final Answer
The correct relationship is \( \frac{9}{16} > \frac{13}{24} \).

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