Afraction having denominator 30 and lying between `(5)/(8) and (7)/(11)` is-

Options

A. `(18)/(30)`
B. `(19)/(30)`
C. `(20)/(30)`
D. `(21)/(30)`

Answer

To find a fraction with a denominator of 30 that lies between \( \frac{5}{8} \) and \( \frac{7}{11} \), we can follow these steps:

### Step 1: Find the LCM of the Denominators
We need to find the least common multiple (LCM) of the denominators 8, 11, and 30 to compare the fractions easily.

- The prime factorization of 8 is \( 2^3 \).
- The prime factorization of 11 is \( 11^1 \).
- The prime factorization of 30 is \( 2^1 \times 3^1 \times 5^1 \).

The LCM is found by taking the highest power of each prime:
- For 2, the highest power is \( 2^3 \) (from 8).
- For 3, the highest power is \( 3^1 \) (from 30).
- For 5, the highest power is \( 5^1 \) (from 30).
- For 11, the highest power is \( 11^1 \) (from 11).

Thus, the LCM is:
\[ LCM = 2^3 \times 3^1 \times 5^1 \times 11^1 = 8 \times 3 \times 5 \times 11 = 1320 \]

### Step 2: Convert the Fractions
Now we convert \( \frac{5}{8} \) and \( \frac{7}{11} \) to have a common denominator of 1320.

- For \( \frac{5}{8} \):
\[ \frac{5}{8} = \frac{5 \times 165}{8 \times 165} = \frac{825}{1320} \]

- For \( \frac{7}{11} \):
\[ \frac{7}{11} = \frac{7 \times 120}{11 \times 120} = \frac{840}{1320} \]

### Step 3: Identify the Range
Now we have:
\[ \frac{825}{1320} < \frac{x}{30} < \frac{840}{1320} \]

### Step 4: Express the Fraction with Denominator 30
To express \( \frac{x}{30} \) with a denominator of 1320, we can find \( x \) such that:
\[ \frac{x}{30} = \frac{44x}{1320} \]

### Step 5: Set Up the Inequality
Now we set up the inequality:
\[ 825 < 44x < 840 \]

### Step 6: Solve for \( x \)
Now we solve the inequalities:

1. For the left side:
\[ 825 < 44x \implies x > \frac{825}{44} \approx 18.75 \]

2. For the right side:
\[ 44x < 840 \implies x < \frac{840}{44} \approx 19.09 \]

### Step 7: Determine Possible Integer Values
Since \( x \) must be an integer, the only integer value that satisfies \( 18.75 < x < 19.09 \) is:
\[ x = 19 \]

### Step 8: Write the Required Fraction
Thus, the required fraction is:
\[ \frac{19}{30} \]

### Final Answer
The fraction having a denominator of 30 and lying between \( \frac{5}{8} \) and \( \frac{7}{11} \) is \( \frac{19}{30} \).

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