Sum of three fractions is `2(11)/(24)`. On dividing the largest fraction by the smallest fraction, `(7)/(6)` is obtained which is `(1)/(3)` greater than the middle fraction. The smallest fraction is

Options

A. `(5)/(8)`
B. `(3)/(4)`
C. `(5)/(6)`
D. `(3)/(7)`

Answer

To solve the problem step-by-step, we will define the fractions and use the information given in the question to find the smallest fraction.

### Step 1: Define the fractions
Let:
- \( x \) = largest fraction
- \( y \) = middle fraction
- \( z \) = smallest fraction

### Step 2: Set up the relationships
From the problem, we know:
1. The sum of the three fractions is \( 2 \frac{11}{24} \).
2. Dividing the largest fraction by the smallest fraction gives \( \frac{7}{6} \).
3. The largest fraction is \( \frac{1}{3} \) greater than the middle fraction.

### Step 3: Convert the mixed number to an improper fraction
Convert \( 2 \frac{11}{24} \) to an improper fraction:
\[ 2 \frac{11}{24} = \frac{2 \times 24 + 11}{24} = \frac{48 + 11}{24} = \frac{59}{24} \]

### Step 4: Establish the equation from the division of fractions
From the second point, we have:
\[ \frac{x}{z} = \frac{7}{6} \]
This implies:
\[ x = \frac{7}{6}z \]

### Step 5: Establish the relationship between the middle and largest fractions
From the third point, we know:
\[ x = y + \frac{1}{3} \]
Substituting \( x \) from the previous equation:
\[ \frac{7}{6}z = y + \frac{1}{3} \]
Rearranging gives:
\[ y = \frac{7}{6}z - \frac{1}{3} \]

### Step 6: Find a common denominator and simplify
To subtract \( \frac{1}{3} \) from \( \frac{7}{6}z \), we convert \( \frac{1}{3} \) to have a denominator of 6:
\[ \frac{1}{3} = \frac{2}{6} \]
Thus:
\[ y = \frac{7}{6}z - \frac{2}{6} = \frac{5}{6}z \]

### Step 7: Substitute into the sum equation
Now we can substitute \( x \), \( y \), and \( z \) into the sum equation:
\[ x + y + z = \frac{59}{24} \]
Substituting the expressions for \( x \) and \( y \):
\[ \frac{7}{6}z + \frac{5}{6}z + z = \frac{59}{24} \]
Combining the fractions:
\[ \left(\frac{7}{6} + \frac{5}{6} + \frac{6}{6}\right)z = \frac{59}{24} \]
This simplifies to:
\[ \frac{18}{6}z = \frac{59}{24} \]
or:
\[ 3z = \frac{59}{24} \]

### Step 8: Solve for \( z \)
Now, divide both sides by 3:
\[ z = \frac{59}{24} \cdot \frac{1}{3} = \frac{59}{72} \]

### Step 9: Conclusion
Thus, the smallest fraction \( z \) is:
\[ \frac{59}{72} \]