Evaluate : `(2 5/4-4 7/6+3 1/3)/(5 2/9-2 4/9-1 1/9)`

Options

A. `1 1/12`
B. `17/20`
C. `1 4/17`
D. `1 2/3`

Answer

To evaluate the expression \((2 \frac{5}{4} - 4 \frac{7}{6} + 3 \frac{1}{3}) / (5 \frac{2}{9} - 2 \frac{4}{9} - 1 \frac{1}{9})\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \(2 \frac{5}{4}\):
\[ 2 \frac{5}{4} = 2 \times 4 + 5 = 8 + 5 = \frac{13}{4} \]

2. Convert \(4 \frac{7}{6}\):
\[ 4 \frac{7}{6} = 4 \times 6 + 7 = 24 + 7 = \frac{31}{6} \]

3. Convert \(3 \frac{1}{3}\):
\[ 3 \frac{1}{3} = 3 \times 3 + 1 = 9 + 1 = \frac{10}{3} \]

4. Convert \(5 \frac{2}{9}\):
\[ 5 \frac{2}{9} = 5 \times 9 + 2 = 45 + 2 = \frac{47}{9} \]

5. Convert \(2 \frac{4}{9}\):
\[ 2 \frac{4}{9} = 2 \times 9 + 4 = 18 + 4 = \frac{22}{9} \]

6. Convert \(1 \frac{1}{9}\):
\[ 1 \frac{1}{9} = 1 \times 9 + 1 = 9 + 1 = \frac{10}{9} \]

### Step 2: Substitute the Improper Fractions into the Expression
Now we can rewrite the expression:
\[ \frac{\frac{13}{4} - \frac{31}{6} + \frac{10}{3}}{\frac{47}{9} - \frac{22}{9} - \frac{10}{9}} \]

### Step 3: Simplify the Denominator
Calculate the denominator:
\[ \frac{47}{9} - \frac{22}{9} - \frac{10}{9} = \frac{47 - 22 - 10}{9} = \frac{15}{9} = \frac{5}{3} \]

### Step 4: Simplify the Numerator
Calculate the numerator:
1. Find a common denominator for \(\frac{13}{4}\), \(\frac{31}{6}\), and \(\frac{10}{3}\). The least common multiple (LCM) of 4, 6, and 3 is 12.
- Convert \(\frac{13}{4}\) to twelfths:
\[ \frac{13 \times 3}{4 \times 3} = \frac{39}{12} \]
- Convert \(\frac{31}{6}\) to twelfths:
\[ \frac{31 \times 2}{6 \times 2} = \frac{62}{12} \]
- Convert \(\frac{10}{3}\) to twelfths:
\[ \frac{10 \times 4}{3 \times 4} = \frac{40}{12} \]

Now substitute these into the numerator:
\[ \frac{39}{12} - \frac{62}{12} + \frac{40}{12} = \frac{39 - 62 + 40}{12} = \frac{17}{12} \]

### Step 5: Combine Numerator and Denominator
Now we have:
\[ \frac{\frac{17}{12}}{\frac{5}{3}} = \frac{17}{12} \times \frac{3}{5} = \frac{17 \times 3}{12 \times 5} = \frac{51}{60} \]

### Step 6: Simplify the Final Result
To simplify \(\frac{51}{60}\):
- The greatest common divisor (GCD) of 51 and 60 is 3.
\[ \frac{51 \div 3}{60 \div 3} = \frac{17}{20} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{17}{20} \]