`(1)/(3(3)/(5)) +(1)/(4(8)/(9)) +(1)/(((3)/(5)))` का मान ज्ञात करो।

Options

A. `2(59)/(396)`
B. `(854)/(396)`
C. `3(854)/(396)`
D. `(29)/(396)`

Answer

To solve the expression \(\frac{1}{3 \cdot \frac{3}{5}} + \frac{1}{4 \cdot \frac{8}{9}} + \frac{1}{\frac{3}{5}}\), we will follow these steps:

### Step 1: Simplify each term
1. **First term**: \(\frac{1}{3 \cdot \frac{3}{5}} = \frac{1}{\frac{9}{5}} = \frac{5}{9}\)
2. **Second term**: \(\frac{1}{4 \cdot \frac{8}{9}} = \frac{1}{\frac{32}{9}} = \frac{9}{32}\)
3. **Third term**: \(\frac{1}{\frac{3}{5}} = \frac{5}{3}\)

### Step 2: Rewrite the expression
Now we can rewrite the expression using the simplified terms:
\[ \frac{5}{9} + \frac{9}{32} + \frac{5}{3} \]

### Step 3: Find a common denominator
The denominators are \(9\), \(32\), and \(3\). The least common multiple (LCM) of these numbers is \(288\).

### Step 4: Convert each fraction to have the common denominator
1. **First term**:
\[ \frac{5}{9} = \frac{5 \cdot 32}{9 \cdot 32} = \frac{160}{288} \]
2. **Second term**:
\[ \frac{9}{32} = \frac{9 \cdot 9}{32 \cdot 9} = \frac{81}{288} \]
3. **Third term**:
\[ \frac{5}{3} = \frac{5 \cdot 96}{3 \cdot 96} = \frac{480}{288} \]

### Step 5: Add the fractions
Now we can add the fractions:
\[ \frac{160}{288} + \frac{81}{288} + \frac{480}{288} = \frac{160 + 81 + 480}{288} = \frac{721}{288} \]

### Step 6: Final answer
The final value of the expression is:
\[ \frac{721}{288} \]