Simplify : `((6)/(7) div (2)/(5)) + ((5)/(9) div (27)/( 25))`

Answer

To simplify the expression \(\frac{6}{7} \div \frac{2}{5} + \frac{5}{9} \div \frac{27}{25}\), we can follow these steps:

### Step 1: Rewrite the Division as Multiplication
We know that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as follows:

\[ \frac{6}{7} \div \frac{2}{5} = \frac{6}{7} \times \frac{5}{2} \]
\[ \frac{5}{9} \div \frac{27}{25} = \frac{5}{9} \times \frac{25}{27} \]

### Step 2: Simplify Each Multiplication
Now, we can simplify each multiplication:

1. For \(\frac{6}{7} \times \frac{5}{2}\):
- Multiply the numerators: \(6 \times 5 = 30\)
- Multiply the denominators: \(7 \times 2 = 14\)
- So, \(\frac{6}{7} \times \frac{5}{2} = \frac{30}{14}\)

2. For \(\frac{5}{9} \times \frac{25}{27}\):
- Multiply the numerators: \(5 \times 25 = 125\)
- Multiply the denominators: \(9 \times 27 = 243\)
- So, \(\frac{5}{9} \times \frac{25}{27} = \frac{125}{243}\)

### Step 3: Add the Two Results
Now we need to add \(\frac{30}{14}\) and \(\frac{125}{243}\). To do this, we need a common denominator.

### Step 4: Find the Least Common Multiple (LCM)
The denominators are 14 and 243. The LCM of 14 and 243 can be calculated as follows:
- The prime factorization of 14 is \(2 \times 7\).
- The prime factorization of 243 is \(3^5\).
- The LCM is \(2 \times 7 \times 3^5 = 1701\).

### Step 5: Convert Each Fraction to the Common Denominator
Now we convert each fraction to have the denominator of 1701:

1. For \(\frac{30}{14}\):
- Multiply both the numerator and denominator by \(\frac{1701}{14}\):
\[ \frac{30 \times 121}{14 \times 121} = \frac{3630}{1701} \]

2. For \(\frac{125}{243}\):
- Multiply both the numerator and denominator by \(\frac{1701}{243}\):
\[ \frac{125 \times 7}{243 \times 7} = \frac{875}{1701} \]

### Step 6: Add the Two Fractions
Now we can add the two fractions:
\[ \frac{3630}{1701} + \frac{875}{1701} = \frac{3630 + 875}{1701} = \frac{4505}{1701} \]

### Final Step: Simplify if Possible
The final answer is \(\frac{4505}{1701}\). We check if this fraction can be simplified further. Since 4505 and 1701 have no common factors other than 1, this is the simplest form.

### Final Answer
\[ \frac{4505}{1701} \]

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