Simplify : `((3)/(5)+(1)/(3))/((2)/(5)+(2)/(5))`

Answer

To simplify the expression \(\frac{\frac{3}{5} + \frac{1}{3}}{\frac{2}{5} + \frac{2}{5}}\), we will follow these steps:

### Step 1: Simplify the Numerator
The numerator is \(\frac{3}{5} + \frac{1}{3}\).

To add these fractions, we need a common denominator. The least common multiple (LCM) of 5 and 3 is 15.

Now, convert each fraction:
\[ \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \]
\[ \frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \]

Now, add the two fractions:
\[ \frac{3}{5} + \frac{1}{3} = \frac{9}{15} + \frac{5}{15} = \frac{9 + 5}{15} = \frac{14}{15} \]

### Step 2: Simplify the Denominator
The denominator is \(\frac{2}{5} + \frac{2}{5}\).

Since both fractions are the same, we can simply add them:
\[ \frac{2}{5} + \frac{2}{5} = \frac{2 + 2}{5} = \frac{4}{5} \]

### Step 3: Combine the Results
Now we have:
\[ \frac{\frac{14}{15}}{\frac{4}{5}} \]

To divide by a fraction, we multiply by its reciprocal:
\[ \frac{14}{15} \div \frac{4}{5} = \frac{14}{15} \times \frac{5}{4} \]

### Step 4: Multiply the Fractions
Now, multiply the fractions:
\[ \frac{14 \times 5}{15 \times 4} = \frac{70}{60} \]

### Step 5: Simplify the Result
Now, simplify \(\frac{70}{60}\):
\[ \frac{70 \div 10}{60 \div 10} = \frac{7}{6} \]

Thus, the simplified result is:
\[ \frac{7}{6} \]

### Final Answer
The simplified form of \(\frac{\frac{3}{5} + \frac{1}{3}}{\frac{2}{5} + \frac{2}{5}}\) is \(\frac{7}{6}\).

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