`(3)/(12)` of `(((2)/(5)+(4)/(15)))/(((3)/(5)-(2)/(5)))=`?

Options

A. `(5)/(7)`
B. `(5)/(6)`
C. `(6)/(5)`
D. `(2)/(6)`

Answer

To solve the equation \(\frac{3}{12} \times \frac{\left(\frac{2}{5} + \frac{4}{15}\right)}{\left(\frac{3}{5} - \frac{2}{5}\right)}\), we will follow these steps:

### Step 1: Simplify the expression inside the parentheses
First, we need to simplify \(\frac{2}{5} + \frac{4}{15}\).

To do this, we need a common denominator. The least common multiple (LCM) of 5 and 15 is 15.

- Convert \(\frac{2}{5}\) to have a denominator of 15:
\[ \frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \]

- Now add \(\frac{6}{15} + \frac{4}{15}\):
\[ \frac{6}{15} + \frac{4}{15} = \frac{6 + 4}{15} = \frac{10}{15} \]

### Step 2: Simplify the denominator
Next, we simplify \(\frac{3}{5} - \frac{2}{5}\).

Since the denominators are the same, we can subtract directly:
\[ \frac{3}{5} - \frac{2}{5} = \frac{3 - 2}{5} = \frac{1}{5} \]

### Step 3: Substitute back into the original expression
Now we substitute back into the original expression:
\[ \frac{3}{12} \times \frac{\frac{10}{15}}{\frac{1}{5}} \]

### Step 4: Simplify the fraction
To simplify \(\frac{\frac{10}{15}}{\frac{1}{5}}\), we can multiply by the reciprocal:
\[ \frac{10}{15} \times \frac{5}{1} = \frac{10 \times 5}{15 \times 1} = \frac{50}{15} \]

### Step 5: Simplify \(\frac{50}{15}\)
Now we simplify \(\frac{50}{15}\):
\[ \frac{50}{15} = \frac{10}{3} \]

### Step 6: Multiply by \(\frac{3}{12}\)
Now we multiply \(\frac{3}{12}\) by \(\frac{10}{3}\):
\[ \frac{3}{12} \times \frac{10}{3} \]

### Step 7: Cancel out common factors
We can cancel the 3 in the numerator and denominator:
\[ \frac{1}{12} \times 10 = \frac{10}{12} \]

### Step 8: Simplify \(\frac{10}{12}\)
Finally, we simplify \(\frac{10}{12}\):
\[ \frac{10}{12} = \frac{5}{6} \]

### Final Answer
The final value is \(\frac{5}{6}\).

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