`(1/4+1/4div5/4)/(1/4xx1/4+(2)1/4=?`

Options

A. (16/25)
B. (32/185)
C. (36/185)
D. none of these

Answer

To solve the expression \((\frac{1}{4} + \frac{1}{4} \div \frac{5}{4}) / (\frac{1}{4} \times \frac{1}{4} + 2 \times \frac{1}{4})\), we will follow these steps:

### Step 1: Simplify the numerator
The numerator is \(\frac{1}{4} + \frac{1}{4} \div \frac{5}{4}\).

First, we need to perform the division:
\[ \frac{1}{4} \div \frac{5}{4} = \frac{1}{4} \times \frac{4}{5} = \frac{1 \times 4}{4 \times 5} = \frac{4}{20} = \frac{1}{5} \]

Now, we can add this to \(\frac{1}{4}\):
\[ \frac{1}{4} + \frac{1}{5} \]

To add these fractions, we need a common denominator. The least common multiple of 4 and 5 is 20:
\[ \frac{1}{4} = \frac{5}{20}, \quad \frac{1}{5} = \frac{4}{20} \]
Thus,
\[ \frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20} \]

### Step 2: Simplify the denominator
The denominator is \(\frac{1}{4} \times \frac{1}{4} + 2 \times \frac{1}{4}\).

First, calculate \(\frac{1}{4} \times \frac{1}{4}\):
\[ \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \]

Now calculate \(2 \times \frac{1}{4}\):
\[ 2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]

Now we need to add \(\frac{1}{16}\) and \(\frac{1}{2}\):
To add these fractions, we need a common denominator. The least common multiple of 16 and 2 is 16:
\[ \frac{1}{2} = \frac{8}{16} \]
Thus,
\[ \frac{1}{16} + \frac{1}{2} = \frac{1}{16} + \frac{8}{16} = \frac{9}{16} \]

### Step 3: Combine the results
Now we have the simplified numerator and denominator:
\[ \frac{\frac{9}{20}}{\frac{9}{16}} \]

To divide these fractions, we multiply by the reciprocal of the denominator:
\[ \frac{9}{20} \div \frac{9}{16} = \frac{9}{20} \times \frac{16}{9} \]

The \(9\) in the numerator and denominator cancels out:
\[ = \frac{16}{20} \]

Now simplify \(\frac{16}{20}\):
\[ \frac{16}{20} = \frac{4}{5} \]

### Final Answer
Thus, the value of the expression is \(\frac{4}{5}\).

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