The value of `5 (1)/(3) + 1 (2)/(9) xx (1)/(4) (10 + (3)/(1 - (1)/(5)))` is

Options

A. `(1373)/(144)`
B. `(67)/(25)`
C. `(128)/(11)`
D. `(128)/(99)`

Answer

To solve the expression \( 5 \frac{1}{3} + 1 \frac{2}{9} \times \frac{1}{4} \left( 10 + \frac{3}{1 - \frac{1}{5}} \right) \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert \( 5 \frac{1}{3} \) and \( 1 \frac{2}{9} \) into improper fractions.

- For \( 5 \frac{1}{3} \):
\[ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3} \]

- For \( 1 \frac{2}{9} \):
\[ 1 \frac{2}{9} = \frac{1 \times 9 + 2}{9} = \frac{9 + 2}{9} = \frac{11}{9} \]

### Step 2: Simplify the Expression Inside the Bracket
Now, simplify the expression inside the bracket \( 10 + \frac{3}{1 - \frac{1}{5}} \).

- First, calculate \( 1 - \frac{1}{5} \):
\[ 1 - \frac{1}{5} = \frac{5 - 1}{5} = \frac{4}{5} \]

- Now, substitute this back into the expression:
\[ 10 + \frac{3}{\frac{4}{5}} = 10 + 3 \times \frac{5}{4} = 10 + \frac{15}{4} \]

- Convert \( 10 \) to a fraction with a denominator of \( 4 \):
\[ 10 = \frac{40}{4} \]

- Now add the fractions:
\[ \frac{40}{4} + \frac{15}{4} = \frac{55}{4} \]

### Step 3: Substitute Back into the Original Expression
Now substitute back into the original expression:
\[ \frac{16}{3} + \frac{11}{9} \times \frac{1}{4} \times \frac{55}{4} \]

### Step 4: Calculate the Multiplication
Calculate \( \frac{11}{9} \times \frac{1}{4} \times \frac{55}{4} \):
\[ \frac{11 \times 55}{9 \times 4 \times 4} = \frac{605}{144} \]

### Step 5: Add the Two Fractions
Now add \( \frac{16}{3} \) and \( \frac{605}{144} \). First, convert \( \frac{16}{3} \) to have a common denominator of \( 144 \):
\[ \frac{16}{3} = \frac{16 \times 48}{3 \times 48} = \frac{768}{144} \]

Now add the two fractions:
\[ \frac{768}{144} + \frac{605}{144} = \frac{768 + 605}{144} = \frac{1373}{144} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{1373}{144} \]