Simplified value of `6-[5/6+{3 7/8-2 1/3+1 7/9}]` is:

Options

A. `3 9/13`
B. `1 61/72`
C. `2 11/72`
D. `1 51/71`

Answer

To simplify the expression \( 6 - \left[ \frac{5}{6} + \left\{ 3 \frac{7}{8} - 2 \frac{1}{3} + 1 \frac{7}{9} \right\} \right] \), we will follow these steps:

### Step 1: Convert mixed numbers to improper fractions
1. Convert \( 3 \frac{7}{8} \) to an improper fraction:
\[ 3 \frac{7}{8} = \frac{3 \times 8 + 7}{8} = \frac{24 + 7}{8} = \frac{31}{8} \]

2. Convert \( 2 \frac{1}{3} \) to an improper fraction:
\[ 2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3} \]

3. Convert \( 1 \frac{7}{9} \) to an improper fraction:
\[ 1 \frac{7}{9} = \frac{1 \times 9 + 7}{9} = \frac{9 + 7}{9} = \frac{16}{9} \]

### Step 2: Substitute back into the expression
Now substitute the improper fractions back into the expression:
\[ 6 - \left[ \frac{5}{6} + \left\{ \frac{31}{8} - \frac{7}{3} + \frac{16}{9} \right\} \right] \]

### Step 3: Simplify inside the braces
To simplify \( \frac{31}{8} - \frac{7}{3} + \frac{16}{9} \), we need a common denominator. The least common multiple (LCM) of 8, 3, and 9 is 72.

1. Convert each fraction:
- \( \frac{31}{8} = \frac{31 \times 9}{72} = \frac{279}{72} \)
- \( \frac{7}{3} = \frac{7 \times 24}{72} = \frac{168}{72} \)
- \( \frac{16}{9} = \frac{16 \times 8}{72} = \frac{128}{72} \)

2. Now substitute back:
\[ \frac{31}{8} - \frac{7}{3} + \frac{16}{9} = \frac{279}{72} - \frac{168}{72} + \frac{128}{72} = \frac{279 - 168 + 128}{72} = \frac{239}{72} \]

### Step 4: Substitute back into the main expression
Now substitute \( \frac{239}{72} \) back into the expression:
\[ 6 - \left[ \frac{5}{6} + \frac{239}{72} \right] \]

### Step 5: Combine the fractions inside the brackets
To combine \( \frac{5}{6} \) and \( \frac{239}{72} \), we need a common denominator. The LCM of 6 and 72 is 72.

1. Convert \( \frac{5}{6} \):
\[ \frac{5}{6} = \frac{5 \times 12}{72} = \frac{60}{72} \]

2. Now combine:
\[ \frac{5}{6} + \frac{239}{72} = \frac{60}{72} + \frac{239}{72} = \frac{299}{72} \]

### Step 6: Final calculation
Now substitute back:
\[ 6 - \frac{299}{72} \]
Convert 6 to a fraction with a denominator of 72:
\[ 6 = \frac{6 \times 72}{72} = \frac{432}{72} \]
Now perform the subtraction:
\[ \frac{432}{72} - \frac{299}{72} = \frac{432 - 299}{72} = \frac{133}{72} \]

### Final Answer
The simplified value of the expression is:
\[ \frac{133}{72} \]