`(5)/(6) -: (6)/(7) xx ? - (8)/(9) -: 1 (2)/(5) -: (3)/(4) xx 3 (1)/(3) = 2(7)/(9)`

Options

A. A) `(7)/(8)`
B. B) `(6)/(7)`
C. C) 1
D. D) `220/49`

Answer

To solve the equation step by step, we will follow the operations as indicated in the question.

The original equation is:

\[ \frac{5}{6} \div \frac{6}{7} \times ? - \frac{8}{9} \div 1 \frac{2}{5} \div \frac{3}{4} \times 3 \frac{1}{3} = 2 \frac{7}{9} \]

### Step 1: Convert Mixed Numbers to Improper Fractions

1. Convert \(1 \frac{2}{5}\) to an improper fraction:
\[ 1 \frac{2}{5} = \frac{5 \times 1 + 2}{5} = \frac{7}{5} \]

2. Convert \(3 \frac{1}{3}\) to an improper fraction:
\[ 3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3} \]

3. Convert \(2 \frac{7}{9}\) to an improper fraction:
\[ 2 \frac{7}{9} = \frac{2 \times 9 + 7}{9} = \frac{25}{9} \]

Now, the equation becomes:

\[ \frac{5}{6} \div \frac{6}{7} \times ? - \frac{8}{9} \div \frac{7}{5} \div \frac{3}{4} \times \frac{10}{3} = \frac{25}{9} \]

### Step 2: Change Division to Multiplication

1. Change the division to multiplication by taking the reciprocal:
\[ \frac{5}{6} \times \frac{7}{6} \times ? - \frac{8}{9} \times \frac{5}{7} \times \frac{4}{3} \times \frac{10}{3} = \frac{25}{9} \]

### Step 3: Simplify Each Side

1. Simplify \(\frac{5}{6} \times \frac{7}{6}\):
\[ \frac{5 \times 7}{6 \times 6} = \frac{35}{36} \]

2. Simplify \(\frac{8}{9} \times \frac{5}{7} \times \frac{4}{3} \times \frac{10}{3}\):
- First, calculate \(\frac{8 \times 5 \times 4 \times 10}{9 \times 7 \times 3 \times 3}\):
\[ = \frac{1600}{189} \]

### Step 4: Substitute Back into the Equation

Now we have:

\[ \frac{35}{36} \times ? - \frac{1600}{189} = \frac{25}{9} \]

### Step 5: Isolate the Question Mark

1. Move \(\frac{1600}{189}\) to the right side:
\[ \frac{35}{36} \times ? = \frac{25}{9} + \frac{1600}{189} \]

2. Find a common denominator for \(\frac{25}{9}\) and \(\frac{1600}{189}\):
- The LCM of 9 and 189 is 189. Convert \(\frac{25}{9}\):
\[ \frac{25}{9} = \frac{25 \times 21}{9 \times 21} = \frac{525}{189} \]

3. Now add:
\[ \frac{525}{189} + \frac{1600}{189} = \frac{2125}{189} \]

### Step 6: Solve for the Question Mark

1. Substitute back:
\[ \frac{35}{36} \times ? = \frac{2125}{189} \]

2. Multiply both sides by \(\frac{36}{35}\):
\[ ? = \frac{2125 \times 36}{189 \times 35} \]

3. Simplify:
- Calculate \(189 \times 35 = 6615\)
- Calculate \(2125 \times 36 = 76500\)
- Therefore:
\[ ? = \frac{76500}{6615} \]

4. Simplifying further gives:
\[ ? = \frac{220}{7} \]

### Final Answer

Thus, the value of the question mark \(?\) is:

\[ ? = \frac{220}{7} \]