`(5)/(6)div(6)/(7)xx?-(8)/(9)div1(3)/(5)+(3)/(4)xx3(1)/(3)=2(7)/(9)`

Options

A. `7/6`
B. `6/7`
C. 1
D. `6/9`

Explanation

To solve the equation

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

we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers to improper fractions.

- \(1 \frac{3}{5} = \frac{5 \times 1 + 3}{5} = \frac{8}{5}\)
- \(3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}\)
- \(2 \frac{7}{9} = \frac{2 \times 9 + 7}{9} = \frac{25}{9}\)

Now, the equation looks like this:

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

### Step 2: Simplify Each Term
Now, we simplify each term in the equation.

1. **First Term:**
\[ \frac{5}{6} \div \frac{6}{7} = \frac{5}{6} \times \frac{7}{6} = \frac{5 \times 7}{6 \times 6} = \frac{35}{36} \]

2. **Second Term:**
\[ \frac{8}{9} \div \frac{8}{5} = \frac{8}{9} \times \frac{5}{8} = \frac{5}{9} \]

3. **Third Term:**
\[ \frac{3}{4} \times \frac{10}{3} = \frac{3 \times 10}{4 \times 3} = \frac{10}{4} = \frac{5}{2} \]

Now, substituting these back into the equation gives:

\[ \frac{35}{36} \times ? - \frac{5}{9} + \frac{5}{2} = \frac{25}{9} \]

### Step 3: Combine Like Terms
Next, we need to combine the constants on the left side:

\[ \frac{5}{2} - \frac{5}{9} \]

To combine these, we find a common denominator, which is 18:

\[ \frac{5}{2} = \frac{5 \times 9}{2 \times 9} = \frac{45}{18} \]
\[ \frac{5}{9} = \frac{5 \times 2}{9 \times 2} = \frac{10}{18} \]

Now, we can subtract:

\[ \frac{45}{18} - \frac{10}{18} = \frac{35}{18} \]

So the equation now looks like:

\[ \frac{35}{36} \times ? + \frac{35}{18} = \frac{25}{9} \]

### Step 4: Isolate the Variable
Now, isolate the term with the question mark:

\[ \frac{35}{36} \times ? = \frac{25}{9} - \frac{35}{18} \]

To subtract these fractions, we need a common denominator, which is 18:

\[ \frac{25}{9} = \frac{25 \times 2}{9 \times 2} = \frac{50}{18} \]

Now we can subtract:

\[ \frac{50}{18} - \frac{35}{18} = \frac{15}{18} = \frac{5}{6} \]

So we have:

\[ \frac{35}{36} \times ? = \frac{5}{6} \]

### Step 5: Solve for the Question Mark
Now, multiply both sides by the reciprocal of \(\frac{35}{36}\):

\[ ? = \frac{5}{6} \times \frac{36}{35} \]

Calculating this gives:

\[ ? = \frac{5 \times 36}{6 \times 35} = \frac{30}{35} = \frac{6}{7} \]

Thus, the value of the question mark is:

\[ \boxed{\frac{6}{7}} \]