If `A = (3)/(4)div (5)/(6) , B = 3 div[ ( 4 div 5) div 6],`
`C = [3 div (4 div5) ] div6 and `
`D =3 div 4 (5 div 6) ,` then

Options

A. A and D are equal
B. All are equal
C. A and B are equal
D. A and Care equal

Answer

To solve the problem, we need to evaluate the expressions for A, B, C, and D step by step.

### Step 1: Calculate A
Given:
\[ A = \frac{3}{4} \div \frac{5}{6} \]

To divide by a fraction, we multiply by its reciprocal:
\[ A = \frac{3}{4} \times \frac{6}{5} \]

Now, multiply the numerators and the denominators:
\[ A = \frac{3 \times 6}{4 \times 5} = \frac{18}{20} \]

Now simplify:
\[ A = \frac{9}{10} \]

### Step 2: Calculate B
Given:
\[ B = 3 \div \left( \frac{4}{5} \div 6 \right) \]

First, calculate the inner division:
\[ \frac{4}{5} \div 6 = \frac{4}{5} \times \frac{1}{6} = \frac{4}{30} = \frac{2}{15} \]

Now substitute this back into B:
\[ B = 3 \div \frac{2}{15} \]

Again, multiply by the reciprocal:
\[ B = 3 \times \frac{15}{2} = \frac{45}{2} \]

### Step 3: Calculate C
Given:
\[ C = \left( 3 \div \left( \frac{4}{5} \right) \right) \div 6 \]

First, calculate the inner division:
\[ 3 \div \frac{4}{5} = 3 \times \frac{5}{4} = \frac{15}{4} \]

Now substitute this back into C:
\[ C = \frac{15}{4} \div 6 = \frac{15}{4} \times \frac{1}{6} = \frac{15}{24} \]

Now simplify:
\[ C = \frac{5}{8} \]

### Step 4: Calculate D
Given:
\[ D = 3 \div 4 \left( \frac{5}{6} \right) \]

This means:
\[ D = 3 \div \left( \frac{4 \times 5}{6} \right) = 3 \div \frac{20}{6} = 3 \times \frac{6}{20} = \frac{18}{20} \]

Now simplify:
\[ D = \frac{9}{10} \]

### Summary of Results
- \( A = \frac{9}{10} \)
- \( B = \frac{45}{2} \)
- \( C = \frac{5}{8} \)
- \( D = \frac{9}{10} \)

### Conclusion
From the calculations, we can see that:
- \( A = D \)
- \( B \) and \( C \) are different.