If `I = (3)/(4) + (5)/(6), II = 3 div [(4 div 5) div 6], III = [3 div (4 div 5)] div 6,` `
` `IV = 3 div 4 (5 div 6)` then

Options

A. I and II are equal
B. I and IV are equal
C. I and III are equal
D. All are equal

Answer

To solve the problem, we will evaluate each expression step by step.

### Step 1: Evaluate I
I = \( \frac{3}{4} + \frac{5}{6} \)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 6 is 12.

Convert each fraction:
- \( \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \)
- \( \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \)

Now add them:
\[ I = \frac{9}{12} + \frac{10}{12} = \frac{19}{12} \]

### Step 2: Evaluate II
II = \( 3 \div \left( \left( 4 \div 5 \right) \div 6 \right) \)

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

Now, calculate \( \left( \frac{4}{5} \div 6 \right) \):
\[ \frac{4}{5} \div 6 = \frac{4}{5} \times \frac{1}{6} = \frac{4}{30} = \frac{2}{15} \]

Now substitute back into II:
\[ II = 3 \div \frac{2}{15} = 3 \times \frac{15}{2} = \frac{45}{2} \]

### Step 3: Evaluate III
III = \( \left( 3 \div \left( 4 \div 5 \right) \right) \div 6 \)

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

Now calculate \( 3 \div \frac{4}{5} \):
\[ 3 \div \frac{4}{5} = 3 \times \frac{5}{4} = \frac{15}{4} \]

Now substitute back into III:
\[ III = \frac{15}{4} \div 6 = \frac{15}{4} \times \frac{1}{6} = \frac{15}{24} = \frac{5}{8} \]

### Step 4: Evaluate IV
IV = \( 3 \div 4 \times \left( 5 \div 6 \right) \)

First, calculate \( 5 \div 6 \):
\[ 5 \div 6 = \frac{5}{6} \]

Now substitute back into IV:
\[ IV = 3 \div 4 \times \frac{5}{6} = \frac{3}{4} \times \frac{5}{6} = \frac{15}{24} = \frac{5}{8} \]

### Summary of Results
- I = \( \frac{19}{12} \)
- II = \( \frac{45}{2} \)
- III = \( \frac{5}{8} \)
- IV = \( \frac{5}{8} \)

### Conclusion
Comparing the results:
- I is not equal to II, III, or IV.
- II is not equal to I, III, or IV.
- III is equal to IV.

Thus, the only equal expressions are III and IV.