If `P = 3/8 div 15/16 ,Q=3 div [(8 div 15)div 16] , R =[3 div (8 div 15)] div 16 and S= 3 div 8 (15 div 16)` then which of the following are equal ?

Options

A. P and Q
B. P and S
C. P and R
D. All are equal

Answer

To solve the question, we need to evaluate the expressions for P, Q, R, and S step by step.

### Step 1: Calculate P
Given:
\[ P = \frac{3}{8} \div \frac{15}{16} \]

To divide by a fraction, we multiply by its reciprocal:
\[ P = \frac{3}{8} \times \frac{16}{15} \]

Now, multiply:
\[ P = \frac{3 \times 16}{8 \times 15} = \frac{48}{120} \]

Now, simplify \( \frac{48}{120} \):
Both 48 and 120 can be divided by 24:
\[ P = \frac{2}{5} \]

### Step 2: Calculate Q
Given:
\[ Q = 3 \div \left( \left( 8 \div 15 \right) \div 16 \right) \]

First, calculate \( 8 \div 15 \):
\[ 8 \div 15 = \frac{8}{15} \]

Now, divide by 16:
\[ \frac{8}{15} \div 16 = \frac{8}{15} \times \frac{1}{16} = \frac{8}{240} = \frac{1}{30} \]

Now, calculate Q:
\[ Q = 3 \div \frac{1}{30} = 3 \times 30 = 90 \]

### Step 3: Calculate R
Given:
\[ R = \left( 3 \div \left( 8 \div 15 \right) \right) \div 16 \]

First, calculate \( 8 \div 15 \):
\[ 8 \div 15 = \frac{8}{15} \]

Now, calculate:
\[ 3 \div \frac{8}{15} = 3 \times \frac{15}{8} = \frac{45}{8} \]

Now, divide by 16:
\[ R = \frac{45}{8} \div 16 = \frac{45}{8} \times \frac{1}{16} = \frac{45}{128} \]

### Step 4: Calculate S
Given:
\[ S = 3 \div 8 \times \left( 15 \div 16 \right) \]

Calculate \( 15 \div 16 \):
\[ 15 \div 16 = \frac{15}{16} \]

Now, calculate S:
\[ S = \frac{3}{8} \times \frac{15}{16} = \frac{3 \times 15}{8 \times 16} = \frac{45}{128} \]

### Conclusion
Now we have:
- \( P = \frac{2}{5} \)
- \( Q = 90 \)
- \( R = \frac{45}{128} \)
- \( S = \frac{45}{128} \)

From the calculations, we see that:
- \( R = S \)

### Final Answer
The expressions that are equal are \( R \) and \( S \).