21/4 : 49/64 * 7/8

To solve the expression \( \frac{21}{4} : \frac{49}{64} \times \frac{7}{8} \), we follow the order of operations:

1. Rewrite the division as multiplication by the reciprocal: \[ \frac{21}{4} \div \frac{49}{64} = \frac{21}{4} \times \frac{64}{49} \]

2. Multiply this result by \( \frac{7}{8} \): \[ \left(\frac{21}{4} \times \frac{64}{49}\right) \times \frac{7}{8} \]

3. Now, let's simplify the expression step-by-step:

   Start with \( \frac{21 \times 64 \times 7}{4 \times 49 \times 8} \)

4. Calculate \( 4 \times 49 = 196 \) and \( 8 = 2 \times 4 \), then rewrite: \[ 21 \times 64 \times 7 = 21 \times 7 \times 64 = 147 \times 64 \]

5. The overall expression becomes: \[ \frac{147 \times 64}{196 \times 8} \]

6. Continue simplifying: Reduce \( 147 \) and \( 49 \) (both divisible by 7): \[ 49 \rightarrow 7 \quad \text{(thus, } 147 \text{ becomes } 21\text{)}, \quad 196 \rightarrow 28 \text{ after dividing both numerator and denominator by 7} \] Therefore, you get: \[ \frac{21 \times 64}{28 \times 8} \]

7. As \( 28 = 4 \times 7 \) and \( 8 = 4 \times 2\), we can further reduce: \[ = \frac{21 \times 64}{(4 \times 7) \times (4 \times 2)} = \frac{21 \times 64}{16 \times 7} \]

   Now, simplifying \( 64 = 16 \times 4 \): \[ = \frac{21 \times 4}{7} \]

8. Cancel \( 7 \): \[ = 12 \times 4 \quad \text{(where } 21/7 = 3) \] Giving: \[ = 12 \]

Thus, the final result of the expression \( \frac{21}{4} : \frac{49}{64} \times \frac{7}{8} = 12 \).