114/225*25/64

To simplify the expression \( \frac{114}{225} \times \frac{25}{64} \), you can follow these steps:

1. Multiply the numerators together: \[ 114 \times 25 = 2850. \]

2. Multiply the denominators together: \[ 225 \times 64 = 14400. \]

3. Now your expression looks like this: \[ \frac{2850}{14400}. \]

4. To simplify \( \frac{2850}{14400} \), find the greatest common divisor (GCD) of 2850 and 14400.

   • The prime factorization of 2850 is: \[ 2850 = 2 \times 3 \times 5^2 \times 19. \]

   • The prime factorization of 14400 is: \[ 14400 = 2^6 \times 3^2 \times 5^2. \]

5. The GCD is found by taking the minimum power of each common factor:

   • \(2^1\) from both.
   • \(3^1\) from both.
   • \(5^2\) from both.

   Therefore, the GCD is: \[ 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150. \]

6. Now divide both the numerator and the denominator by their GCD: \[ \frac{2850 \div 150}{14400 \div 150} = \frac{19}{96}. \]

Therefore, the simplified result of \( \frac{114}{225} \times \frac{25}{64} \) is \( \frac{19}{96} \).