Step 1: Understand the problem. The problem asks to find the actual value (वास्तविक मान) of the fraction $\frac{882}{922}$. This means simplifying the fraction to its lowest terms. Step 2: Find the prime factorization of the numerator. The numerator is 882. $$ 882 = 2 \times 441 $$ $$ 441 = 3 \times 147 $$ $$ 147 = 3 \times 49 $$ $$ 49 = 7 \times 7 $$ So, the prime factorization of 882 is $2 \times 3^2 \times 7^2$. Step 3: Find the prime factorization of the denominator. The denominator is 922. $$ 922 = 2 \times 461 $$ To determine if 461 is a prime number, we test for divisibility by prime numbers up to $\sqrt{461} \approx 21.47$. The prime numbers to check are 2, 3, 5, 7, 11, 13, 17, 19. • 461 is not divisible by 2 (it is an odd number). • The sum of its digits $4+6+1=11$, which is not divisible by 3. • It does not end in 0 or 5, so it is not divisible by 5. • $461 \div 7 = 65$ with a remainder of 6. • $461 \div 11 = 41$ with a remainder of 10. • $461 \div 13 = 35$ with a remainder of 6. • $461 \div 17 = 27$ with a remainder of 2. • $461 \div 19 = 24$ with a remainder of 5. Since 461 is not divisible by any prime number up to its square root, 461 is a prime number. So, the prime factorization of 922 is $2 \times 461$. Step 4: Find the greatest common divisor (GCD). Comparing the prime factorizations: $882 = 2 \times 3^2 \times 7^2$ $922 = 2 \times 461$ The only common prime factor is 2. Therefore, the GCD of 882 and 922 is 2. Step 5: Simplify the fraction. Divide both the numerator and the denominator by their GCD. $$ \frac{882}{922} = \frac{882 \div 2}{922 \div 2} $$ $$ \frac{882}{922} = \frac{441}{461} $$ The fraction $\frac{441}{461}$ is in its simplest form because 441 and 461 have no common factors other than 1. The actual value of $\frac{882}{922}$ is $\boxed{\frac{441}{461}}$.