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Understanding the reciprocal of a fraction is key to solving many fraction problems. The reciprocal of a fraction just means flipping it upside down. For instance, if we have a fraction \(\frac{a}{b}\), the reciprocal will be \(\frac{b}{a}\). This is true for any fraction as long as the numerator and denominator are not zero! To find the reciprocal of \(\frac{485}{581}\) as in the original problem, we simply flip it and get \(\frac{581}{485}\). We use reciprocals to turn division into multiplication which is a simpler operation to manage.

Once we have the reciprocal, we can proceed to multiply the fractions. Multiplying fractions is fairly easy: just multiply the numerators together to get the new numerator, and the denominators together to get the new denominator. For example, to multiply \(\frac{8633}{7387}\) by \(\frac{581}{485}\), we perform the following steps:

• Multiply the numerators: \(8633 \times 581 = 501,517\)
• Multiply the denominators: \(7387 \times 485 = 3,583,295\)

This results in the fraction \(\frac{501,517}{3,583,295}\). Now, our next job is to see if this fraction can be simplified.

Prime factors are the building blocks of our numbers—any number can be broken down into prime factors, which are the prime numbers that multiply together to result in the original number. A prime number is a number greater than 1 that only has two divisors: 1 and itself. To simplify fractions, we look for common prime factors in the numerator and denominator. In our example, we have the fraction \(\frac{501,517}{3,583,295}\). By breaking down both the numerator and denominator into their prime factors, we can identify any common factors that can be divided out to simplify the fraction further. If no common prime factors are found, the fraction is already in its simplest form.
Let's practice finding prime factors: to find the prime factors of a number, we can use a factor tree or repeated division by prime numbers until we are left only with prime numbers.