The Correct Answer and Explanation is:
The correct simplified form of the fraction 20/110 is 2/11.
To simplify a fraction, the goal is to find an equivalent fraction where the numerator (the top number) and the denominator (the bottom number) are the smallest possible whole numbers. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is the largest number that divides both of them without leaving a remainder.
For the fraction 20/110, we need to find the greatest common divisor of 20 and 110.
One common method is to look for obvious factors. Since both the numerator, 20, and the denominator, 110, end in zero, we know they are both divisible by 10. This is a good starting point. Let’s divide both numbers by this common factor:
- Divide the numerator by 10: 20 ÷ 10 = 2
- Divide the denominator by 10: 110 ÷ 10 = 11
This gives us the new fraction: 2/11.
Now, we must verify if this fraction is in its simplest form. We need to check if there are any common factors (other than 1) between the new numerator, 2, and the new denominator, 11. The factors of 2 are 1 and 2. The factors of 11 are 1 and 11. Since 11 is a prime number, its only factors are itself and 1. The only common factor between 2 and 11 is 1, which means the fraction cannot be reduced any further. Numbers that have no common factors other than 1 are called coprime.
Another more systematic way to ensure you have found the greatest common divisor is through prime factorization. We break down each number into its prime factors:
- Prime factorization of 20: 2 × 2 × 5
- Prime factorization of 110: 2 × 5 × 11
The common prime factors are 2 and 5. The greatest common divisor is the product of these common factors: 2 × 5 = 10. Dividing the original fraction’s numerator and denominator by this GCD (10) confirms our result of 2/11.
