How do we write a fraction of numerator 2 as the sum of three unit fractions?
lots of ways, if I understand what you want.
For example,
1/3 + 1/4 + 1/12 = 2/3
To write a fraction with a numerator of 2 as the sum of three unit fractions, you need to find three fractions that have 1 as their numerator (unit fractions) and add them up to get a fraction with a numerator of 2.
Let's denote the three unit fractions as 1/a, 1/b, and 1/c, where a, b, and c are positive integers.
To find the values of a, b, and c, we need to solve the equation:
1/a + 1/b + 1/c = 2
This is known as an Egyptian fraction, where the sum of unit fractions equals a given fraction.
To solve this equation, we can use a method called "Greedy Algorithm".
1. Start with a = 3, b = 3, and c = 3 (you can choose any number greater than 2 for a, b, and c).
2. Substitute these values into the equation and check if the sum equals 2. If the sum is less than 2, increment a, b, or c by 1 (starting with a).
3. Repeat step 2 until the sum equals 2.
Let's go through an example:
Starting with a = 3, b = 3, and c = 3:
1/3 + 1/3 + 1/3 = 1
Since the sum is less than 2, we increment a by 1:
1/4 + 1/3 + 1/3 = 1 + 1/12
Now the sum is greater than 2, so we need to adjust the values of b and c.
1/4 + 1/5 + 1/5 = 1 + 1/4 + 1/25
Now the sum is still greater than 2, so we continue adjusting the values of a, b, and c.
1/4 + 1/5 + 1/6 = 1 + 1/4 + 1/5 + 1/120
Finally, we have found the values of a, b, and c that satisfy the equation:
1/4 + 1/5 + 1/6 = 2
Therefore, we can write the fraction 2/1 as the sum of the unit fractions 1/4, 1/5, and 1/6.