Write the sum of unit fraction for 2/3
The Correct Answer and Explanation is:
Sum of Unit Fractions for 2/3
The fraction 2/3 can be expressed as the sum of two distinct unit fractions:
2/3 = 1/2 + 1/6
Explanation
A unit fraction is a fraction where the numerator is 1 and the denominator is a positive integer, such as 1/2, 1/7, or 1/25. The process of expressing a given fraction as a sum of distinct unit fractions is a mathematical concept that dates back to ancient Egypt. These representations are often called Egyptian Fractions. The standard convention is to use unit fractions with unique denominators.
To find the unit fraction representation for 2/3, a common method is the greedy algorithm. This involves finding the largest possible unit fraction that is smaller than the target fraction, subtracting it, and then repeating the process with the remainder.
Step 1: Find the largest unit fraction less than 2/3.
To find this, we look for the smallest integer denominator, let’s call it n, such that 1/n is less than or equal to 2/3.
- If n = 1, we have 1/1 = 1, which is greater than 2/3.
- If n = 2, we have 1/2 = 0.5. Since 2/3 is approximately 0.667, 1/2 is indeed less than 2/3.
Therefore, the first unit fraction in our sum is 1/2.
Step 2: Subtract this unit fraction from the original fraction.
Now, we calculate the remainder by subtracting 1/2 from 2/3. To do this, we must find a common denominator, which is 6.
2/3 – 1/2 = 4/6 – 3/6 = 1/6
Step 3: Analyze the remainder.
The result of our subtraction is 1/6. This remainder is, conveniently, already a unit fraction. This means our process is complete. We have successfully broken 2/3 into two parts.
By combining the results from our steps, we can confidently state that the sum of the unit fractions 1/2 and 1/6 is equal to 2/3. This can be verified by adding them back together:
1/2 + 1/6 = 3/6 + 1/6 = 4/6 = 2/3
