Understanding the Problem
The expression 1/2 + 1/3 + 1/6 may seem simple, but it requires careful attention to detail to add the fractions correctly. The first step is to find the least common multiple (LCM) of the denominators 2, 3, and 6.
It's essential to have a solid understanding of fractions and their properties to tackle this problem. If you're new to fractions, take a few minutes to review the basics before proceeding.
Step 1: Find the Least Common Multiple (LCM)
The LCM of 2, 3, and 6 is 6. To find the LCM, we need to list the multiples of each denominator:
- 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, ...
- 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...
- 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...
As you can see, the smallest number that appears in all three lists is 6, so the LCM of 2, 3, and 6 is 6.
Step 2: Convert Each Fraction to Have the LCM as the Denominator
Now that we know the LCM is 6, we need to convert each fraction to have 6 as the denominator. To do this, we multiply the numerator and denominator of each fraction by the necessary number to make the denominator 6:
- 1/2 = (1 x 3)/(2 x 3) = 3/6
- 1/3 = (1 x 2)/(3 x 2) = 2/6
- 1/6 (no change needed)
Now we have the fractions 3/6, 2/6, and 1/6 with the same denominator, which is 6.
Step 3: Add the Fractions
With the fractions now having the same denominator, we can add them together by adding the numerators:
(3 + 2 + 1)/6 = 6/6
Since the numerator and denominator are the same, the result is a whole number: 1.
Step 4: Simplify the Result (Optional)
Since the result is a whole number, there's no need to simplify further. However, if we were to simplify a fraction, we would divide the numerator and denominator by their greatest common divisor (GCD).
For example, if the result were 2/4, we could simplify it by dividing both numbers by 2, resulting in 1/2.
Comparing Fractions
| Denominator | LCM | Result |
|---|---|---|
| 2, 3, 6 | 6 | 1 |
| 2, 4, 6 | 12 | 2 |
| 3, 4, 6 | 12 | 2 |
The table above shows that the LCM and result change depending on the denominators used. If the denominators are 2, 4, and 6, the LCM is 12, and the result is 2. If the denominators are 3, 4, and 6, the LCM is also 12, and the result is still 2.
Practical Tips and Variations
When adding fractions, it's essential to find the LCM of the denominators to ensure accurate results. If the denominators are not related (e.g., 2, 5, and 7), you may need to list the multiples of each denominator to find the LCM.
Another important tip is to simplify fractions when possible to avoid unnecessary calculations. In the case of 1/2 + 1/3 + 1/6, the result is a whole number, so there's no need to simplify further.
Remember, practice makes perfect! Try adding different sets of fractions with various denominators to solidify your understanding of this concept.