Question

Use your fraction pieces or another method to help fill in each blank. \(\frac{1}{2}+\frac{1}{3}+\quad=1 \frac{1}{6}\)

Short answer

The missing fraction is \(\frac{1}{6}\).

Step-by-step solution

Find the common denominator

Convert the fractions to have a common denominator. The denominators are 2 and 3. The least common multiple (LCM) of 2 and 3 is 6.

Convert each fraction

Convert \(\frac{1}{2}\) to sixths: \(\frac{1}{2} = \frac{3}{6}\). Convert \(\frac{1}{3}\) to sixths: \(\frac{1}{3} = \frac{2}{6}\).

Add the fractions

Add \(\frac{3}{6}\) and \(\frac{2}{6}\): \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).

Subtract from the total

The total is \(\frac{1}{6}\) from the mixed number. We need to complete the sum to get \(\frac{11}{6}\) (which is equal to 1 \(\frac{1}{6}\)). Thus, \(\frac{11}{6} - \frac{5}{6} = \frac{6}{6} = 1\).

Find the missing fraction

The missing fraction we need to add is \(\frac{1}{6}\). Therefore, the complete sum will be \(\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1 \frac{1}{6}\).

Key concepts

least common multiple

To add fractions, we often need to find a common denominator. This common denominator helps us turn fractions into forms that are easier to add. One key concept in finding a common denominator is the Least Common Multiple (LCM). The LCM of two or more numbers is the smallest number that is a multiple of each of these numbers.

For example, let's consider the denominators 2 and 3. We need to determine the smallest number that both 2 and 3 can divide evenly into.

• Listing multiples of 2: 2, 4, 6, 8, 10...
• Listing multiples of 3: 3, 6, 9, 12...

Observing the lists, 6 is the smallest number that appears in both. Therefore, 6 is the LCM of 2 and 3.

Finding the LCM is crucial because it allows us to convert fractions to have the same denominator, which simplifies the addition process.

convert fractions

Once we know the LCM, we must convert fractions to have this common denominator. This step ensures fractions have a common base, making them easier to add.

Let's illustrate this with our fractions \(\frac{1}{2}\) and \(\frac{1}{3}\). We have already determined that the LCM is 6. Now, let's convert these fractions to have a denominator of 6:

\(\frac{1}{2}\) needs to be multiplied by 3 (both numerator and denominator):

\(\frac{1}{2} \times \frac{3}{3} = \frac{3}{6}\)

Similarly, \(\frac{1}{3}\) needs to be multiplied by 2:

\(\frac{1}{3} \times \frac{2}{2} = \frac{2}{6}\)

Now, the fractions \(\frac{1}{2}\) and \(\frac{1}{3}\) are converted to \(\frac{3}{6}\) and \(\frac{2}{6}\), respectively. This conversion is essential for adding fractions effectively.

This step ensures that our fractions are equivalent to the original, but set to a common denominator.

mixed numbers

Mixed numbers combine a whole number with a fraction. In our exercise, we have a mixed number 1 \(\frac{1}{6}\). To solve problems involving mixed numbers, we often convert between mixed numbers and improper fractions.

An improper fraction has a numerator larger than its denominator. To convert the mixed number 1 \(\frac{1}{6}\) into an improper fraction, we follow these steps:

• Multiply the whole number (1) by the denominator (6).
• Add the result to the numerator (1).

So, 1 \(\times 6 = 6\), and then we add the 1 (numerator of the fraction part), resulting in 7. This makes the improper fraction \(\frac{7}{6}\).

After working with improper fractions, we might need to convert the sum back to a mixed number. For example, adding \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\) and then recognizing we need a full \(\frac{6}{6}\) to make a whole number. Consequently, \(\frac{11}{6}\) converts to 1 \(\frac{5}{6}\).

Understanding mixed numbers helps us ensure our final answer is put into easy, readable forms.