Question

Use your fraction pieces or another method to help fill in each blank. \(\frac{5}{6}+\frac{1}{3}+\frac{1}{2}+\quad=2\)

Short answer

\(\frac{1}{3}\)

Step-by-step solution

Find a common denominator

The fractions have different denominators: 6, 3, and 2. Determine the least common multiple (LCM) of these denominators, which is 6.

Convert each fraction to the common denominator

Convert \(\frac{1}{3}\) and \(\frac{1}{2}\) to fractions with the denominator of 6. So \(\frac{1}{3}=\frac{2}{6}\) and \(\frac{1}{2}=\frac{3}{6}\).

Add the fractions

Now add \(\frac{5}{6} + \frac{2}{6} + \frac{3}{6}\):\[\frac{5}{6} + \frac{2}{6} + \frac{3}{6} = \frac{10}{6} = \frac{5}{3}\]

Determine the missing fraction

Subtract the sum of the fractions from 2: \(2 - \frac{5}{3} = \frac{6}{3} - \frac{5}{3} = \frac{1}{3}\).\Therefore, the missing fraction is \frac{1}{3}.

Key concepts

Least Common Multiple

Understanding the least common multiple (LCM) is crucial when adding fractions with different denominators. The LCM is the smallest number that is a multiple of each of the denominators involved. For the fractions \(\frac{5}{6}, \frac{1}{3}, \frac{1}{2}\), we need to find the LCM of 6, 3, and 2.

List the multiples of each number:

• Multiples of 6: 6, 12, 18, 24, ...
• Multiples of 3: 3, 6, 9, 12, ...
• Multiples of 2: 2, 4, 6, 8, 10, ...

The smallest number that appears in all lists is 6, hence the LCM is 6. This tells us that we can use 6 as the common denominator for all fractions in the addition problem.

Fraction Addition

Once we have the common denominator, we can proceed with fraction addition. Converting fractions to the same denominator helps us add them easily. Here’s how you convert and add:

Given fractions: \(\frac{5}{6}, \frac{1}{3}$$$$, \frac{1}{2}\)
Convert to common denominator:

• Keep \(\frac{5}{6}\) as is, since its denominator is already 6.
• Convert \(\frac{1}{3}\) to \(\frac{2}{6}\) because \(\frac{1 \times 2}{3 \times 2} = \frac{2}{6}\).
• Convert \(\frac{1}{2}\) to \(\frac{3}{6}\) because \(\frac{1 \times 3}{2 \times 3} = \frac{3}{6}\).

Now, add the fractions: \(\frac{5}{6} + \frac{2}{6} + \frac{3}{6} = \frac{10}{6}\).
To make it simpler, \(\frac{10}{6}\) can be reduced to \(\frac{5}{3}\) by dividing the numerator and denominator by 2.

Common Denominator

A common denominator is needed to add fractions properly. It acts like a common ground for the fractions, making their denominators the same. Here’s why it’s essential:

A common denominator allows you to combine fractions by focusing only on the numerators, as the denominators are made equal.
Once we converted \(\frac{1}{3}\) to \(\frac{2}{6}\) and \(\frac{1}{2}\) to \(\frac{3}{6}\), we noticed all fractions had the denominator of 6. Therefore, the numerators (5, 2, 3) could be directly added. Adding them gave us: \(\frac{5}{6} + \frac{2}{6} + \frac{3}{6} = \frac{10}{6}\).

Subtracting fraction from a whole number involves same principle. Convert the whole number to fraction format: 2 in fraction terms is \(\frac{6}{3}\). As calculated, if the sum of current fractions is \(\frac{5}{3}\), missing part is simply the difference: \(\frac{6}{3} - \frac{5}{3} = \frac{1}{3}\).