$2 \frac{4}{5}+3 \frac{1}{2}$
To calculate the sum of two mixed numbers \(2 \frac{4}{5}\) and \(3 \frac{1}{2}\), we first analyze their structure. A mixed number \(a \frac{b}{c}\) represents the sum \(a + \frac{b}{c}\). Thus, we can express the sum as \((2 + \frac{4}{5}) + (3 + \frac{1}{2})\). We can regroup these parts by adding the whole numbers separately and the fractions separately.
First, we add the whole number parts together:
\[2 + 3 = 5.\]
Next, we add the fractional parts:
\[\frac{4}{5} + \frac{1}{2}.\]
To add these fractions, we find a common denominator, which is 10. We rewrite the fractions:
\[\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10},\]
\[\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}.\]
Adding these gives:
\[\frac{8}{10} + \frac{5}{10} = \frac{13}{10}.\]
The sum of the fractions \(\frac{13}{10}\) is an improper fraction, which can be written as a mixed number:
\[\frac{13}{10} = 1 \frac{3}{10}.\]
Now, we add the whole number sum from the previous step to this result:
\[5 + 1 \frac{3}{10} = 6 \frac{3}{10}.\]
Alternatively, expressed as a decimal, this is \(6 + 0.3 = 6.3\). Thus, the sum is \(6 \frac{3}{10}\).
In everyday life, adding mixed numbers is quite common when dealing with measurements. For instance, if you are building shelves and have one board measuring \(2 \frac{4}{5}\) meters and another measuring \(3 \frac{1}{2}\) meters, you need to find the total length of wood required. Calculating \(2 \frac{4}{5} + 3 \frac{1}{2} = 6 \frac{3}{10}\) meters helps you determine exactly how much lumber to purchase, ensuring you have enough for your project without unnecessary waste.