- Separate the whole number and fractional parts of the mixed numbers: \(4+3+10=17\) and \(\frac{5}{8}+\frac{2}{3}+\frac{1}{4}\).
- Find a common denominator for the fractions to sum them: \(\frac{15}{24} + \frac{16}{24} + \frac{6}{24} = \frac{37}{24} = 1 \frac{13}{24}\).
- Add the sum of the whole numbers to the fractional result: \(17 + 1 \frac{13}{24} = 18 \frac{13}{24}\).
- The total length of the ribbon is \(\boxed{18 \frac{13}{24}}\) yd.
Explanation
- Understanding the Problem
To find the total length of the ribbon sold, we need to add the lengths of the three individual pieces: \(4 \frac{5}{8}\) yards, \(3 \frac{2}{3}\) yards, and \(10 \frac{1}{4}\) yards.
The expression for the total length is:
\[4 \frac{5}{8} + 3 \frac{2}{3} + 10 \frac{1}{4}\]
- Adding the Fractional Parts
To add the mixed numbers, we first sum the whole number parts and the fractional parts separately.
Sum of the whole numbers:
\[4 + 3 + 10 = 17\]
Sum of the fractional parts:
\[\frac{5}{8} + \frac{2}{3} + \frac{1}{4}\]
To add these fractions, we find the least common denominator (LCD) for 8, 3, and 4. The LCD is 24.
Now, express each fraction with a denominator of 24:
\[\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}\]
\[\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}\]
\[\frac{1}{4} = \frac{1 \times 6}{4 \times 6} = \frac{6}{24}\]
Adding them together gives:
\[\frac{15}{24} + \frac{16}{24} + \frac{6}{24} = \frac{37}{24}\]
Since \(\frac{37}{24}\) is an improper fraction, we convert it into a mixed number:
\[\frac{37}{24} = 1 \frac{13}{24}\]
- Final Calculation and Conclusion
Finally, we add the sum of the whole numbers to the result of the fractional addition:
\[17 + 1 \frac{13}{24} = 18 \frac{13}{24}\]
Alternatively, as an improper fraction, the sum is:
\[17 + \frac{37}{24} = \frac{408}{24} + \frac{37}{24} = \frac{445}{24}\]
The total length of the ribbon sold is \(18 \frac{13}{24}\) yards.
Examples
This method of adding lengths is useful in many real-world situations, such as construction or sewing. For example, if you are buying wood for a DIY shelf project and you need several pieces of different lengths like 4 5/8 feet, 3 2/3 feet, and 10 1/4 feet, you can use this exact approach to determine the minimum length of lumber you need to buy to ensure you have enough material to cut all the pieces you require. Ensuring accurate measurements like this helps reduce waste and saves money on your projects.