To find out how long the original roll of paper was before Sami cut off a piece, we need to add the length of the piece that Sami cut off to the remaining length of the roll. We can represent this mathematically.
1. Identify given values:
- Cut length: [tex]\(6 \frac{3}{4}\)[/tex] inches
- Remaining length: [tex]\(36 \frac{1}{3}\)[/tex] inches
2. Convert the mixed numbers to improper fractions:
- For [tex]\(6 \frac{3}{4}\)[/tex]:
[tex]\[
6 \frac{3}{4} = 6 + \frac{3}{4} = \frac{24}{4} + \frac{3}{4} = \frac{27}{4}
\][/tex]
- For [tex]\(36 \frac{1}{3}\)[/tex]:
[tex]\[
36 \frac{1}{3} = 36 + \frac{1}{3} = \frac{108}{3} + \frac{1}{3} = \frac{109}{3}
\][/tex]
3. Find the least common denominator (LCD) to add the fractions:
- The denominators are 4 and 3. The LCD is 12.
- Convert [tex]\(\frac{27}{4}\)[/tex] to a fraction with denominator 12:
[tex]\[
\frac{27}{4} = \frac{27 \times 3}{4 \times 3} = \frac{81}{12}
\][/tex]
- Convert [tex]\(\frac{109}{3}\)[/tex] to a fraction with denominator 12:
[tex]\[
\frac{109}{3} = \frac{109 \times 4}{3 \times 4} = \frac{436}{12}
\][/tex]
4. Add the fractions with a common denominator:
[tex]\[
\frac{81}{12} + \frac{436}{12} = \frac{517}{12}
\][/tex]
5. Convert the improper fraction back to a mixed number:
- Divide 517 by 12 to get the whole number and remainder:
[tex]\[
517 \div 12 = 43 \text{ R } 1 \quad \text{(43 remainder 1)}
\][/tex]
- Thus, the improper fraction [tex]\(\frac{517}{12}\)[/tex] can be written as the mixed number:
[tex]\[
43 \frac{1}{12}
\][/tex]
6. Verify the result by comparing it with the given options:
- [tex]\(29 \frac{7}{12}\)[/tex] inches
- [tex]\(30 \frac{7}{12}\)[/tex] inches
- [tex]\(42 \frac{1}{12}\)[/tex] inches
- [tex]\(43 \frac{1}{12}\)[/tex] inches
The calculated original length of the roll of paper is [tex]\(43 \frac{1}{12}\)[/tex] inches, which matches the fourth option given.
Therefore, the original length of the roll of paper was [tex]\(43 \frac{1}{12}\)[/tex] inches.
