Let's solve this problem step-by-step.
1. Representing the Quantities in Mixed Fractions:
Daisuke starts with [tex]\( 14 \frac{1}{3} \)[/tex] yards of rope, and he uses [tex]\( 11 \frac{5}{6} \)[/tex] yards.
2. Converting Mixed Fractions to Improper Fractions:
To perform subtraction, it is easier to work with improper fractions first.
- Convert [tex]\( 14 \frac{1}{3} \)[/tex] to an improper fraction:
[tex]\[
14 \frac{1}{3} = 14 + \frac{1}{3} = \frac{42}{3} + \frac{1}{3} = \frac{43}{3}
\][/tex]
- Convert [tex]\( 11 \frac{5}{6} \)[/tex] to an improper fraction:
[tex]\[
11 \frac{5}{6} = 11 + \frac{5}{6} = \frac{66}{6} + \frac{5}{6} = \frac{71}{6}
\][/tex]
3. Finding a Common Denominator:
To subtract these fractions, we need a common denominator. The least common multiple of 3 and 6 is 6.
- Convert [tex]\( \frac{43}{3} \)[/tex] to a fraction with a denominator of 6:
[tex]\[
\frac{43}{3} \times \frac{2}{2} = \frac{86}{6}
\][/tex]
4. Subtracting the Fractions:
Now, we can subtract [tex]\( \frac{71}{6} \)[/tex] from [tex]\( \frac{86}{6} \)[/tex]:
[tex]\[
\frac{86}{6} - \frac{71}{6} = \frac{15}{6}
\][/tex]
5. Simplifying the Fraction:
Simplify [tex]\( \frac{15}{6} \)[/tex]:
[tex]\[
\frac{15}{6} = 2 \frac{3}{6} = 2 \frac{1}{2}
\][/tex]
This means Daisuke has [tex]\( 2 \frac{1}{2} \)[/tex] yards of rope left.
So, after cutting off the rope, Daisuke has approximately 2.81 yards of rope left.
