Paolo helped in the community garden for $2\frac{3}{4}$ hours this week. That was $1\frac{5}{6}$ equal-length shifts, because Paolo stopped early one day when it started to rain. How long is a single shift?

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Q. Paolo helped in the community garden for

2\frac{3}{4}

hours this week. That was

1\frac{5}{6}

equal-length shifts, because Paolo stopped early one day when it started to rain. How long is a single shift?

Convert to Improper Fractions: Convert the mixed numbers to improper fractions. $\newline$ $2 \frac{3}{4}$ hours can be converted to an improper fraction by multiplying the whole number by the denominator and adding the numerator, then placing the result over the original denominator. $\newline$ $2 \frac{3}{4} = (2 \times 4 + 3)/4 = (8 + 3)/4 = \frac{11}{4}$ hours. $\newline$ $1 \frac{5}{6}$ shifts can be converted similarly. $\newline$ $1 \frac{5}{6} = (1 \times 6 + 5)/6 = (6 + 5)/6 = \frac{11}{6}$ shifts.
Find Shift Length: Divide the total hours worked by the number of shifts to find the length of one shift. $\newline$ To find the length of one shift, we divide the total hours worked by the number of shifts. $\newline$ $(11/4)$ hours $\div$ $(11/6)$ shifts = $(11/4) \times (6/11)$ = $(11 \times 6) / (4 \times 11)$ .
Simplify Fraction: Simplify the fraction. $\newline$ $(11 \times 6) / (4 \times 11)$ simplifies because the $11$ s in the numerator and denominator cancel each other out. $\newline$ $(11 \times 6) / (4 \times 11) = 6/4$ .
Divide by GCD: Simplify the fraction further by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ . $\frac{6}{4} = \frac{(6/2)}{(4/2)} = \frac{3}{2}$ .
Convert to Mixed Number: Convert the improper fraction back to a mixed number if necessary. $\newline$ $\frac{3}{2}$ is an improper fraction and can be written as a mixed number. Since $2$ goes into $3$ once with a remainder of $1$ , the mixed number is: $\newline$ $\frac{3}{2} = 1 \frac{1}{2}$ .