A car repair shop offers its customers free coffee while they wait. By the end of the day, the coffee urn, which had started out with 8 1/2 gallons of coffee, was left with 2 1/3 g... A car repair shop offers its customers free coffee while they wait. By the end of the day, the coffee urn, which had started out with 8 1/2 gallons of coffee, was left with 2 1/3 gallons. How many gallons of coffee had been dispensed? Write your answer as a fraction or as a whole or mixed number. Read more

Steps to Solve

1. Convert mixed numbers to improper fractions

   Convert $8 \frac{1}{2}$ and $2 \frac{1}{3}$ to improper fractions:

   $8 \frac{1}{2} = \frac{17}{2}$ (calculated as $8 \times 2 + 1 = 17$)

   $2 \frac{1}{3} = \frac{7}{3}$ (calculated as $2 \times 3 + 1 = 7$)

2. Find a common denominator

   The least common denominator (LCD) of the fractions $\frac{17}{2}$ and $\frac{7}{3}$ is $6$.

   Convert the fractions:

   $$ \frac{17}{2} = \frac{17 \times 3}{2 \times 3} = \frac{51}{6} $$

   $$ \frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6} $$

3. Subtract the fractions

   Now subtract the two fractions:

   $$ \frac{51}{6} - \frac{14}{6} = \frac{51 - 14}{6} = \frac{37}{6} $$

4. Convert the result back to a mixed number

   To convert $\frac{37}{6}$ to a mixed number, divide $37$ by $6$:

   $$ 37 \div 6 = 6 \text{ remainder } 1 $$

   So,

   $$ \frac{37}{6} = 6 \frac{1}{6} $$