To find the total distance the delivery truck drives, we need to add the lengths of the three routes together:
- First Route: \( 12 \frac{2}{3} \) miles
- Second Route: \( 10 \frac{3}{4} \) miles
- Third Route: \( 15 \frac{1}{2} \) miles
We will first convert each mixed number into an improper fraction:
-
Convert \( 12 \frac{2}{3} \): \[ 12 \frac{2}{3} = \left(12 \times 3 + 2\right) / 3 = \frac{36 + 2}{3} = \frac{38}{3} \]
-
Convert \( 10 \frac{3}{4} \): \[ 10 \frac{3}{4} = \left(10 \times 4 + 3\right) / 4 = \frac{40 + 3}{4} = \frac{43}{4} \]
-
Convert \( 15 \frac{1}{2} \): \[ 15 \frac{1}{2} = \left(15 \times 2 + 1\right) / 2 = \frac{30 + 1}{2} = \frac{31}{2} \]
Next, we need to add these three improper fractions: \[ \frac{38}{3} + \frac{43}{4} + \frac{31}{2} \]
To perform this addition, we need a common denominator. The least common multiple of \(3\), \(4\), and \(2\) is \(12\).
Now we will convert all the fractions:
-
Convert \( \frac{38}{3} \): \[ \frac{38}{3} = \frac{38 \times 4}{3 \times 4} = \frac{152}{12} \]
-
Convert \( \frac{43}{4} \): \[ \frac{43}{4} = \frac{43 \times 3}{4 \times 3} = \frac{129}{12} \]
-
Convert \( \frac{31}{2} \): \[ \frac{31}{2} = \frac{31 \times 6}{2 \times 6} = \frac{186}{12} \]
Now we can add these fractions together: \[ \frac{152}{12} + \frac{129}{12} + \frac{186}{12} = \frac{152 + 129 + 186}{12} = \frac{467}{12} \]
Now we convert the improper fraction \( \frac{467}{12} \) back to a mixed number. We do this by dividing \(467\) by \(12\): \[ 467 \div 12 = 38 \quad \text{(with a remainder of } 11\text{)} \]
So, \( \frac{467}{12} \) can be expressed as: \[ 38 \frac{11}{12} \]
Thus, the total number of miles the delivery truck drives is: \[ \boxed{38 \frac{11}{12}} \]
1 answer