Question

A delivery truck drives three routes in one day. The first route is 12 2/3
miles long, the second route is 10 3/4
miles long, and the third route is 15 1/2
miles long.

What is the total number of miles the delivery truck drives?

Enter your answer as a mixed number in simplest form by filling in the boxes.

Answer 1

To find the total distance the delivery truck drives, we need to add the lengths of the three routes together:

1. First Route: \( 12 \frac{2}{3} \) miles
2. Second Route: \( 10 \frac{3}{4} \) miles
3. Third Route: \( 15 \frac{1}{2} \) miles

We will first convert each mixed number into an improper fraction:

1. **Convert \( 12 \frac{2}{3} \)**:
\[
12 \frac{2}{3} = \left(12 \times 3 + 2\right) / 3 = \frac{36 + 2}{3} = \frac{38}{3}
\]

2. **Convert \( 10 \frac{3}{4} \)**:
\[
10 \frac{3}{4} = \left(10 \times 4 + 3\right) / 4 = \frac{40 + 3}{4} = \frac{43}{4}
\]

3. **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:

1. **Convert \( \frac{38}{3} \)**:
\[
\frac{38}{3} = \frac{38 \times 4}{3 \times 4} = \frac{152}{12}
\]

2. **Convert \( \frac{43}{4} \)**:
\[
\frac{43}{4} = \frac{43 \times 3}{4 \times 3} = \frac{129}{12}
\]

3. **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:
\[
\boxed{38 \frac{11}{12}}
\]