Question

Using the paths shown, how long is the shortest route from Hampton to Lexington?

Answer 1

In this case, we have to add mixed numbers to find the shortest path from the shown alternatives. We need to work with mixed numbers, fractions, and even decimals.

To answer this question, we know that:

1. The path from Hampton to Middletown is 8 + 1/4 mi.

2. Now, we have two alternatives to go from Middletown to Campbell:

First, we have a path that is equivalent to 6 + 1/8 mi (direct). On the other path, we need to go from Middletown to Danville, and then to Campbell. The measure of the latter path is then:

$3\frac{3}{8}+3\frac{3}{8}=(3+\frac{3}{8})+(3+\frac{3}{8})=(\frac{8\cdot3+3\cdot1}{8})+(\frac{8\cdot3+3\cdot1}{8})$

Thus, we have:

$\frac{(24+3)}{8}+\frac{(24+3)}{8}=\frac{27+27}{8}=\frac{2\cdot(27)}{8}=\frac{2}{8}\cdot27=\frac{1}{4}\cdot27=\frac{27}{4}$

Which is equivalent to:

$\frac{27}{4}=\frac{24}{4}+\frac{3}{4}=6+\frac{3}{4}=6\frac{3}{4}=6.75mi$

If we compare the measures of the two paths, we have that:

a. The path from Middletown to Campbell (directly) is equal to:

$6\frac{1}{8}=6+\frac{1}{8}=6.125mi$

b. The path from Middletown to Danville, and then from Danville to Campbell is equal to 6.75 miles (it is longer).

Therefore, the shortest path, in this part of the "journey", is equal to 6.125 miles or 6 + 1/8 miles.

Now, the complete measure of the path from Hamptom to Lexington is the sum of:

1. The measure of the path from Hamptom to Middletown (8 + 1/4 miles) plus

2. The measure of the path from Middletown to Campbell (6 + 1/8 miles) (directly) plus

3. The measure of the path from Campbell to Lexington (5 + 3/4 miles).

And this is, numerically, as follows:

$(8+\frac{1}{4})+(6+\frac{1}{8})+(5+\frac{3}{4})=8+6+5+\frac{1}{4}+\frac{3}{4}+\frac{1}{8}$$19+\frac{4}{4}+\frac{1}{8}=19+1+\frac{1}{8}=20+\frac{1}{8}=20\frac{1}{8}mi$

We can express the latter result as a fraction as follows:

$20+\frac{1}{8}=\frac{20\cdot8+1\cdot1}{8}=\frac{160+1}{8}=\frac{161}{8}mi$

Therefore, the shortest route from Hamptom to Lexington is:

1. As a fraction:

$\frac{161}{8}mi$

2. Or as a mixed number:

$20\frac{1}{8}mi$

[Notice that we sum two fractions with the same denominator above:

$\frac{1}{4}+\frac{3}{4}=\frac{3+1}{4}=\frac{4}{4}=1$

.]