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**Hint:**We first try to use the binary operation of addition to find the total path crossed. We explain the process of changing the mixed fractions to improper fraction for the addition. We use the LCM of the denominators as the multiplication of those numbers as they are co-prime. Then we find the addition value.

**Complete step-by-step solution:**

Jill walked $8\dfrac{1}{8}$ miles to a park and then $7\dfrac{2}{5}$ miles home. We need to find the total path that she walked.

We will use the binary operation of addition to get the total. We first convert the mixed fractions to improper fractions.

Let the mixed fraction be $x\dfrac{c}{b}$. The condition is $c < b$. $x\dfrac{c}{b}$ can be expressed as $x+\dfrac{c}{b}$. Now we express it in the form of improper fraction. Let’s assume the improper fraction is $\dfrac{a}{b}$ where $a>b$.

Then the equational condition will be $\dfrac{a}{b}=x+\dfrac{c}{b}$.

The solution of the equation $x+\dfrac{c}{b}=\dfrac{bx+c}{b}$.

Therefore, $8\dfrac{1}{8}=\dfrac{8\times 8+1}{8}=\dfrac{65}{8}$ and $7\dfrac{2}{5}=\dfrac{7\times 5+2}{5}=\dfrac{37}{5}$.

Now we add them to get $\dfrac{65}{8}+\dfrac{37}{5}$. The LCM of the denominators will be $5\times 8=40$ as the numbers are co-prime.

So, $\dfrac{65}{8}+\dfrac{37}{5}=\dfrac{5\times 65+8\times 37}{40}=\dfrac{621}{40}$.

The total path that she walked is $\dfrac{621}{40}$ miles.

**Note:**We can always complete the addition using the integer and the proper fraction part of the total mixed fractions. We add them separately to find the solution. In case of finding the LCM of the numbers the multiplication of the numbers is the way when they are co-primes. In other cases, we have to first divide the numbers with their GCD. Then we multiply those quotients with the GCD to find the LCM.

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