Solving Fraction Exceedance Problem

This problem requires us to calculate the difference between two fractional expressions. We need to evaluate each expression individually and then find out by how much the first value is greater than the second.

Step 1: Evaluate the First Expression

The first expression given is $\frac{12}{\left(\frac{13}{14}\right)}$.

To simplify this, we can multiply the numerator by the reciprocal of the denominator:

$ \frac{12}{\left(\frac{13}{14}\right)} = 12 \times \frac{14}{13} $

Performing the multiplication:

$ \frac{12 \times 14}{13} = \frac{168}{13} $

Step 2: Evaluate the Second Expression

The second expression is $\frac{\left(\frac{12}{13}\right)}{14}$.

This can be rewritten as:

$ \frac{12}{13} \div 14 = \frac{12}{13} \times \frac{1}{14} $

Multiplying these fractions gives:

$ \frac{12 \times 1}{13 \times 14} = \frac{12}{182} $

Simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2:

$ \frac{12 \div 2}{182 \div 2} = \frac{6}{91} $

Step 3: Calculate the Difference (Exceedance)

We need to find the amount by which $\frac{168}{13}$ exceeds $\frac{6}{91}$. This is done by subtraction:

$ \frac{168}{13} - \frac{6}{91} $

To subtract these fractions, find a common denominator. The least common multiple (LCM) of 13 and 91 is 91, because $13 \times 7 = 91$.

Convert $\frac{168}{13}$ into an equivalent fraction with the denominator 91:

$ \frac{168}{13} = \frac{168 \times 7}{13 \times 7} = \frac{1176}{91} $

Now, perform the subtraction:

$ \frac{1176}{91} - \frac{6}{91} = \frac{1176 - 6}{91} = \frac{1170}{91} $

Step 4: Convert to a Mixed Number

The result $\frac{1170}{91}$ is an improper fraction. Convert it into a mixed number.

Divide 1170 by 91:

$ 1170 \div 91 = 12 \text{ with a remainder of } 78 $

This gives the mixed number $12\frac{78}{91}$.

Step 5: Simplify the Fractional Part

The fractional part of the mixed number, $\frac{78}{91}$, can be simplified. Find the greatest common divisor (GCD) of 78 and 91, which is 13.

Divide both the numerator and the denominator by 13:

$ \frac{78 \div 13}{91 \div 13} = \frac{6}{7} $

So, the final simplified answer is $12\frac{6}{7}$.