Special Cases (0 and 1, Inverse, Fraction Line): In combination with other operations

To solve the expression $\frac{\frac{10}{7}}{2}+\frac{2}{\frac{7}{8}}$, we need to perform operations in the correct order as per the rules of the order of operations (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Step 1: Simplify the complex fraction $\frac{\frac{10}{7}}{2}$
A complex fraction can be simplified by multiplying the numerator by the reciprocal of the denominator. In this case, the numerator is $\frac{10}{7}$ and the denominator is 2 (which means $\frac{2}{1}$).

$\frac{\frac{10}{7}}{2} = \frac{10}{7} \times \frac{1}{2} = \frac{10 \cdot 1}{7 \cdot 2} = \frac{10}{14}$

Simplify $\frac{10}{14}$ by dividing both the numerator and the denominator by their greatest common divisor (2):

$\frac{10}{14} = \frac{5}{7}$

Step 2: Simplify the complex fraction $\frac{2}{\frac{7}{8}}$
Again, multiply the numerator by the reciprocal of the denominator:
The reciprocal of $\frac{7}{8}$ is $\frac{8}{7}$.

$\frac{2}{\frac{7}{8}} = 2 \times \frac{8}{7} = \frac{2 \cdot 8}{7} = \frac{16}{7}$

Step 3: Add the simplified fractions $\frac{5}{7} + \frac{16}{7}$
Since the fractions have like denominators, we can add the numerators directly:

$\frac{5}{7} + \frac{16}{7} = \frac{5 + 16}{7} = \frac{21}{7}$

Simplify $\frac{21}{7}$ by dividing the numerator by the denominator:

$\frac{21}{7} = 3$

Thus, the solution to the expression is $3$.

To simplify the fraction exercise, we will multiply $\frac{4}{7}$ by $\frac{1}{2}$.

We will then rearrange the exercise accordingly and following the order of operations rules, we will first solve the multiplication exercise:

$5+\frac{4}{7}\times\frac{1}{2}=$

Note that in the multiplication exercise, we can reduce 4 in the numerator and 2 in the denominator by 2:

$5+\frac{2}{7}\times\frac{1}{1}=5+\frac{2}{7}+1$

Finally we will combine the whole numbers to get:

$5+1+\frac{2}{7}=6\frac{2}{7}$

When we are presented with a fraction over a fraction (in this case one-third over one-sixth) We can convert it into a more manageable form.

$1/3 : 1/6$

It's important to remember that a fraction is actually another sign of division, hence the given exercise is in fact equivalent to one-third divided by one-sixth.
When dealing with division of fractions, the easiest method for solving them is by performing "multiplication by the reciprocal" as shown below:

$1/3\times6/1$

Multiply the numerator by the numerator and the denominator by the denominator to obtain the following result:

$\frac{6}{3}$

Which when reduced equals

$\frac{2}{1}$

Now let's return to the original exercise. In order to solve it we need to take the mixed fraction and convert it to an improper fraction.
We can achieve this by simply moving the whole numbers back to the numerator.

To do this we'll multiply the whole number by the denominator and then proceed to add it to the numerator

$3\times2=6$

$6+1=7$

Therefore the resulting fraction is:

$\frac{7}{2}$

We want to proceed to perform the subtraction exercise.
When both fractions have the same denominator we subtract them.
Therefore in order to achieve this we'll expand the fraction $\frac{2}{1}$ to a denominator of 2, and obtain the following:

$\frac{4}{2}$

We can now proceed to perform subtraction -

$\frac{7}{2}-\frac{4}{2}=\frac{3}{2}$

Convert this back to a mixed fraction in order to obtain the following result:

To solve the expression $\frac{\frac{3}{5}}{\frac{9}{10}}+\frac{\frac{7}{9}}{\frac{1}{3}}$, we need to apply the division of fractions and simplify the resulting expressions.

First, consider the expression $\frac{\frac{3}{5}}{\frac{9}{10}}$:

• When dividing by a fraction, multiply by its reciprocal. The reciprocal of $\frac{9}{10}$ is $\frac{10}{9}$.
• Therefore, $\frac{\frac{3}{5}}{\frac{9}{10}} = \frac{3}{5} \times \frac{10}{9}$.
• Multiplying the numerators and the denominators, we get $\frac{3 \times 10}{5 \times 9} = \frac{30}{45}$.
• Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 15: $\frac{30 \div 15}{45 \div 15} = \frac{2}{3}$.

Next, consider the expression $\frac{\frac{7}{9}}{\frac{1}{3}}$:

• The reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$.
• Therefore, $\frac{\frac{7}{9}}{\frac{1}{3}} = \frac{7}{9} \times \frac{3}{1}$.
• Multiplying the numerators and the denominators, we get $\frac{7 \times 3}{9 \times 1} = \frac{21}{9}$.
• Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3: $\frac{21 \div 3}{9 \div 3} = \frac{7}{3}$.

Now add the simplified fractions: $\frac{2}{3} + \frac{7}{3}$.

• The fractions have a common denominator, 3, so we can simply add the numerators: $\frac{2 + 7}{3} = \frac{9}{3}$.
• Simplify $\frac{9}{3}$ by dividing both the numerator and the denominator by 3: $\frac{9 \div 3}{3 \div 3} = 3$.

Therefore, the final solution to the expression is $3$.