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

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

We will arrange 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:

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

We will combine the whole numbers and get:

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

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 $3$.

When we have a fraction over a fraction, in this case one-third over one-sixth, we can convert it to a form that might be more familiar to us:

$1/3 : 1/6$

It's important to remember that a fraction is actually another sign of division, so the exercise we have is one-third divided by one-sixth.
When dealing with division of fractions, the easiest method for solving is performing "multiplication by the reciprocal", meaning:

$1/3\times6/1$

Multiply numerator by numerator and denominator by denominator and get:

$\frac{6}{3}$

Which when reduced equals

$\frac{2}{1}$

Now let's return to the original exercise, to solve it we need to take the mixed fraction and convert it to an improper fraction,
meaning move the whole numbers back to the numerator.

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

$3\times2=6$

$6+1=7$

And therefore the fraction is:

$\frac{7}{2}$

Now we want to do the subtraction exercise, but we see that we have another step on the way.
We subtract fractions when both fractions have the same denominator,
so we'll expand the fraction $\frac{2}{1}$ to a denominator of 2, and we'll get:

$\frac{4}{2}$

And now we can perform subtraction -

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

We'll convert this back to a mixed fraction and we'll see that the result is

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 $3$.