All Operations in Fractions: In combination with other operations

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

Therefore, we'll start by simplifying the expressions in parentheses first:
$(\frac{1}{4}+\frac{7}{4}-\frac{5}{4}-\frac{1}{4})\cdot10:7:5=\\ \frac{1+7-5-1}{4}\cdot10:7:5 =\\ \frac{2}{4}\cdot10:7:5 = \\ \frac{1}{2}\cdot10:7:5$

We calculated the expression inside the parentheses by adding the fractions, which we did by creating one fraction using the common denominator (4) which in this case is the denominator in all fractions, so we only added/subtracted the numerators (according to the fraction sign), then we reduced the resulting fraction,

We'll continue and note that between multiplication and division operations there is no defined precedence for either operation, therefore we'll calculate the result of the expression obtained in the last step step by step from left to right (which is the regular order in arithmetic operations), meaning we'll first perform the multiplication operation, which is the first from the left, and then we'll perform the division operation that comes after it, and so on:

$\frac{1}{2}\cdot10:7:5 =\\ \frac{1\cdot10}{2}:7:5 =\\ \frac{10}{2}:7:5 =\\ 5:7:5 =\\ \frac{5}{7}:5$

In the first step, we performed the multiplication of the fraction by the whole number, remembering that multiplying by a fraction means multiplying by the fraction's numerator, then we simplified the resulting fraction and reduced it (effectively performing the division operation that results from it), in the final step we wrote the division operation as a simple fraction, since this division operation yields a non-whole result,

We'll continue and to perform the final division operation, we'll remember that dividing by a number is the same as multiplying by its reciprocal, and therefore we'll replace the division operation with multiplication by the reciprocal:

$\frac{5}{7}:5 =\\ \frac{5}{7}\cdot\frac{1}{5}$

In this case we preferred to multiply by the reciprocal because the divisor in the expression is a fraction and it's more convenient to perform multiplication between fractions,

We will then perform the multiplication between the fractions we got in the last step, while remembering that multiplication between fractions is performed by multiplying numerator by numerator and denominator by denominator while maintaining the fraction line, then we'll simplify the resulting expression by reducing it:

$\frac{5}{7}\cdot\frac{1}{5} =\\ \frac{5\cdot1}{7\cdot5}=\\ \frac{5}{35}=\\ \frac{1}{7}$

Let's summarize the solution steps, we got that:

$(\frac{1}{4}+\frac{7}{4}-\frac{5}{4}-\frac{1}{4})\cdot10:7:5=\\ \frac{1+7-5-1}{4}\cdot10:7:5 =\\ \frac{1}{2}\cdot10:7:5 =\\ 5:7:5 =\\ \frac{5}{7}\cdot\frac{1}{5} =\\ \frac{1}{7}$

Therefore the correct answer is answer B.

According to the order of operations, we will first solve the expression in parentheses.

Note that since the denominators are not common, we will look for a number that is both divisible by 2 and 3. That is 6.

We will multiply one-third by 2 and one-half by 3, now we will get the expression:

$\frac{1}{4}\times(\frac{2+3}{6})=$

Let's solve the numerator of the fraction:

$\frac{1}{4}\times\frac{5}{6}=$

We will combine the fractions into a multiplication expression:

$\frac{1\times5}{4\times6}=\frac{5}{24}$

According to the order of operations rules, we will first address the expression in parentheses.

The common denominator between the fractions is 4, so we will multiply each numerator by the number needed for its denominator to reach 4.

We will multiply the first fraction's numerator by 2 and the second fraction's numerator by 1:

$(\frac{9}{2}-\frac{3}{4})=\frac{2\times9-1\times3}{4}=\frac{18-3}{4}=\frac{15}{4}$

Now we have the expression:

$\frac{1}{3}\times\frac{15}{4}=$

Note that we can reduce 15 and 3:

$\frac{1}{1}\times\frac{5}{4}=$

Now we multiply numerator by numerator and denominator by denominator:

$\frac{1\times5}{1\times4}=\frac{5}{4}=1\frac{1}{4}$

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 $3$.

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 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:

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