Division of Fractions: In combination with other operations

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:

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

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 solve the problem, $\frac{1}{2}:\frac{1}{2}-\frac{1}{4}$, follow these steps:

• Step 1: First, interpret the division $\frac{1}{2} : \frac{1}{2}$ as multiplication by the reciprocal. This becomes $\frac{1}{2} \times \frac{2}{1}$.
• Step 2: Perform the multiplication: $\frac{1}{2} \times \frac{2}{1} = \frac{1 \times 2}{2 \times 1} = \frac{2}{2} = 1.$
• Step 3: Next, subtract $\frac{1}{4}$ from 1: $1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}.$

Therefore, the solution to the problem is $\frac{3}{4}$.

To solve the problem $\frac{1}{4} : \frac{1}{2} + \frac{1}{4}$, follow these steps:

Step 1: Perform the division $\frac{1}{4} : \frac{1}{2}$.

• When dividing by a fraction, multiply by the reciprocal. Hence, $\frac{1}{4} : \frac{1}{2}$ becomes $\frac{1}{4} \times \frac{2}{1}$.
• Multiply the numerators: $1 \times 2 = 2$.
• Multiply the denominators: $4 \times 1 = 4$.
• So, $\frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2}$ after simplification.

Step 2: Now add the result from Step 1 to $\frac{1}{4}$.

• We need to add $\frac{1}{2} + \frac{1}{4}$.
• Convert $\frac{1}{2}$ to have a common denominator with $\frac{1}{4}$. $\frac{1}{2} = \frac{2}{4}$.
• Add the fractions: $\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$.

Therefore, the solution to the problem is $\frac{3}{4}$.

To solve the problem, we'll follow the outlined steps:

• Step 1: Calculate the division $\frac{3}{5} : \frac{5}{6}$.
• Step 2: Add $\frac{1}{5}$ to the result from Step 1.
• Step 3: Simplify the resulting fraction.

Now, let's work through each step:

Step 1: To divide $\frac{3}{5}$ by $\frac{5}{6}$, we multiply by the reciprocal of $\frac{5}{6}$. This gives us:

$\frac{3}{5} \times \frac{6}{5} = \frac{3 \times 6}{5 \times 5} = \frac{18}{25}$

Step 2: Now, add $\frac{1}{5}$ to $\frac{18}{25}$. First, we convert $\frac{1}{5}$ to the same denominator as $\frac{18}{25}$:

$\frac{1}{5} = \frac{5}{25}$

Step 3: Add $\frac{18}{25}$ and $\frac{5}{25}$:

$\frac{18}{25} + \frac{5}{25} = \frac{18 + 5}{25} = \frac{23}{25}$

Thus, the solution to the problem is $\frac{23}{25}$.

To solve this fraction problem, follow these steps:

• Step 1: Evaluate the Division:
  The expression starts with $\frac{2}{3} \div \frac{3}{4}$. To divide fractions, multiply by the reciprocal:
  $\frac{2}{3} \times \frac{4}{3} = \frac{2 \times 4}{3 \times 3} = \frac{8}{9}$
• Step 2: Add Fractions:
  Now add $\frac{8}{9}$ to $\frac{1}{9}$. Since the denominators are the same, add the numerators directly:
  $\frac{8}{9} + \frac{1}{9} = \frac{8 + 1}{9} = \frac{9}{9} = 1$

Thus, the solution to the expression $\frac{2}{3}:\frac{3}{4} + \frac{1}{9}$ is $1$.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the division $\frac{3}{4} : \frac{5}{4}$.
• Step 2: Use the formula $\frac{a}{b} : \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.
• Step 3: Simplify the division result.
• Step 4: Add $\frac{1}{2}$ to the result of the division.
• Step 5: Simplify the addition result to get the final answer.

Now, let's work through each step:

Step 1: We need to calculate $\frac{3}{4} : \frac{5}{4}$. Using the division of fractions formula, this becomes:

$\frac{3}{4} \times \frac{4}{5} = \frac{3 \times 4}{4 \times 5} = \frac{12}{20}$.

Step 2: Simplify $\frac{12}{20}$. Divide the numerator and the denominator by their greatest common divisor, which is 4:

$\frac{12 \div 4}{20 \div 4} = \frac{3}{5}$.

Step 3: Add $\frac{1}{2}$ to the result $\frac{3}{5}$:

The common denominator for addition is 10. Therefore:

$\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}$ and $\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$.

Add these two fractions:

$\frac{6}{10} + \frac{5}{10} = \frac{6 + 5}{10} = \frac{11}{10}$.

Therefore, the solution to the problem is $\frac{11}{10}$.

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 $3$.

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: