Dividing Mixed Numbers and Fractions: Dividing a mixed fraction by a fraction

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

• Step 1: Convert the mixed number to an improper fraction.
• Step 2: Apply the division rule for fractions.
• Step 3: Simplify the resulting fraction if necessary.

Now, let's work through each step:
Step 1: Convert the mixed number $2\frac{1}{2}$ to an improper fraction.
A mixed number like $2\frac{1}{2}$ can be converted to an improper fraction by multiplying the whole number part (2) by the denominator of the fractional part (2) and then adding the numerator of the fractional part (1).
Thus, $2\frac{1}{2}$ becomes $\frac{2 \times 2 + 1}{2} = \frac{5}{2}$.

Step 2: Divide $\frac{5}{2}$ by $\frac{1}{2}$.
Dividing by a fraction is equivalent to multiplying by its reciprocal.
The reciprocal of $\frac{1}{2}$ is $2$. So, we multiply:
$\frac{5}{2} \div \frac{1}{2} = \frac{5}{2} \times 2 = \frac{5}{2} \times \frac{2}{1} = \frac{5 \times 2}{2 \times 1} = \frac{10}{2}$.

Step 3: Simplify $\frac{10}{2}$.
Here, $\frac{10}{2} = 5$.

Therefore, the solution to the problem is $5$, which corresponds to choice 3.

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

• Step 1: Convert the mixed number $1\frac{1}{4}$ to an improper fraction.
• Step 2: Calculate the reciprocal of $\frac{1}{3}$.
• Step 3: Multiply the improper fraction by the reciprocal.
• Step 4: Simplify the resulting fraction, if necessary, to achieve the final result.

Now, let’s work through each step:

Step 1: Convert $1\frac{1}{4}$ to an improper fraction.

To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction part, and then add the numerator. Keep the same denominator:

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

Step 2: Calculate the reciprocal of $\frac{1}{3}$.

The reciprocal of $\frac{1}{3}$ is $3$ or $\frac{3}{1}$.

Step 3: Multiply $\frac{5}{4}$ by $\frac{3}{1}$:

$\frac{5}{4} \times \frac{3}{1} = \frac{5 \times 3}{4 \times 1} = \frac{15}{4}$.

Step 4: Simplify and convert $\frac{15}{4}$ back to a mixed number:

Divide the numerator by the denominator: $15 \div 4 = 3$ with a remainder of $3$.

The mixed number is $3\frac{3}{4}$.

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

To solve the given problem, we need to divide the mixed number by the fraction $\frac{2}{3}$. Here's a detailed step-by-step solution:

Step 1: Convert Mixed Number to Improper Fraction.
The mixed number $1\frac{1}{2}$ can be converted into an improper fraction. Multiply the whole number 1 by the denominator 2, and add the numerator 1. This gives us:

$1 \times 2 + 1 = 2 + 1 = 3$ Thus, the improper fraction is $\frac{3}{2}$.

Step 2: Divide by the Fraction $\frac{2}{3}$.
Dividing by a fraction is equivalent to multiplying by its reciprocal. So, $\frac{3}{2} \div \frac{2}{3}$ becomes $\frac{3}{2} \times \frac{3}{2}$.

Step 3: Perform the Multiplication.
Multiply the numerators together and the denominators together:

$\frac{3}{2} \times \frac{3}{2} = \frac{3 \times 3}{2 \times 2} = \frac{9}{4}$

Step 4: Convert to a Mixed Number if Needed.
Convert $\frac{9}{4}$ back to a mixed number. Divide 9 by 4, which gives 2 with a remainder of 1:

$9 \div 4 = 2 \quad \text{(whole number)}, \quad \text{remainder } 1$ Thus, $\frac{9}{4} = 2\frac{1}{4}$.

Therefore, the solution to the problem is $\boxed{2\frac{1}{4}}$.

We are required to divide the mixed number $1\frac{1}{2}$ by the fraction $\frac{2}{3}$.

Step 1: Convert the mixed number $1\frac{1}{2}$ into an improper fraction.
1 whole is equal to $\frac{2}{2}$, so $1\frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2}$.
Therefore, $1\frac{1}{2} = \frac{3}{2}$.

Step 2: Find the reciprocal of $\frac{2}{3}$.
The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

Step 3: Multiply the improper fraction by the reciprocal.
$\frac{3}{2} \div \frac{2}{3} = \frac{3}{2} \times \frac{3}{2} = \frac{9}{4}$.

Step 4: Convert the improper fraction back to a mixed number.
$\frac{9}{4} = 2 \frac{1}{4}$ because 9 divided by 4 is 2 with a remainder of 1. This gives $2\frac{1}{4}$.

Therefore, the solution to the problem is $\boxed{2\frac{1}{4}}$.

To solve the problem of dividing the mixed number $1\frac{3}{5}$ by the fraction $\frac{1}{2}$, follow these steps:

• Step 1: Convert the mixed number to an improper fraction.
• Step 2: Take the reciprocal of the divisor fraction.
• Step 3: Multiply the improper fraction by the reciprocal.
• Step 4: Simplify the resulting fraction if necessary.

Now, let's execute each step:

Step 1: Convert $1\frac{3}{5}$ to an improper fraction. To do this, multiply the whole number 1 by the denominator 5 and add the numerator 3:

$1 \times 5 + 3 = 5 + 3 = 8$

This gives us $\frac{8}{5}$.

Step 2: Take the reciprocal of $\frac{1}{2}$, which is $\frac{2}{1}$.

Step 3: Multiply $\frac{8}{5}$ by $\frac{2}{1}$:

$\frac{8}{5} \times \frac{2}{1} = \frac{8 \times 2}{5 \times 1} = \frac{16}{5}$

Step 4: Simplify $\frac{16}{5}$ by converting it back to a mixed number:

Divide 16 by 5, which results in 3 with a remainder of 1, giving us $3\frac{1}{5}$.

Therefore, the result of $1\frac{3}{5} \div \frac{1}{2}$ is $3\frac{1}{5}$.

To solve the problem of dividing the mixed number $2\frac{1}{6}$ by the fraction $\frac{2}{3}$, follow these steps:

• Step 1: Convert the mixed number $2\frac{1}{6}$ into an improper fraction.
  This is done by multiplying the whole number 2 by the denominator 6, then adding the numerator 1. Therefore, $2 \times 6 + 1 = 13$. The improper fraction is $\frac{13}{6}$.
• Step 2: Convert the division into multiplication by using the reciprocal of $\frac{2}{3}$.
  The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.
• Step 3: Multiply the improper fraction $\frac{13}{6}$ by the reciprocal $\frac{3}{2}$.
  Performing the multiplication gives $\frac{13}{6} \times \frac{3}{2} = \frac{13 \times 3}{6 \times 2} = \frac{39}{12}$.
• Step 4: Simplify the resulting fraction $\frac{39}{12}$.
  This fraction simplifies by dividing both the numerator and denominator by their greatest common divisor, which is 3. Thus $\frac{39 \div 3}{12 \div 3} = \frac{13}{4}$.
• Step 5: Convert the improper fraction $\frac{13}{4}$ back to a mixed number.
  By dividing 13 by 4, we get 3 with a remainder of 1. This results in the mixed number $3\frac{1}{4}$.

Therefore, the solution to the problem is $3\frac{1}{4}$.

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

• Step 1: Convert the mixed number $2\frac{1}{3}$ to an improper fraction.
• Step 2: Take the reciprocal of $\frac{1}{2}$.
• Step 3: Multiply the improper fraction by the reciprocal.
• Step 4: Simplify the resulting fraction if necessary, and convert back to a mixed number.

Now, let's work through each step:
Step 1: Convert the mixed number $2\frac{1}{3}$ to an improper fraction.
$2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$.
Step 2: The reciprocal of $\frac{1}{2}$ is $2$.
Step 3: Multiply $\frac{7}{3}$ by $2$:
$\frac{7}{3} \times 2 = \frac{7 \times 2}{3} = \frac{14}{3}$.
Step 4: Convert the improper fraction $\frac{14}{3}$ to a mixed number.
Divide 14 by 3, which is 4 with a remainder of 2, so we have $4\frac{2}{3}$.

Therefore, the solution to the problem is $4\frac{2}{3}$.

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

• Step 1: Convert the mixed number to an improper fraction.
• Step 2: Find the reciprocal of the divisor fraction.
• Step 3: Multiply the two fractions.
• Step 4: Convert the result back to a mixed number.

Now, let's work through each step:

Step 1: Convert the mixed number.
The mixed number $2\frac{1}{2}$ is converted to an improper fraction as follows:
$2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}$

Step 2: Find the reciprocal of the divisor.
The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.

Step 3: Multiply the two fractions.
Now we multiply $\frac{5}{2}$ by $\frac{3}{2}$:
$\frac{5}{2} \times \frac{3}{2} = \frac{5 \times 3}{2 \times 2} = \frac{15}{4}$

Step 4: Convert the result back to a mixed number.
Convert $\frac{15}{4}$ to a mixed number:
15 divided by 4 gives a quotient of 3 and a remainder of 3, so:
$\frac{15}{4} = 3\frac{3}{4}$

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

To solve the division problem $2\frac{2}{7}:\frac{3}{4}$, follow these steps:

• Convert the mixed number to an improper fraction:

$2\frac{2}{7}$ can be rewritten as $\frac{2 \times 7 + 2}{7} = \frac{14 + 2}{7} = \frac{16}{7}$.

• Rewrite the division as multiplication by the reciprocal:

Instead of dividing by $\frac{3}{4}$, multiply by its reciprocal: $\frac{16}{7} \times \frac{4}{3}$.

• Perform the multiplication:

Calculate: $\frac{16 \times 4}{7 \times 3} = \frac{64}{21}$.

• Convert the improper fraction to a mixed number.

Divide 64 by 21: the quotient is 3 and the remainder is 1, giving us the mixed number:

• $64 \div 21 = 3$ remainder $1$.
• The mixed number is $3\frac{1}{21}$.

Therefore, the result of $2\frac{2}{7} \div \frac{3}{4}$ is $\boxed{3\frac{1}{21}}$.

To solve the division of the mixed fraction $1\frac{2}{7}$ by the fraction $\frac{3}{4}$, follow these steps:

• Step 1: Convert the mixed fraction to an improper fraction: $1\frac{2}{7} = \frac{9}{7}$.
• Step 2: Determine the reciprocal of the divisor fraction: $\text{Reciprocal of } \frac{3}{4} = \frac{4}{3}$.
• Step 3: Multiply the improper fraction by the reciprocal of the divisor: $\frac{9}{7} \times \frac{4}{3} = \frac{36}{21}$.
• Step 4: Simplify the resulting fraction: The greatest common divisor of 36 and 21 is 3. Thus, $\frac{36}{21} = \frac{12}{7}$.
• Step 5: Convert back to a mixed number: The improper fraction $\frac{12}{7} = 1\frac{5}{7}$.

Therefore, the solution to the problem is $1\frac{5}{7}$.

To solve the division $2\frac{2}{7} \\colon \frac{3}{4}$, follow these steps:

• Step 1: Convert the mixed number $2\frac{2}{7}$ to an improper fraction.

The whole number is 2 and the fraction is $\frac{2}{7}$. Convert this by using the formula: $2\frac{2}{7} = \frac{2 \times 7 + 2}{7} = \frac{14 + 2}{7} = \frac{16}{7}$

• Step 2: Invert the divisor $\frac{3}{4}$ to its reciprocal and multiply.

The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. Thus, the division changes to multiplication: $\frac{16}{7} \times \frac{4}{3}$

• Step 3: Multiply the fractions.

Multiply the numerators and the denominators: $\frac{16 \times 4}{7 \times 3} = \frac{64}{21}$

• Step 4: Convert the improper fraction back to a mixed number.

Divide the numerator by the denominator: $\text{21 goes into 64 three times (since } 21 \times 3 = 63\text{). The remainder is } 64 - 63 = 1.$ Thus, the improper fraction can be written as the mixed number: $3\frac{1}{21}$

Therefore, the solution to the problem is $3\frac{1}{21}$.

To solve the problem $2\frac{1}{7} : \frac{5}{7}$, we proceed as follows:

• Step 1: Convert the mixed number $2\frac{1}{7}$ to an improper fraction.

To do this, multiply the whole number part by the denominator and add the numerator: $(2 \times 7) + 1 = 14 + 1 = 15$. Therefore, $2\frac{1}{7} = \frac{15}{7}$.

• Step 2: Find the reciprocal of the fraction $\frac{5}{7}$.

The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$.

• Step 3: Multiply $\frac{15}{7}$ by $\frac{7}{5}$.

The multiplication is: $\frac{15}{7} \times \frac{7}{5} = \frac{15 \times 7}{7 \times 5} = \frac{105}{35}$.

• Step 4: Simplify $\frac{105}{35}$.

To simplify, divide both the numerator and the denominator by their greatest common factor, which is 35: $\frac{105 \div 35}{35 \div 35} = \frac{3}{1} = 3$.

Thus, the solution to the problem $2\frac{1}{7} : \frac{5}{7}$ is $3$.

To solve the problem of dividing the mixed number $2\frac{4}{7}$ by the fraction $\frac{3}{4}$, follow these steps:

• Step 1: Convert the mixed number $2\frac{4}{7}$ into an improper fraction.
• Step 2: Calculate the reciprocal of the divisor, $\frac{3}{4}$.
• Step 3: Multiply the improper fraction by the reciprocal.
• Step 4: Simplify the resulting fraction and convert it to a mixed number if applicable.

Now, let us solve the problem step by step:

Step 1: Convert the mixed number $2\frac{4}{7}$ into an improper fraction.

$2\frac{4}{7} = \frac{2 \times 7 + 4}{7} = \frac{14 + 4}{7} = \frac{18}{7}$

Step 2: Calculate the reciprocal of the divisor $\frac{3}{4}$.

The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$.

Step 3: Multiply the improper fraction $\frac{18}{7}$ by the reciprocal $\frac{4}{3}$.

$\frac{18}{7} \times \frac{4}{3} = \frac{18 \times 4}{7 \times 3} = \frac{72}{21}$

Step 4: Simplify $\frac{72}{21}$ and convert it back to a mixed number.

We find the greatest common divisor of 72 and 21, which is 3, and divide both the numerator and the denominator by 3:

$\frac{72}{21} = \frac{72 \div 3}{21 \div 3} = \frac{24}{7}$

Finally, convert $\frac{24}{7}$ back to a mixed number.

$\frac{24}{7} = 3\frac{24 - 21}{7} = 3\frac{3}{7}$

However, the final answer should match one of the given choices, specifically in a typographical form equivalent to one of them:

The equivalent expression of $3\frac{9}{21}$ (from choice 4) when $\frac{3}{7}$ is simplified and matches the correct result:

$\frac{9}{21}$ is indeed $\frac{3}{7}$.

Therefore, the correct answer is $\boxed{3\frac{9}{21}}$, as demonstrated by matching all criteria set by the problem statement.

To solve the problem of dividing the mixed number $2\frac{5}{7}$ by the fraction $\frac{4}{5}$, we will follow the outlined steps:

• Step 1: Convert the mixed number to an improper fraction.
• For $2\frac{5}{7}$: Multiply the whole number $2$ by the denominator $7$, giving $2 \times 7 = 14$.
• Add the numerator $5$ to this result, giving $14 + 5 = 19$. Thus, $2\frac{5}{7} = \frac{19}{7}$.
• Step 2: Divide the improper fraction by $\frac{4}{5}$ by multiplying by the reciprocal of $\frac{4}{5}$, which is $\frac{5}{4}$.
• Perform the multiplication: $\frac{19}{7} \times \frac{5}{4} = \frac{19 \times 5}{7 \times 4} = \frac{95}{28}$.
• Step 3: Convert the result back to a mixed number if needed.
• Divide $95$ by $28$ to find the whole part; $28$ goes into $95$ three times, with a remainder.
• The remainder is $95 - (28 \times 3) = 95 - 84 = 11$.
• Therefore, $\frac{95}{28}$ is written as the mixed number $3\frac{11}{28}$.

Therefore, the solution is $3\frac{11}{28}$.

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

• Step 1: Convert the mixed number to an improper fraction.
• Step 2: Multiply by the reciprocal of the fraction.
• Step 3: Simplify and express the result as a decimal if necessary.

Now, let's work through each step:
Step 1: Convert $1 \frac{5}{13}$ to an improper fraction. To do this, multiply the whole number 1 by the denominator 13 and add the numerator 5:
$1 \frac{5}{13} = \frac{1 \cdot 13 + 5}{13} = \frac{13 + 5}{13} = \frac{18}{13}$
Step 2: To divide by $\frac{5}{13}$, multiply by the reciprocal of the fraction $\frac{5}{13}$, which is $\frac{13}{5}$:
$\frac{18}{13} \times \frac{13}{5} = \frac{18 \cdot 13}{13 \cdot 5} = \frac{18}{5}$
Step 3: Simplify the fraction $\frac{18}{5}$ by dividing 18 by 5:
$\frac{18}{5} = 3.6$

Therefore, the solution to the problem is $3.6$.