All Operations in Fractions: Multiplication by a reciprocal

To solve the division of fractions problem $\frac{3}{4}:\frac{1}{6}$, we will use multiplication by the reciprocal. Let us break down the steps:

• Step 1: Identify the fractions. The first fraction is $\frac{3}{4}$ and the second fraction is $\frac{1}{6}$.
• Step 2: Find the reciprocal of the second fraction. The reciprocal of $\frac{1}{6}$ is $\frac{6}{1}$ or just $6$.
• Step 3: Multiply the first fraction by the reciprocal of the second fraction:

$\frac{3}{4} \times \frac{6}{1} = \frac{3 \times 6}{4 \times 1} = \frac{18}{4}$

• Step 4: Simplify the resulting fraction. Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 2:

$\frac{18}{4} = \frac{18 \div 2}{4 \div 2} = \frac{9}{2}$

• Step 5: Convert the improper fraction to a mixed number. Divide 9 by 2, which gives 4 with a remainder of 1:

$\frac{9}{2} = 4\frac{1}{2}$

Thus, the solution to the problem $\frac{3}{4}:\frac{1}{6}$ is $4\frac{1}{2}$.

To solve this problem, let's follow these steps:

• Step 1: Simplify the fraction $\frac{2}{4}$.
• Step 2: Find the reciprocal of $\frac{1}{3}$.
• Step 3: Multiply the simplified fraction by the reciprocal.
• Step 4: Simplify the resulting fraction if needed.

Now, let's work through each step:
Step 1: Simplify $\frac{2}{4}$ to $\frac{1}{2}$ by dividing both the numerator and the denominator by 2.
Step 2: The reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$.
Step 3: Multiply $\frac{1}{2} \times \frac{3}{1}$. This gives us $\frac{1 \times 3}{2 \times 1} = \frac{3}{2}$.
Step 4: $\frac{3}{2}$ is an improper fraction. Convert it to a mixed number, $1\frac{1}{2}$.

Therefore, the solution to the division $\frac{2}{4}:\frac{1}{3}$ is $1\frac{1}{2}$.

To solve the division $\frac{1}{2} \div \frac{3}{5}$, we will follow the multiplication by the reciprocal method. Here are the steps:

• Step 1: Find the reciprocal of the divisor $\frac{3}{5}$, which is $\frac{5}{3}$.
• Step 2: Multiply the dividend $\frac{1}{2}$ by the reciprocal found in Step 1: $\frac{1}{2} \times \frac{5}{3}$.
• Step 3: Multiply the numerators: $1 \times 5 = 5$.
• Step 4: Multiply the denominators: $2 \times 3 = 6$.
• Step 5: Combine the results to form the fraction $\frac{5}{6}$.

The simplified result of $\frac{1}{2} \div \frac{3}{5}$ is $\frac{5}{6}$.

To solve this problem, we need to divide the fraction $\frac{1}{5}$ by the fraction $\frac{1}{10}$. When dividing fractions, the procedure involves multiplying by the reciprocal of the divisor (the second fraction).

Let's start with the solution:

• First, determine the reciprocal of $\frac{1}{10}$. The reciprocal of a fraction is obtained by swapping its numerator and denominator. Thus, the reciprocal of $\frac{1}{10}$ is $\frac{10}{1}$.
• Now multiply $\frac{1}{5}$ by the reciprocal of $\frac{1}{10}$, which is $\frac{10}{1}$:

$\frac{1}{5} \times \frac{10}{1} = \frac{1 \times 10}{5 \times 1} = \frac{10}{5}$

Simplify the fraction $\frac{10}{5}$:

$\frac{10}{5} = 2$

Therefore, the result of $\frac{1}{5} : \frac{1}{10}$ is $2$.

To solve the division of two fractions $\frac{1}{2} : \frac{1}{4}$, follow these steps:

• Step 1: Identify the operation: The problem involves dividing $\frac{1}{2}$ by $\frac{1}{4}$.
• Step 2: Use the reciprocal: In fraction division, multiply by the reciprocal of the second fraction. Thus, $\frac{1}{2} : \frac{1}{4} = \frac{1}{2} \times \frac{4}{1}$.
• Step 3: Perform the multiplication: Now compute the multiplication by multiplying the numerators and the denominators: $\frac{1 \times 4}{2 \times 1} = \frac{4}{2}$.
• Step 4: Simplify the fraction: The fraction $\frac{4}{2}$ simplifies to $2$.

Thus, the solution to the division $\frac{1}{2} : \frac{1}{4}$ is $2$. Therefore, the correct answer choice is $2$ (Choice 1).

To solve the division of fractions problem $\frac{1}{2} \div \frac{2}{3}$, we follow these steps:

• Step 1: Rewrite the division problem as multiplication by the reciprocal: $\frac{1}{2} \div \frac{2}{3}$ becomes $\frac{1}{2} \times \frac{3}{2}$.
• Step 2: Multiply the numerators together: $1 \times 3 = 3$.
• Step 3: Multiply the denominators together: $2 \times 2 = 4$.
• Step 4: Form the new fraction from the resulting numerator and denominator: $\frac{3}{4}$.

Thus, the result of dividing $\frac{1}{2}$ by $\frac{2}{3}$ is $\frac{3}{4}$.

The correct answer is $\frac{3}{4}$.

To solve the problem of dividing $\frac{2}{5}$ by $\frac{1}{4}$, we need to follow these steps:

• Step 1: Identify the reciprocal of $\frac{1}{4}$.
  This reciprocate is $\frac{4}{1}$.
• Step 2: Convert the division of fractions to multiplication with the reciprocal:
  $\frac{2}{5} \div \frac{1}{4} = \frac{2}{5} \times \frac{4}{1}$.
• Step 3: Perform the multiplication:
  $\frac{2 \times 4}{5 \times 1} = \frac{8}{5}$.
• Step 4: Simplify $\frac{8}{5}$ into a mixed number:
  $\frac{8}{5}$ can be written as $1\frac{3}{5}$ because 5 goes into 8 once with 3 remaining.

Therefore, the solution to the problem is $1\frac{3}{5}$.

To find the result of dividing $\frac{2}{4}$ by $\frac{4}{3}$, follow these steps:

• Step 1: Simplify the fraction $\frac{2}{4}$. This becomes $\frac{1}{2}$ because both the numerator and the denominator can be divided by 2.
• Step 2: Find the reciprocal of the fraction $\frac{4}{3}$. The reciprocal is $\frac{3}{4}$ because it exchanges the numerator and the denominator.
• Step 3: Multiply $\frac{1}{2}$ by $\frac{3}{4}$. $\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}$
• Step 4: The result is $\frac{3}{8}$, which is already in its simplest form.

Therefore, the solution to the problem $\frac{2}{4}:\frac{4}{3}$ is $\frac{3}{8}$.

To solve the division of fractions $\frac{2}{3}$ by $\frac{3}{8}$, we follow these steps:

• Step 1: Identify the reciprocal of the second fraction $\frac{3}{8}$. The reciprocal is $\frac{8}{3}$.

• Step 2: Multiply the first fraction $\frac{2}{3}$ by the reciprocal $\frac{8}{3}$:

$\frac{2}{3} \times \frac{8}{3}$

• Step 3: Multiply the numerators together and the denominators together:

$\frac{2 \times 8}{3 \times 3} = \frac{16}{9}$

• Step 4: Simplify the resulting fraction, if possible. In this case, $\frac{16}{9}$ is already in its simplest form.

• Step 5: Convert the improper fraction $\frac{16}{9}$ to a mixed number:

$16 \div 9 = 1$ remainder $7$, so the mixed number is $1\frac{7}{9}$.

Therefore, the solution to the problem is ${1\frac{7}{9}}$.

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

• Step 1: Identify the fractions involved and their reciprocal.
• Step 2: Convert the division into multiplication using the reciprocal.
• Step 3: Simplify the resulting fraction, if needed, converting any improper fraction to a mixed number.

Now, let's work through each step:
Step 1: We are given the fraction $\frac{1}{3}$ and we are dividing by $\frac{1}{4}$. The reciprocal of $\frac{1}{4}$ is $\frac{4}{1}$.
Step 2: Multiply $\frac{1}{3}$ by the reciprocal of $\frac{1}{4}$:

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

Step 3: Simplify $\frac{4}{3}$. Since $\frac{4}{3}$ is an improper fraction, convert it to a mixed number:
Divide 4 by 3, which goes 1 time with a remainder of 1. Therefore, $\frac{4}{3} = 1\frac{1}{3}$.

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

To solve this problem, we'll employ the division of fractions technique:

• Step 1: Find the reciprocal of the divisor. Here, the divisor is $\frac{4}{7}$. Its reciprocal is $\frac{7}{4}$.
• Step 2: Multiply the dividend $\frac{2}{5}$ by the reciprocal of the divisor.
• Step 3: Calculate $\frac{2}{5} \times \frac{7}{4}$.
• Step 4: Simplify the result, if possible.

Now, let's work through these steps:

Step 1: The reciprocal of $\frac{4}{7}$ is $\frac{7}{4}$.

Step 2: Multiply the fractions: $\frac{2}{5} \times \frac{7}{4} = \frac{2 \times 7}{5 \times 4} = \frac{14}{20}$.

Step 3: Simplify $\frac{14}{20}$. The greatest common divisor (GCD) of 14 and 20 is 2, so divide both the numerator and the denominator by 2:

$\frac{14}{20} = \frac{14 \div 2}{20 \div 2} = \frac{7}{10}$.

Therefore, the solution to the problem is $\frac{7}{10}$.

To solve the problem $\frac{3}{7} \div \frac{3}{6}$, we'll perform division of fractions by multiplying with the reciprocal:

• Step 1: Identify the reciprocal of the second fraction, $\frac{3}{6}$. The reciprocal is $\frac{6}{3}$.
• Step 2: Multiply the first fraction, $\frac{3}{7}$, by the reciprocal of the second fraction, $\frac{6}{3}$:

$\frac{3}{7} \times \frac{6}{3} = \frac{3 \times 6}{7 \times 3}$

Step 3: Simplify the fraction $\frac{18}{21}$. Notice that both the numerator and the denominator are divisible by 3.

$\frac{18}{21} = \frac{18 \div 3}{21 \div 3} = \frac{6}{7}$

Therefore, the solution to the problem is $\frac{6}{7}$.

To solve the fraction division problem $\frac{4}{6} : \frac{3}{7}$, follow these steps:

• Step 1: Simplify the first fraction $\frac{4}{6}$.

Simplify by finding the greatest common divisor (GCD) of 4 and 6, which is 2. Divide both the numerator and denominator by 2:

$\frac{4}{6} = \frac{4/2}{6/2} = \frac{2}{3}$

• Step 2: Find the reciprocal of the second fraction $\frac{3}{7}$.

The reciprocal of $\frac{3}{7}$ is $\frac{7}{3}$.

• Step 3: Multiply the simplified first fraction by the reciprocal of the second.

$\frac{2}{3} \times \frac{7}{3} = \frac{2 \times 7}{3 \times 3} = \frac{14}{9}$

• Step 4: Convert the improper fraction $\frac{14}{9}$ to a mixed number.

Divide 14 by 9, which gives a quotient of 1 and a remainder of 5:

$\frac{14}{9} = 1\frac{5}{9}$

Therefore, the solution to the problem is $1\frac{5}{9}$.

To solve the problem of dividing $\frac{4}{9}$ by $\frac{1}{3}$, we follow these steps:

• Step 1: Convert the division operation into multiplication by the reciprocal of the second fraction.
• Step 2: Perform the multiplication of the fractions.
• Step 3: Simplify the resulting fraction, if possible, and convert it into a mixed number.

Let's proceed with the steps:

Step 1: The expression $\frac{4}{9} : \frac{1}{3}$ is equivalent to multiplying $\frac{4}{9}$ by the reciprocal of $\frac{1}{3}$. The reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$.

Step 2: Multiply the fractions:

$\frac{4}{9} \times \frac{3}{1} = \frac{4 \times 3}{9 \times 1} = \frac{12}{9}.$

Step 3: Simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3:

$\frac{12}{9} = \frac{12 \div 3}{9 \div 3} = \frac{4}{3}.$

Step 4: Convert the improper fraction $\frac{4}{3}$ into a mixed number. $\frac{4}{3}$ equals $1$ with a remainder of $1$, so it can be written as the mixed number $1\frac{1}{3}$.

Therefore, the solution to the problem $\frac{4}{9} : \frac{1}{3}$ is $\mathbf{1\frac{1}{3}}$.

To solve this problem, we'll use the reciprocal method for dividing fractions.

Step 1: Identify the reciprocal of the divisor.
The reciprocal of $\frac{7}{5}$ is $\frac{5}{7}$.

Step 2: Multiply the dividend by the reciprocal of the divisor.
Multiply $\frac{2}{3}$ by $\frac{5}{7}$:

$\frac{2}{3} \times \frac{5}{7} = \frac{2 \times 5}{3 \times 7} = \frac{10}{21}$

There is no common factor between 10 and 21 other than 1, meaning $\frac{10}{21}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{10}{21}$.

To solve the problem of dividing the fractions $\frac{1}{2}$ by $\frac{5}{3}$, we proceed as follows:

We can simplify a division of fractions by multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator.

First, we find the reciprocal of $\frac{5}{3}$, which is $\frac{3}{5}$.

Next, we multiply the fractions $\frac{1}{2}$ and $\frac{3}{5}$:

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

This results in

$\frac{3}{10}.$

Thus, the solution to $\frac{1}{2}:\frac{5}{3}$ is $\frac{3}{10}$.

To solve the problem $\frac{2}{5} : \frac{6}{7}$, we can follow these steps:

• Step 1: Identify the operation and fractions involved. We are dividing $\frac{2}{5}$ by $\frac{6}{7}$.
• Step 2: Convert the division of fractions into multiplication by the reciprocal. We rewrite $\frac{2}{5} : \frac{6}{7}$ as $\frac{2}{5} \times \frac{7}{6}$.
• Step 3: Perform the multiplication of fractions. Multiply the numerators together and the denominators together:
• Numerator: $2 \times 7 = 14$
• Denominator: $5 \times 6 = 30$
• Step 4: Form the resulting fraction: $\frac{14}{30}$.
• Step 5: Simplify the fraction $\frac{14}{30}$ by finding the greatest common divisor (GCD) of 14 and 30, which is 2:
• Divide both 14 and 30 by their GCD:
• Simplified numerator: $14 \div 2 = 7$
• Simplified denominator: $30 \div 2 = 15$
• Resulting simplified fraction: $\frac{7}{15}$

Therefore, the solution to the problem is $\mathbf{\frac{7}{15}}$.

To solve the problem of dividing two fractions, we apply the method of multiplication by the reciprocal:

• Step 1: The given problem is $\frac{3}{4} \div \frac{5}{6}$.
• Step 2: Find the reciprocal of the divisor $\frac{5}{6}$, which is $\frac{6}{5}$.
• Step 3: Multiply the dividend $\frac{3}{4}$ by the reciprocal of the divisor: $\frac{3}{4} \times \frac{6}{5}$
• Step 4: Perform the multiplication: $\frac{3 \times 6}{4 \times 5} = \frac{18}{20}$
• Step 5: Simplify the resulting fraction $\frac{18}{20}$: $\frac{18 \div 2}{20 \div 2} = \frac{9}{10}$

Therefore, the correct result of the division $\frac{3}{4} \div \frac{5}{6}$ is $\frac{9}{10}$.

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

• Step 1: Identify the given fractions.
• Step 2: Take the reciprocal of the second fraction.
• Step 3: Perform the multiplication using the reciprocal.
• Step 4: Simplify the resultant fraction if possible.
• Step 5: Convert to a mixed number if needed.

First, let's restate the problem equation:
We are to compute $\frac{5}{6} \div \frac{3}{5}$.

Step 1: Identify the fractions: $\frac{5}{6} \text{ and } \frac{3}{5}$.

Step 2: Take the reciprocal of $\frac{3}{5}$, which is $\frac{5}{3}$.

Step 3: Multiply $\frac{5}{6}$ by $\frac{5}{3}$:
$\frac{5}{6} \times \frac{5}{3} = \frac{5 \times 5}{6 \times 3} = \frac{25}{18}$.

Step 4: Convert $\frac{25}{18}$ to a mixed number:
$25 \div 18 = 1$ remainder $7$. Therefore, $\frac{25}{18} = 1\frac{7}{18}$.

Therefore, the result of $\frac{5}{6} \div \frac{3}{5}$ is $1\frac{7}{18}$.

To solve this problem, we'll use division of fractions by multiplying by the reciprocal:

• Step 1: Identify the reciprocal of the divisor.
  The divisor is $\frac{1}{4}$, and its reciprocal is $4$.
• Step 2: Multiply the dividend by the reciprocal of the divisor.
  Compute $\frac{5}{2} \times 4$.
• Step 3: Perform the multiplication.
  $\frac{5}{2} \times 4 = \frac{5 \times 4}{2} = \frac{20}{2} = 10$.

Therefore, the solution to the problem is $10$.