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

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

• Step 1: Convert the mixed number $2\frac{1}{3}$ into an improper fraction.
• Step 2: Take the reciprocal of this improper fraction.
• Step 3: Multiply $\frac{3}{5}$ by this reciprocal.
• Step 4: Simplify the resulting fraction if needed.

Now, let's work through each step:

Step 1: Convert $2\frac{1}{3}$ to an improper fraction.
A mixed number like $2\frac{1}{3}$ means $2 + \frac{1}{3}$. To convert it, multiply the whole number part by the denominator of the fractional part and add the numerator: $2 \times 3 + 1 = 6 + 1 = 7$. Thus, the improper fraction is $\frac{7}{3}$.

Step 2: Find the reciprocal of $\frac{7}{3}$.
The reciprocal of $\frac{7}{3}$ is $\frac{3}{7}$.

Step 3: Multiply $\frac{3}{5}$ by $\frac{3}{7}$.
When multiplying fractions, multiply numerators and denominators: $\frac{3}{5} \times \frac{3}{7} = \frac{3 \times 3}{5 \times 7} = \frac{9}{35}$.

Step 4: Simplify $\frac{9}{35}$ if necessary.
The fraction $\frac{9}{35}$ is already in its simplest form, as 9 and 35 have no common factors other than 1.

Therefore, the solution to the problem is $\frac{9}{35}$.

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

• Step 1: Convert the mixed number to an improper fraction.
• Step 2: Find the reciprocal of the improper fraction.
• Step 3: Multiply the first fraction by this reciprocal.
• Step 4: Simplify the result to get the final answer.

Let's go through each step in detail:

Step 1: Convert $2\frac{1}{2}$ to an improper fraction:

$2\frac{1}{2}$ is equal to $2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$.

Step 2: Find the reciprocal of $\frac{5}{2}$:

The reciprocal of $\frac{5}{2}$ is $\frac{2}{5}$.

Step 3: Multiply $\frac{3}{10}$ by $\frac{2}{5}$:

$\frac{3}{10} \times \frac{2}{5} = \frac{3 \times 2}{10 \times 5} = \frac{6}{50}$.

Step 4: Simplify $\frac{6}{50}$:

To simplify, we find the greatest common divisor of 6 and 50, which is 2:

$\frac{6}{50} = \frac{3}{25}$ after dividing the numerator and denominator by 2.

Therefore, the solution to the problem is $\frac{3}{25}$. This matches choice 2.

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

• Convert the mixed number $1\frac{4}{5}$ to an improper fraction.
• Divide $\frac{1}{2}$ by the improper fraction using multiplication by the reciprocal.
• Simplify the resulting expression.

Now, let's work through each step:
Step 1: Convert the mixed number to an improper fraction.
$1\frac{4}{5}$: Multiply 1 by 5 and add 4, resulting in $\frac{9}{5}$.

Step 2: Divide $\frac{1}{2}$ by $\frac{9}{5}$.
Use the division of fractions rule by multiplying by the reciprocal: $\frac{1}{2} \times \frac{5}{9} = \frac{1 \times 5}{2 \times 9} = \frac{5}{18}$.

Therefore, the solution to the problem is $\frac{5}{18}$.

To solve this problem, we will follow the steps below:

• Step 1: Convert the mixed number to an improper fraction.
• Step 2: Divide by using multiplication with the reciprocal.
• Step 3: Simplify the resulting fraction, if necessary.

Step 1: Convert $1\frac{1}{2}$ to an improper fraction.
The mixed number $1\frac{1}{2}$ is converted by multiplying the whole number 1 by the denominator 2, adding the numerator 1, resulting in $\frac{3}{2}$.

Step 2: Divide by multiplying with the reciprocal.
The division $\frac{3}{5} \div \frac{3}{2}$ is performed by multiplying $\frac{3}{5}$ with the reciprocal of $\frac{3}{2}$, which is $\frac{2}{3}$.

Thus, we have:

$\frac{3}{5} \times \frac{2}{3} = \frac{3 \times 2}{5 \times 3} = \frac{6}{15}$.

Step 3: Simplify the result.
The fraction $\frac{6}{15}$ can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 3:
$\frac{6 \div 3}{15 \div 3} = \frac{2}{5}$.

Therefore, the final solution to the problem is $\frac{2}{5}$.

To solve this problem, we'll break it down into manageable steps:

• Step 1: Convert the mixed number $2\frac{4}{5}$ into an improper fraction.
• Step 2: Divide the fraction $\frac{4}{5}$ by the improper fraction obtained in Step 1.
• Step 3: Multiply by the reciprocal to perform the division.

Let's work through each step:

Step 1: Convert the mixed number to an improper fraction.
The mixed number $2\frac{4}{5}$ can be converted as follows:

$2\frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{10 + 4}{5} = \frac{14}{5}$

Step 2: Change the division into multiplication by using the reciprocal of $\frac{14}{5}$.
The reciprocal of $\frac{14}{5}$ is $\frac{5}{14}$.

Step 3: Multiply $\frac{4}{5}$ by $\frac{5}{14}$:

$\frac{4}{5} \times \frac{5}{14} = \frac{4 \times 5}{5 \times 14}$

Simplify the expression:

$= \frac{20}{70}$

Reduce $\frac{20}{70}$ by dividing the numerator and the denominator by their greatest common divisor, which is 10:

$= \frac{2}{7}$

Therefore, the solution to the problem is $\boxed{\frac{2}{7}}$, which corresponds to choice 2.

To solve the problem $\frac{3}{4}:1\frac{2}{3}$, follow these steps:

• Step 1: Convert the mixed number $1\frac{2}{3}$ to an improper fraction.
• Step 2: Rewrite the division problem using the reciprocal of the divisor.
• Step 3: Multiply the fractions.

Let’s proceed through each step:

Step 1: Convert the mixed number to an improper fraction.

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

Step 2: The division $\frac{3}{4} : \frac{5}{3}$ becomes $\frac{3}{4} \times \frac{3}{5}$ (multiply by the reciprocal).

Step 3: Multiply the two fractions:

$\frac{3}{4} \times \frac{3}{5} = \frac{3 \times 3}{4 \times 5} = \frac{9}{20}$

Thus, the answer is $\frac{9}{20}$.

The correct answer is choice 3: $\frac{9}{20}$.

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

• Step 1: Convert the mixed number $1\frac{1}{5}$ into an improper fraction.
• Step 2: Divide $\frac{1}{4}$ by the improper fraction using the reciprocal.
• Step 3: Simplify the resulting fraction.

Now, let's work through each step:

Step 1: Convert the mixed number $1\frac{1}{5}$ into an improper fraction. The integer part is 1 and the fraction part is $\frac{1}{5}$. The improper fraction becomes:

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

Step 2: Divide $\frac{1}{4}$ by the improper fraction $\frac{6}{5}$. This is equivalent to multiplying $\frac{1}{4}$ by the reciprocal of $\frac{6}{5}$:

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

Perform the multiplication:

$\frac{1 \times 5}{4 \times 6} = \frac{5}{24}$

Step 3: The resulting fraction $\frac{5}{24}$ is already in its simplest form.

Therefore, the solution to the problem is $\frac{5}{24}$. The correct choice is option 4.

To solve the problem of dividing the fraction $\frac{3}{7}$ by the mixed number $1\frac{2}{3}$, we'll follow these steps:

• Step 1: Convert the mixed number $1\frac{2}{3}$ into an improper fraction.
• Step 2: Find the reciprocal of this improper fraction.
• Step 3: Multiply the initial fraction by this reciprocal.

Let's proceed with each step:

Step 1: Convert $1\frac{2}{3}$ to an improper fraction:
A mixed number $a\frac{b}{c}$ can be converted to an improper fraction by multiplying the whole number part by the denominator and adding the numerator. Hence:
$1 \cdot 3 + 2 = 3 + 2 = 5$
This gives us the improper fraction $\frac{5}{3}$.

Step 2: Find the reciprocal of $\frac{5}{3}$:
The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. Therefore, the reciprocal of $\frac{5}{3}$ is $\frac{3}{5}$.

Step 3: Multiply $\frac{3}{7}$ by the reciprocal $\frac{3}{5}$:
$\frac{3}{7} \times \frac{3}{5} = \frac{3 \cdot 3}{7 \cdot 5}$
Compute the multiplication:
$\frac{9}{35}$

Therefore, the solution to the problem is $\frac{9}{35}$.

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

• Step 1: Convert the mixed number $1\frac{3}{4}$ to an improper fraction.

• Step 2: Replace the division with multiplication by the reciprocal of the improper fraction.

• Step 3: Perform the multiplication and simplify if necessary.

Now, let's work through each step:
Step 1: Convert $1\frac{3}{4}$ to an improper fraction.
A mixed number $a \frac{b}{c}$ is converted to an improper fraction $\frac{ac + b}{c}$.
So, $1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}$.

Step 2: Replace the division with multiplication by the reciprocal of $\frac{7}{4}$.
The reciprocal of $\frac{7}{4}$ is $\frac{4}{7}$, so the expression becomes: $\frac{2}{3} \times \frac{4}{7}$.

Step 3: Perform the multiplication:
Multiply the numerators: $2 \times 4 = 8$.
Multiply the denominators: $3 \times 7 = 21$.
Thus, the product is $\frac{8}{21}$.

No further simplification is needed as 8 and 21 have no common factors.

Therefore, the solution to the problem is $\frac{8}{21}$.

To solve this division problem, we will follow these steps:

• Step 1: Convert the mixed number to an improper fraction.
• Step 2: Replace division with multiplication by the reciprocal.
• Step 3: Perform the multiplication and simplify the result.

Let's begin with the solution:

Step 1: Convert the mixed number $1\frac{1}{3}$ into an improper fraction.

To convert, multiply 1 (the whole number) by 3 (the denominator) and add 1 (the numerator):

$1 \times 3 + 1 = 3 + 1 = 4$

Therefore, $1\frac{1}{3} = \frac{4}{3}$.

Step 2: Write the division problem as multiplication by the reciprocal of $\frac{4}{3}$.

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

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

Step 3: Multiply the fractions and simplify if possible.

Multiply the numerators and the denominators:

$\text{Numerators: } 3 \times 3 = 9$

$\text{Denominators: } 4 \times 4 = 16$

Thus, $\frac{3}{4} \times \frac{3}{4} = \frac{9}{16}$.

No further simplification is required as 9 and 16 are relatively prime.

Therefore, the solution to the problem is $\frac{9}{16}$.

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

• Step 1: Convert the mixed number, $1 \frac{3}{4}$, into an improper fraction.
• Step 2: Rewrite the division as a multiplication by the reciprocal of the improper fraction.
• Step 3: Perform the multiplication and simplify the resulting fraction.

Now, let's work through each step:
Step 1: Convert the mixed number. $1 \frac{3}{4}$ can be converted as follows:

The whole number $1$ times the denominator $4$ plus the numerator $3$ gives $7$.
Thus, $1 \frac{3}{4} = \frac{7}{4}$.

Step 2: Divide $\frac{5}{8}$ by $\frac{7}{4}$ by multiplying by its reciprocal:
$\frac{5}{8} \div \frac{7}{4} = \frac{5}{8} \times \frac{4}{7}$.

Step 3: Multiply across the numerators and the denominators:
$\frac{5 \times 4}{8 \times 7} = \frac{20}{56}$.

Simplify $\frac{20}{56}$ by finding the greatest common divisor (GCD), which is $4$:
$\frac{20}{56} = \frac{20 \div 4}{56 \div 4} = \frac{5}{14}$.

Therefore, the solution to the problem is $\frac{5}{14}$.

To solve this problem, we will follow these steps:

• Step 1: Convert the mixed number $2\frac{3}{8}$ into an improper fraction.
• Step 2: Divide $\frac{7}{8}$ by the improper fraction by multiplying with its reciprocal.
• Step 3: Simplify the resulting fraction.

Now, let's go through each step:

Step 1: Convert the mixed number $2\frac{3}{8}$ to an improper fraction. We do this by multiplying the whole number part by the denominator and adding the numerator:

$2\frac{3}{8} = \frac{2 \times 8 + 3}{8} = \frac{16 + 3}{8} = \frac{19}{8}$.

Step 2: To divide by a fraction $\frac{19}{8}$, we multiply by its reciprocal $\frac{8}{19}$:

$\frac{7}{8} : \frac{19}{8} = \frac{7}{8} \times \frac{8}{19}$.

Step 3: Simplify the resulting expression. Multiply the numerators and the denominators:

$\frac{7 \times 8}{8 \times 19} = \frac{56}{152}$.

The $8$s cancel out, leaving us with:

$\frac{56}{152} = \frac{7}{19}$.

Therefore, the solution to the problem is $\frac{7}{19}$.

To solve the problem $\frac{5}{6} : 3\frac{1}{3}$, we follow these steps:

• Step 1: Convert the mixed number to an improper fraction.
• Step 2: Multiply the dividend by the reciprocal of the improper fraction.
• Step 3: Simplify the resulting fraction.

Let's carry out these steps:
Step 1: Convert $3\frac{1}{3}$ into an improper fraction. We do this by multiplying the whole number $3$ by the denominator $3$ and adding the numerator $1$, giving us:

This yields the improper fraction $\frac{10}{3}$.

Step 2: Divide $\frac{5}{6}$ by $\frac{10}{3}$ involves multiplying $\frac{5}{6}$ by the reciprocal of $\frac{10}{3}$, which is $\frac{3}{10}$:

Step 3: Simplify $\frac{15}{60}$ by dividing both the numerator and denominator by their greatest common divisor, which is 15:

Thus, the solution to $\frac{5}{6} : 3\frac{1}{3}$ is $\frac{1}{4}$.