Multiplying and Dividing Mixed Numbers: Only division

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

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Divide the improper fractions by multiplying by the reciprocal of the second fraction.
• Step 3: Simplify the resulting fraction if possible.
• Step 4: Convert the simplified improper fraction back to a mixed number if needed.

Let's work through each step:

Step 1: Convert $3\frac{1}{2}$ to an improper fraction.

$3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}$

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

$1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4}$

Step 2: Divide $\frac{7}{2}$ by $\frac{5}{4}$.

The division of fractions is done by multiplying by the reciprocal: $\frac{7}{2} \times \frac{4}{5}$.

Step 3: Perform the multiplication.

$\frac{7 \times 4}{2 \times 5} = \frac{28}{10}$

Step 4: Simplify $\frac{28}{10}$.

Divide both the numerator and the denominator by 2: $\frac{28 \div 2}{10 \div 2} = \frac{14}{5}$.

Convert $\frac{14}{5}$ back to a mixed number.

$\frac{14}{5} = 2\frac{4}{5}$ because 14 divided by 5 is 2 with a remainder of 4.

Therefore, the solution to the problem $3\frac{1}{2} : 1\frac{1}{4}$ is $2\frac{4}{5}$.

To solve this problem, we'll convert the mixed numbers to improper fractions and then perform the division.

• Step 1: Convert $6\frac{1}{2}$ to an improper fraction:

$6\frac{1}{2} = \frac{13}{2}$.

• Step 2: Convert $1\frac{1}{4}$ to an improper fraction:

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

• Step 3: Divide the two improper fractions:

$\frac{13}{2} \div \frac{5}{4} = \frac{13}{2} \times \frac{4}{5}$.

• Step 4: Multiply the fractions:

$\frac{13}{2} \times \frac{4}{5} = \frac{13 \times 4}{2 \times 5} = \frac{52}{10}$.

• Step 5: Simplify $\frac{52}{10}$:

$\frac{52}{10} = \frac{26}{5} = 5\frac{1}{5}$.

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

To solve this division of mixed numbers, follow these detailed steps:

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Divide by multiplying by the reciprocal of the second fraction.
• Step 3: Simplify the resulting fraction and convert it back to a mixed number, if necessary.

Let's perform each step with the given numbers:

Step 1: Convert the mixed numbers to improper fractions.
$1\frac{5}{7} = \frac{1 \times 7 + 5}{7} = \frac{12}{7}$
$1\frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3}$

Step 2: Instead of dividing by $\frac{4}{3}$, multiply by its reciprocal, which is $\frac{3}{4}$.
$\frac{12}{7} \times \frac{3}{4} = \frac{12 \times 3}{7 \times 4} = \frac{36}{28}$

Step 3: Simplify the fraction $\frac{36}{28}$.
Both the numerator and the denominator are divisible by 4:
$\frac{36 \div 4}{28 \div 4} = \frac{9}{7}$

Convert $\frac{9}{7}$ back to a mixed number:
Since $9$ divided by $7$ is $1$ with a remainder of $2$, it becomes:
$1\frac{2}{7}$

Therefore, the solution to the problem is $1\frac{2}{7}$.

To solve this problem, we need to proceed through the following steps:

• Step 1: Convert the mixed numbers to improper fractions.
• Step 2: Divide the improper fractions using multiplication by the reciprocal.
• Step 3: Simplify the resulting fraction and convert back to a mixed number.

Now, let's work through each step:

Step 1: Convert
For $3\frac{5}{8}$, multiply the whole number 3 by the denominator 8 and add the numerator 5:
$3 \times 8 + 5 = 24 + 5 = 29$
$<=>$ This gives us the improper fraction $\frac{29}{8}$.

For $1\frac{1}{2}$, multiply the whole number 1 by the denominator 2 and add the numerator 1:
$1 \times 2 + 1 = 2 + 1 = 3$
$<=>$ This gives us the improper fraction $\frac{3}{2}$.

Step 2: Divide the improper fractions.
$\frac{29}{8} \div \frac{3}{2}$ becomes $\frac{29}{8} \times \frac{2}{3}$.
Multiply the numerators and the denominators:
$\frac{29 \times 2}{8 \times 3} = \frac{58}{24}$.

Step 3: Simplify the resulting fraction.
Divide both the numerator and the denominator by their greatest common divisor, which is 2:
$\frac{58 \div 2}{24 \div 2} = \frac{29}{12}$.
Convert $\frac{29}{12}$ into a mixed number:
$29 \div 12 = 2 \text{ remainder } 5$
Thus, it becomes $2\frac{5}{12}$.

Therefore, the division of $3\frac{5}{8}$ by $1\frac{1}{2}$ yields $2\frac{5}{12}$.

To solve this problem, follow these steps:

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Divide by multiplying by the reciprocal.
• Step 3: Simplify the result.

Now, let's work through the steps:

Step 1: Convert the mixed numbers to improper fractions.
For $7\frac{1}{2}$, convert it as follows: $7\frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{15}{2}$.
For $2\frac{1}{4}$, convert it as follows: $2\frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}$.

Step 2: To divide $\frac{15}{2}$ by $\frac{9}{4}$, multiply $\frac{15}{2}$ by the reciprocal of $\frac{9}{4}$:
$\frac{15}{2} \times \frac{4}{9}$.

Step 3: Multiply and simplify the fractions:
$\frac{15 \times 4}{2 \times 9} = \frac{60}{18}$.

Now simplify $\frac{60}{18}$ by dividing both the numerator and denominator by their greatest common divisor, which is 6:
$\frac{60 \div 6}{18 \div 6} = \frac{10}{3}$.

Convert $\frac{10}{3}$ back to a mixed number:

The whole number part is $10 \div 3 = 3$ with a remainder of 1. Thus, the mixed number is $3\frac{1}{3}$.

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

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

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Divide the fractions by multiplying by the reciprocal of the divisor.
• Step 3: Simplify the resulting improper fraction and convert it back to a mixed number.

Let's work through these steps:

Step 1: Convert the mixed numbers:
$6\frac{2}{3} = \frac{6 \times 3 + 2}{3} = \frac{20}{3}$
$2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}$

Step 2: Divide the fractions:
To divide $\frac{20}{3}$ by $\frac{5}{2}$, we multiply by the reciprocal of $\frac{5}{2}$:
$\frac{20}{3} \times \frac{2}{5} = \frac{20 \times 2}{3 \times 5} = \frac{40}{15}$

Step 3: Simplify and convert back:
Simplify $\frac{40}{15}$:
The greatest common divisor of 40 and 15 is 5:
$\frac{40}{15} = \frac{8}{3}$
Convert $\frac{8}{3}$ to a mixed number:
$8\div3 = 2$ remainder $2$, so $\frac{8}{3} = 2\frac{2}{3}$.

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

Let's solve the problem step by step:

• Step 1: Convert the mixed numbers to improper fractions.
• Step 2: Divide the first improper fraction by the second.
• Step 3: Simplify the result and convert back to a mixed number if needed.

Step 1: Convert the mixed numbers to improper fractions.
For $3\frac{3}{4}$:
- The whole number is 3, the numerator is 3, and the denominator is 4.
- Convert: $3 \frac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}$.

For $1\frac{1}{3}$:
- The whole number is 1, the numerator is 1, and the denominator is 3.
- Convert: $1 \frac{1}{3} = \frac{(1 \times 3) + 1}{3} = \frac{3 + 1}{3} = \frac{4}{3}$.

Step 2: Divide the fractions by multiplying by the reciprocal of the second fraction.
$\frac{15}{4} \div \frac{4}{3} = \frac{15}{4} \times \frac{3}{4} = \frac{15 \times 3}{4 \times 4} = \frac{45}{16}$.

Step 3: Convert the improper fraction back to a mixed number.
Divide 45 by 16:
- 45 divided by 16 is 2, remainder 13, giving us $2\frac{13}{16}$.

Thus, the solution to the problem is $2\frac{13}{16}$, which matches choice (1).

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

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Perform the division by multiplying by the reciprocal of the divisor.
• Step 3: Simplify the resulting fraction if necessary.

Now, let's work through each step:
Step 1: Convert the mixed number $4\frac{1}{4}$ to an improper fraction. This is done by multiplying the whole number by the denominator and adding the numerator: $4 \times 4 + 1 = 17$. So, $4\frac{1}{4} = \frac{17}{4}$.
Convert $2\frac{1}{8}$ to an improper fraction: $2 \times 8 + 1 = 17$. Therefore, $2\frac{1}{8} = \frac{17}{8}$.

Step 2: Divide the fractions by multiplying by the reciprocal of the divisor. In this case, divide $\frac{17}{4}$ by $\frac{17}{8}$, which is equivalent to multiplying $\frac{17}{4}$ by $\frac{8}{17}$:
$\frac{17}{4} \times \frac{8}{17} = \frac{17 \times 8}{4 \times 17} = \frac{136}{68}$.

Step 3: Simplify the resulting fraction $\frac{136}{68}$. Since the numerator and the denominator are both divisible by 68, simplification gives: $\frac{136 \div 68}{68 \div 68} = \frac{2}{1} = 2$.

Therefore, the solution to the problem is $2$.

To solve the problem of dividing the mixed numbers $7\frac{1}{2}$ by $2\frac{1}{2}$, follow these steps:

• Step 1: Convert the mixed numbers to improper fractions.
• Step 2: Divide the fractions by multiplying by the reciprocal.
• Step 3: Simplify the resulting fraction to find the answer.

Now, let’s break it down step-by-step:

Step 1: Convert $7\frac{1}{2}$ to an improper fraction. To do this, multiply the whole number 7 by the denominator 2 and add the numerator 1:
$7 \times 2 + 1 = 14 + 1 = 15$.

This makes the improper fraction $\frac{15}{2}$.

Convert $2\frac{1}{2}$ in a similar manner:
$2 \times 2 + 1 = 4 + 1 = 5$, giving $\frac{5}{2}$.

Step 2: Divide $\frac{15}{2}$ by $\frac{5}{2}$ by multiplying $\frac{15}{2}$ by the reciprocal of $\frac{5}{2}$, which is $\frac{2}{5}$:
$\frac{15}{2} \times \frac{2}{5} = \frac{15 \times 2}{2 \times 5} = \frac{30}{10}$.

Step 3: Simplify $\frac{30}{10}$: divide the numerator and the denominator by their greatest common divisor, 10,
$\frac{30}{10} = \frac{30 \div 10}{10 \div 10} = \frac{3}{1} = 3$.

Therefore, the solution to $7\frac{1}{2} \div 2\frac{1}{2}$ is $3$, which corresponds to choice 3.

To solve this problem, we will divide the mixed numbers by following these steps:

• Step 1: Convert Mixed Numbers to Improper Fractions
  First, let's convert the mixed numbers to improper fractions.
  For $5\frac{1}{4}$:
  Convert using $\text{Whole part} \times \text{Denominator} + \text{Numerator}$:
  $5\frac{1}{4} = \frac{(5 \times 4) + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4}$.

• For $2\frac{3}{8}$:
  Convert using the same method:
  $2\frac{3}{8} = \frac{(2 \times 8) + 3}{8} = \frac{16 + 3}{8} = \frac{19}{8}$.

• Step 2: Calculate the Division Using Reciprocal
  Now, we divide $\frac{21}{4}$ by $\frac{19}{8}$ by multiplying by the reciprocal:
  $\frac{21}{4} \div \frac{19}{8} = \frac{21}{4} \times \frac{8}{19} = \frac{21 \times 8}{4 \times 19}$.

• Simplify:
  Before multiplication, simplify where possible. Factor the numerator and denominator:
  $\frac{168}{76}$.

• Step 3: Simplify and Convert to Mixed Number
  Simplify $\frac{168}{76}$ by finding the greatest common divisor (GCD), which is 4:
  $\frac{168}{76} = \frac{168 \div 4}{76 \div 4} = \frac{42}{19}.$
  Convert $\frac{42}{19}$ to a mixed number:
  Divide 42 by 19 gives 2 with a remainder of 4:
  Thus, $\frac{42}{19} = 2\frac{4}{19}$.

Therefore, the solution to the problem is $2\frac{4}{19}$, which corresponds to choice 4.

To solve the problem $5\frac{4}{5} \div 2\frac{1}{10}$, we follow these steps:

• Step 1: Convert mixed numbers to improper fractions.
  $5\frac{4}{5} = \frac{5 \times 5 + 4}{5} = \frac{25 + 4}{5} = \frac{29}{5}$
• $2\frac{1}{10} = \frac{2 \times 10 + 1}{10} = \frac{20 + 1}{10} = \frac{21}{10}$
• Step 2: Divide $\frac{29}{5}$ by $\frac{21}{10}$ by multiplying by the reciprocal.
  $\frac{29}{5} \div \frac{21}{10} = \frac{29}{5} \times \frac{10}{21}$
• Step 3: Multiply the fractions and simplify.
  $\frac{29}{5} \times \frac{10}{21} = \frac{29 \times 10}{5 \times 21} = \frac{290}{105}$
• Step 4: Simplify $\frac{290}{105}$.
  Find the greatest common divisor (GCD) of 290 and 105. The GCD is 5.
  $\frac{290 \div 5}{105 \div 5} = \frac{58}{21}$
• Step 5: Convert the improper fraction $\frac{58}{21}$ into a mixed number.
  $58 \div 21 = 2$ remainder $16$, so $\frac{58}{21} = 2\frac{16}{21}$

Therefore, the solution is $2\frac{16}{21}$.

To solve the problem, follow these steps:

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Divide by multiplying by the reciprocal of the second fraction.
• Step 3: Simplify the result, and if required, convert back to a mixed number.

Let's proceed with each step:
Step 1: Convert $10\frac{3}{7}$ and $4\frac{1}{14}$ to improper fractions.
The first mixed number $10\frac{3}{7}$ becomes: $\frac{10 \times 7 + 3}{7} = \frac{73}{7}$.
The second mixed number $4\frac{1}{14}$ becomes: $\frac{4 \times 14 + 1}{14} = \frac{57}{14}$.

Step 2: Divide by multiplying by the reciprocal.
$\frac{73}{7} \div \frac{57}{14} = \frac{73}{7} \times \frac{14}{57}$.

Step 3: Perform the multiplication and simplify.
The multiplication of fractions gives: $\frac{73 \times 14}{7 \times 57} = \frac{1022}{399}$.

Step 4: Convert to a mixed number, if necessary.
$1022 \div 399$ gives a quotient of 2 and a remainder of 224: $1022 = 2 \times 399 + 224$.
Thus, the mixed number is $2\frac{224}{399}$.

The solution to the problem is $2\frac{224}{399}$, which corresponds to choice 4.