Multiplying and Dividing Mixed Numbers - Examples, Exercises and Solutions

To solve the problem of multiplying the mixed numbers $2\frac{5}{6}$ and $1\frac{1}{4}$, we will follow these steps:

• Step 1: Convert mixed numbers to improper fractions.

For $2\frac{5}{6}$:
Multiply the whole number 2 by the denominator 6, resulting in 12. Add the numerator 5 to get 17.
Thus, $2\frac{5}{6} = \frac{17}{6}$.

For $1\frac{1}{4}$:
Multiply the whole number 1 by the denominator 4, resulting in 4. Add the numerator 1 to get 5.
Thus, $1\frac{1}{4} = \frac{5}{4}$.

• Step 2: Multiply the improper fractions.

Multiply $\frac{17}{6}$ by $\frac{5}{4}$:
The result is $\frac{17 \times 5}{6 \times 4} = \frac{85}{24}$.

• Step 3: Convert the result back to a mixed number.

To convert $\frac{85}{24}$ to a mixed number, divide 85 by 24:
85 divided by 24 is 3, with a remainder of 13.
Hence, $\frac{85}{24} = 3\frac{13}{24}$.

Therefore, the product of the mixed numbers $2\frac{5}{6}$ and $1\frac{1}{4}$ is $3\frac{13}{24}$.

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

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Multiply the improper fractions.
• Step 3: Convert the result back to a mixed number.

Now, let's work through each step:

Step 1: Convert each mixed number to an improper fraction.
For $1\frac{1}{4}$:
- Whole number is 1, denominator is 4, and numerator is 1.
- Convert to improper fraction: $1\frac{1}{4} = \frac{4 \times 1 + 1}{4} = \frac{5}{4}$.

For $1\frac{6}{8}$:
- Whole number is 1, denominator is 8, and numerator is 6.
- Convert to improper fraction: $1\frac{6}{8} = \frac{8 \times 1 + 6}{8} = \frac{14}{8}$.
- Simplify $\frac{14}{8}$ to $\frac{7}{4}$ by dividing both the numerator and the denominator by 2.

Step 2: Multiply the improper fractions:
$\frac{5}{4} \times \frac{7}{4} = \frac{5 \times 7}{4 \times 4} = \frac{35}{16}$.

Step 3: Convert the improper fraction back to a mixed number:
Divide 35 by 16. This gives 2 as the quotient with a remainder of 3.
Thus, $\frac{35}{16} = 2\frac{3}{16}$.

Therefore, the product of $1\frac{1}{4} \times 1\frac{6}{8}$ is $2\frac{3}{16}$.

To solve the problem of multiplying the mixed numbers $2\frac{1}{4}$ and $1\frac{2}{3}$, we proceed as follows:

• Step 1: Convert Mixed Numbers to Improper Fractions

  • Convert $2\frac{1}{4}$ to an improper fraction: $2\frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}$

  • Convert $1\frac{2}{3}$ to an improper fraction: $1\frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3}$

• Step 2: Multiply the Improper Fractions

• Now, multiply $\frac{9}{4}$ by $\frac{5}{3}$: $\frac{9}{4} \times \frac{5}{3} = \frac{9 \times 5}{4 \times 3} = \frac{45}{12}$

• Step 3: Simplify the Fraction

• Simplify $\frac{45}{12}$ by finding the greatest common divisor of 45 and 12, which is 3: $\frac{45 \div 3}{12 \div 3} = \frac{15}{4}$

• Step 4: Convert Back to a Mixed Number

• Convert $\frac{15}{4}$ into a mixed number: $15 \div 4 = 3 \quad \text{remainder} \quad 3$ So, $\frac{15}{4} = 3\frac{3}{4}$.

Based on the calculations, the product of $2\frac{1}{4}$ and $1\frac{2}{3}$ is $3\frac{3}{4}$.

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

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

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Multiply the improper fractions.
• Step 3: Simplify the resulting fraction, if needed, and convert it back to a mixed number.

Now, let's work through each step:
Step 1: Convert the mixed numbers to improper fractions.
For $1\frac{4}{5}$:
$1\frac{4}{5} = \frac{1 \times 5 + 4}{5} = \frac{9}{5}$.
For $2\frac{1}{2}$:
$2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}$.

Step 2: Multiply the improper fractions:
$\frac{9}{5} \times \frac{5}{2} = \frac{9 \times 5}{5 \times 2} = \frac{45}{10}$.

Step 3: Simplify the fraction and convert it back to a mixed number:
$\frac{45}{10} = \frac{9}{2} = 4\frac{1}{2}$.

Therefore, the product of $1\frac{4}{5} \times 2\frac{1}{2}$ is $4\frac{1}{2}$, which corresponds to choice 2.

To solve this problem, we'll convert the mixed numbers into improper fractions, multiply them, and simplify the result:

• Step 1: Convert mixed numbers to improper fractions.
  • $1\frac{4}{5}$ becomes $\frac{1 \times 5 + 4}{5} = \frac{9}{5}$.
  • $1\frac{1}{3}$ becomes $\frac{1 \times 3 + 1}{3} = \frac{4}{3}$.
• Step 2: Multiply the improper fractions.
  • $\frac{9}{5} \times \frac{4}{3} = \frac{9 \times 4}{5 \times 3} = \frac{36}{15}$.
• Step 3: Simplify the fraction $\frac{36}{15}$.
  • The greatest common divisor of 36 and 15 is 3.
  • $\frac{36 \div 3}{15 \div 3} = \frac{12}{5}$.
• Step 4: Convert the improper fraction $\frac{12}{5}$ back to a mixed number.
  • $12 \div 5$ is 2 with a remainder of 2.
  • The mixed number is $2\frac{2}{5}$.

Therefore, the product of $1\frac{4}{5} \times 1\frac{1}{3}$ is $2\frac{2}{5}$.

To solve the problem $1\frac{3}{9} \times 2\frac{2}{4}$, we will follow these steps:

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Multiply the improper fractions.
• Step 3: Convert the resulting improper fraction back into a mixed number.

Let’s begin with each step in detail:

Step 1: Convert $1\frac{3}{9}$ and $2\frac{2}{4}$ to improper fractions.
- For $1\frac{3}{9}$: Convert the fraction $\frac{3}{9}$ to its simplest form, which is $\frac{1}{3}$. Then, the mixed number $1\frac{1}{3}$ becomes $1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}$.
- For $2\frac{2}{4}$: The fraction $\frac{2}{4}$ simplifies to $\frac{1}{2}$. Then, the mixed number $2\frac{1}{2}$ becomes $2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$.

Step 2: Multiply the improper fractions:
$\frac{4}{3} \times \frac{5}{2} = \frac{4 \times 5}{3 \times 2} = \frac{20}{6}$.

Simplify $\frac{20}{6}$:
Find the greatest common divisor (GCD) of 20 and 6, which is 2. Then $\frac{20}{6} = \frac{20 \div 2}{6 \div 2} = \frac{10}{3}$.

Step 3: Convert the improper fraction $\frac{10}{3}$ back to a mixed number:
Divide 10 by 3 to get 3 with a remainder of 1, thus $\frac{10}{3} = 3\frac{1}{3}$.

Therefore, the product is $3\frac{1}{3}$.

To solve the problem of multiplying $3 \frac{2}{5}$ by $1 \frac{1}{6}$, follow these steps:

• Step 1: Convert Mixed Numbers to Improper Fractions.
  - For $3 \frac{2}{5}$, convert by multiplying 3 (the whole number) by 5 (the denominator) and adding 2 (the numerator):
  $(3 \times 5) + 2 = 15 + 2 = 17$. Thus, $3 \frac{2}{5} = \frac{17}{5}$.
  - For $1 \frac{1}{6}$, convert by multiplying 1 (the whole number) by 6 (the denominator) and adding 1 (the numerator):
  $(1 \times 6) + 1 = 6 + 1 = 7$. Thus, $1 \frac{1}{6} = \frac{7}{6}$.
• Step 2: Multiply the Improper Fractions.
  Multiply $\frac{17}{5}$ by $\frac{7}{6}$:
  $\frac{17}{5} \times \frac{7}{6} = \frac{17 \times 7}{5 \times 6} = \frac{119}{30}$.
• Step 3: Convert the Result to a Mixed Number.
  Divide 119 by 30. The quotient is 3 with a remainder of 29, so the mixed number is $3 \frac{29}{30}$.

Conclusion: $3 \frac{2}{5} \times 1 \frac{1}{6} = 3 \frac{29}{30}$.

The correct choice from the options provided is Choice 4: $3 \frac{29}{30}$.

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

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Multiply the improper fractions.
• Step 3: Simplify the resulting fraction, if necessary.
• Step 4: Convert the result back to a mixed number.

Now, let's work through each step:

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

For $2\frac{2}{6}$:

$2\frac{2}{6} = \frac{2 \times 6 + 2}{6} = \frac{12 + 2}{6} = \frac{14}{6}$

Simplifying $\frac{14}{6}$ by dividing the numerator and the denominator by 2, we get $\frac{14}{6} = \frac{7}{3}$.

For $1\frac{4}{10}$:

$1\frac{4}{10} = \frac{1 \times 10 + 4}{10} = \frac{10 + 4}{10} = \frac{14}{10}$

Simplifying $\frac{14}{10}$ by dividing the numerator and the denominator by 2, we get $\frac{14}{10} = \frac{7}{5}$.

Step 2: Multiply the improper fractions:

$\frac{7}{3} \times \frac{7}{5} = \frac{7 \times 7}{3 \times 5} = \frac{49}{15}$

Step 3: Simplify the resulting fraction, if necessary. In this case, $\frac{49}{15}$ is already in its simplest form.

Step 4: Convert the result back to a mixed number:

$\frac{49}{15}$ can be rewritten as $3\frac{4}{15}$, since 49 divided by 15 is 3 with a remainder of 4.

Therefore, the solution to the problem is $3\frac{4}{15}$.

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

• Step 1: Convert each mixed number to an improper fraction.
• Step 2: Multiply the improper fractions.
• Step 3: Simplify the resulting fraction.
• Step 4: Convert the improper fraction back to a mixed number if necessary.

Let's begin:

Step 1: Convert to improper fractions
Convert $1\frac{6}{8}$ to an improper fraction:
$\frac{8 \times 1 + 6}{8} = \frac{14}{8}$.

Convert $2\frac{2}{6}$ to an improper fraction:
$\frac{6 \times 2 + 2}{6} = \frac{14}{6}$.

Step 2: Multiply the fractions
Multiply $\frac{14}{8}$ and $\frac{14}{6}$:
$\frac{14}{8} \times \frac{14}{6} = \frac{196}{48}$.

Step 3: Simplify the fraction
Find the greatest common divisor (GCD) of 196 and 48, which is 4. Simplify $\frac{196}{48}$:
$\frac{196 \div 4}{48 \div 4} = \frac{49}{12}$.

Step 4: Convert to a mixed number
$\frac{49}{12}$ as a mixed number is $4\frac{1}{12}$ since 49 divided by 12 is 4 with a remainder of 1.

Therefore, the solution to the problem is $4\frac{1}{12}$.

To solve this problem, we will convert the mixed numbers to improper fractions and multiply them.

• Step 1: Convert Mixed Numbers to Improper Fractions

For the first mixed number $2\frac{10}{20}$:
- Convert $2\frac{10}{20}$ to an improper fraction:
Initially, $10/20$ can be simplified to $1/2$ (since $\gcd(10, 20) = 10$). Therefore, the mixed number $2\frac{1}{2}$ is equal to:
$2 \times 2 + 1 = 4 + 1 = 5$
Thus, $2\frac{1}{2}$ becomes $\frac{5}{2}$.

For the second mixed number $1\frac{4}{16}$:
- Simplify $4/16$ to $1/4$ (since $\gcd(4, 16) = 4$). Hence, $1\frac{4}{16}$ simplifies to $1\frac{1}{4}$:
$1 \times 4 + 1 = 4 + 1 = 5$
Thus, $1\frac{1}{4}$ becomes $\frac{5}{4}$.

• Step 2: Multiply the Improper Fractions

Multiply the fractions $\frac{5}{2} \times \frac{5}{4}$:
$\frac{5 \times 5}{2 \times 4} = \frac{25}{8}$

• Step 3: Simplify and Convert to Mixed Number

Convert $\frac{25}{8}$ back to a mixed number by dividing:
- Divide 25 by 8, which goes 3 times (remainder 1).
Thus, $\frac{25}{8} = 3\frac{1}{8}$.

Therefore, the solution to the problem is $3\frac{1}{8}$.

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

• Step 1: Convert $4\frac{2}{3}$ and $1\frac{1}{5}$ to improper fractions.
• Step 2: Multiply these improper fractions.
• Step 3: Simplify the result, if necessary, and convert it back to a mixed number.

Now, let's work through each step:
Step 1: Convert mixed numbers to improper fractions:
For $4\frac{2}{3}$:
Multiply the whole number 4 by the denominator 3 and add the numerator 2:
$4 \times 3 + 2 = 12 + 2 = 14$.
Thus, $4\frac{2}{3} = \frac{14}{3}$.
For $1\frac{1}{5}$:
Multiply the whole number 1 by the denominator 5 and add the numerator 1:
$1 \times 5 + 1 = 5 + 1 = 6$.
Thus, $1\frac{1}{5} = \frac{6}{5}$.

Step 2: Multiply the improper fractions $\frac{14}{3}$ and $\frac{6}{5}$:
$\frac{14}{3} \times \frac{6}{5} = \frac{14 \times 6}{3 \times 5} = \frac{84}{15}$.

Step 3: Simplify the resulting fraction $\frac{84}{15}$ and convert it to a mixed number if necessary:
Find the greatest common divisor (GCD) of 84 and 15, which is 3.
Divide both numerator and denominator by their GCD:
$\frac{84 \div 3}{15 \div 3} = \frac{28}{5}$.

Convert the improper fraction $\frac{28}{5}$ to a mixed number:
Divide 28 by 5: Quotient is 5, remainder is 3.
Thus, $\frac{28}{5} = 5\frac{3}{5}$.

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

To solve this problem, we'll convert the given mixed numbers to improper fractions, multiply them, and simplify the result. Let's proceed step by step:

• Step 1: Convert the mixed numbers to improper fractions.
  For $1\frac{4}{6}$: Multiply the whole number by the denominator and add the numerator: $1\frac{4}{6} = \frac{1 \times 6 + 4}{6} = \frac{10}{6}$.
  For $1\frac{2}{8}$: Similarly, multiply the whole number by the denominator and add the numerator: $1\frac{2}{8} = \frac{1 \times 8 + 2}{8} = \frac{10}{8}$.
• Step 2: Multiply the improper fractions.
  $\frac{10}{6} \times \frac{10}{8} = \frac{10 \times 10}{6 \times 8} = \frac{100}{48}$.
• Step 3: Simplify the resulting fraction.
  Find the GCD of 100 and 48, which is 4. Divide both the numerator and the denominator by 4:
  $\frac{100}{48} = \frac{100 \div 4}{48 \div 4} = \frac{25}{12}$.
• Step 4: Convert the simplified improper fraction back to a mixed number.
  Divide 25 by 12: 25 divided by 12 is 2 with a remainder of 1. So, $\frac{25}{12} = 2\frac{1}{12}$.

Therefore, the solution to the problem is $2\frac{1}{12}$. This matches the correct answer choice 2.

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

• Convert the mixed numbers to improper fractions.
• Multiply the improper fractions.
• Simplify the result and convert back to a mixed number if necessary.

Let's work through these steps:

1. Convert each mixed number to an improper fraction:

• For $2\frac{4}{12}$, first simplify the fraction $\frac{4}{12}$ to $\frac{1}{3}$. So, $2\frac{1}{3}$ becomes:

$\frac{2 \times 3 + 1}{3} = \frac{7}{3}$.

• For $1\frac{2}{4}$, first simplify the fraction $\frac{2}{4}$ to $\frac{1}{2}$. So, $1\frac{1}{2}$ becomes:

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

2. Multiply the improper fractions:

The multiplication of $\frac{7}{3}$ and $\frac{3}{2}$ is:

$\frac{7}{3} \times \frac{3}{2} = \frac{21}{6}$.

3. Simplify the resulting fraction $\frac{21}{6}$:

$\frac{21}{6} = \frac{7}{2}$ after dividing the numerator and denominator by 3.

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

$\frac{7}{2} = 3\frac{1}{2}$.

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

Let's solve the problem by following these steps:

• Step 1: Convert the mixed numbers to improper fractions.
• Step 2: Multiply the improper fractions.
• Step 3: Simplify the resulting fraction and convert back to a mixed number if needed.

Step 1:
Convert $1\frac{4}{12}$ to an improper fraction.

Calculate the improper fraction: $1\frac{4}{12} = \frac{1 \times 12 + 4}{12} = \frac{16}{12}$

Convert $1\frac{4}{14}$ to an improper fraction.

Calculate the improper fraction: $1\frac{4}{14} = \frac{1 \times 14 + 4}{14} = \frac{18}{14}$

Step 2:
Multiply the two improper fractions:

$\frac{16}{12} \times \frac{18}{14} = \frac{16 \times 18}{12 \times 14}$

Simplify the multiplication:

• Numerator: $16 \times 18 = 288$
• Denominator: $12 \times 14 = 168$

The resulting fraction is $\frac{288}{168}$.

Step 3:
Simplify $\frac{288}{168}$.

Find the greatest common divisor (GCD) of 288 and 168, which is 24.

• Divide numerator by GCD: $288 \div 24 = 12$
• Divide denominator by GCD: $168 \div 24 = 7$

The simplified fraction is $\frac{12}{7}$.

Convert $\frac{12}{7}$ back to a mixed number:

12 divided by 7 is 1 with a remainder of 5, so the mixed number is $1\frac{5}{7}$.

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

To solve this problem, we will follow these steps:

• Convert the mixed numbers to improper fractions.
• Simplify the fractions when possible.
• Multiply the improper fractions.
• Convert the resulting improper fraction back to a mixed number and simplify.

Here's the step-by-step solution:

Step 1: Convert $3 \frac{6}{9}$ and $3 \frac{4}{20}$ to improper fractions.

For $3 \frac{6}{9}$, which can be simplified to $3 \frac{2}{3}$: $3 + \frac{6}{9} = 3 + \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}$.

For $3 \frac{4}{20}$, which can be simplified to $3 \frac{1}{5}$: $3 + \frac{4}{20} = 3 + \frac{1}{5} = \frac{3 \times 5 + 1}{5} = \frac{15 + 1}{5} = \frac{16}{5}$.

Step 2: Simplify fractions if possible. The fractions $\frac{11}{3}$ and $\frac{16}{5}$ are already in simplest form.

Step 3: Multiply the fractions:

$\frac{11}{3} \times \frac{16}{5} = \frac{11 \times 16}{3 \times 5} = \frac{176}{15}$.

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

Divide 176 by 15: $176 \div 15 = 11$ remainder $11$. Thus, $\frac{176}{15} = 11 \frac{11}{15}$.

Therefore, the solution to the problem is $11 \frac{11}{15}$, which matches choice 3.

Therefore, the final answer is $11 \frac{11}{15}$.