Multiplication of Integers by a Fraction and a Mixed number: Multiplying mixed fractions by whole numbers

To solve this mathematical problem, let's follow the outlined approach:

• Step 1: Convert the mixed number to an improper fraction.
• Step 2: Multiply the improper fraction by the whole number.
• Step 3: Simplify the result, if needed.

Let's begin with Step 1 by converting the mixed number $1\frac{3}{4}$ to an improper fraction. To do this, multiply the whole number (1) by the denominator (4) of the fraction, and then add the numerator (3):

$1 \times 4 + 3 = 4 + 3 = 7$

This gives us the improper fraction $\frac{7}{4}$.

Next, proceed to Step 2: Multiply the whole number, 4, by the improper fraction we found:

$4 \times \frac{7}{4} = \frac{4 \times 7}{4} = \frac{28}{4}$

For Step 3, simplify $\frac{28}{4}$:

$\frac{28}{4} = 7$

Thus, the product of the whole number 4 and the mixed number $1\frac{3}{4}$ is $7$.

To solve the problem, let's proceed with the following steps:

• Step 1: Convert the mixed number $2 \frac{1}{3}$ to an improper fraction.
• Step 2: Perform the multiplication with the whole number 2.
• Step 3: Simplify and, if required, convert the result back to a mixed number.

Now, let's work through each step:

Step 1: Convert $2 \frac{1}{3}$ to an Improper Fraction

The mixed number $2 \frac{1}{3}$ can be converted to an improper fraction by multiplying the whole number 2 by the denominator 3 and adding the numerator 1. Thus, $2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$.

Step 2: Multiply by 2

Next, multiply the improper fraction $\frac{7}{3}$ by the whole number 2. Treat 2 as $\frac{2}{1}$ for multiplication: $\frac{7}{3} \times \frac{2}{1} = \frac{7 \times 2}{3 \times 1} = \frac{14}{3}$.

Step 3: Simplify or Convert the Improper Fraction

Finally, convert the improper fraction $\frac{14}{3}$ back to a mixed number, if desired:

• Divide the numerator by the denominator: $14 \div 3 = 4$ with a remainder of 2.
• This results in the mixed number $4\frac{2}{3}$.

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

To solve this problem, let us proceed with the outlined steps:

• Step 1: Convert the mixed number $1\frac{1}{4}$ into an improper fraction. Use the formula: $\frac{ac + b}{c}$, where $a = 1$, $b = 1$, and $c = 4$. Therefore, $1\frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4}$.
• Step 2: Multiply the whole number $5$ by the improper fraction $\frac{5}{4}$:
  $5 \times \frac{5}{4} = \frac{5 \times 5}{4} = \frac{25}{4}$.
• Step 3: Convert the improper fraction $\frac{25}{4}$ back into a mixed number. Divide $25$ by $4$ to get $6$ as the whole number and the remainder is $1$, thus the mixed number is $6\frac{1}{4}$.

Therefore, the result of $5 \times 1\frac{1}{4}$ is $6\frac{1}{4}$.

The correct multiple-choice answer, corresponding to this result, is choice 4: $6\frac{1}{4}$.

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

• Convert the mixed number $1 \frac{1}{3}$ into an improper fraction.
• Multiply the whole number by this improper fraction.

Let's execute these steps:
Step 1: Convert the mixed number $1 \frac{1}{3}$ to an improper fraction.
First, multiply the whole number part by the denominator: $1 \times 3 = 3$.
Then, add the numerator: $3 + 1 = 4$.
So, $1 \frac{1}{3}$ converts to $\frac{4}{3}$.

Step 2: Multiply the whole number $3$ by $\frac{4}{3}$.
The multiplication is performed as follows: $3 \times \frac{4}{3} = \frac{3 \times 4}{3} = \frac{12}{3} = 4$.

Therefore, the solution to the problem is $4$.

To solve the problem $2 \times 2\frac{1}{2}$, we will approach it step by step:

• Step 1: Convert the mixed number $2\frac{1}{2}$ to an improper fraction.
• Step 2: Multiply the improper fraction by the integer $2$.
• Step 3: Simplify the result if necessary.

Let's execute these steps:

Step 1: Convert $2\frac{1}{2}$ to an improper fraction:
The denominator is $2$. Multiply the whole number part, $2$, by the denominator, $2$, and add the numerator, $1$:

$2 \times 2 + 1 = 5$

So, the improper fraction is $\frac{5}{2}$.

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

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

Step 3: Simplify $\frac{10}{2}$:

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

Therefore, the solution to the problem is $5$.

To solve the problem, we will follow these steps:

• Step 1: Convert the mixed number $1\frac{1}{5}$ to an improper fraction.
• Step 2: Multiply this improper fraction by the whole number 4.
• Step 3: Convert the resulting improper fraction back to a mixed number.

Now, let's work through each step:

Step 1: Convert $1\frac{1}{5}$ to an improper fraction.

The mixed number $1\frac{1}{5}$ can be converted as follows:
Multiply the whole number 1 by the denominator 5, and add the numerator 1:
$1 \times 5 + 1 = 6$.
So, $1\frac{1}{5}$ is equivalent to $\frac{6}{5}$.

Step 2: Multiply $4$ by $\frac{6}{5}$.

We perform the multiplication: $4 \times \frac{6}{5} = \frac{4 \times 6}{5} = \frac{24}{5}$.

Step 3: Convert $\frac{24}{5}$ back to a mixed number.

Divide 24 by 5 to find how many whole parts there are:
$24 \div 5 = 4$ with a remainder of 4.
Thus, $\frac{24}{5}$ can be expressed as $4\frac{4}{5}$.

Therefore, the solution to the problem is $4\frac{4}{5}$.

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

• Step 1: Convert the mixed number into an improper fraction.
• Step 2: Multiply the whole number by the improper fraction.
• Step 3: Convert the result back to a mixed number if needed and simplify.

Now, let's work through each step:

Step 1: Convert the mixed number.
The mixed number given is $1\frac{1}{2}$.
To convert $1\frac{1}{2}$ into an improper fraction, multiply the whole number part (1) by the denominator (2) and add the numerator (1):
$1 \times 2 + 1 = 3$.
So, $1\frac{1}{2}$ converts to $\frac{3}{2}$.

Step 2: Multiply the whole number by the improper fraction.
Now we multiply the whole number 3 by the improper fraction $\frac{3}{2}$:
$3 \times \frac{3}{2} = \frac{3 \times 3}{2} = \frac{9}{2}$.

Step 3: Convert the result to a mixed number.
The improper fraction $\frac{9}{2}$ can be converted into a mixed number by performing the division $9 \div 2$.
9 divided by 2 equals 4 with a remainder of 1, so $\frac{9}{2}$ is the same as $4\frac{1}{2}$.

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

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

• Step 1: Convert the mixed number $3\frac{2}{3}$ into an improper fraction. The whole number part is 3, and the fractional part is $\frac{2}{3}$. Use the formula: $\text{Improper Fraction} = \frac{(3 \times 3) + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}.$
• Step 2: Multiply the improper fraction $\frac{11}{3}$ by the whole number 4: $4 \times \frac{11}{3} = \frac{4 \times 11}{3} = \frac{44}{3}.$
• Step 3: Convert the improper fraction $\frac{44}{3}$ back to a mixed number. Divide 44 by 3: - 44 divided by 3 is 14 with a remainder of 2. - Therefore, $\frac{44}{3} = 14\frac{2}{3}$.

Thus, the product of $4$ and $3\frac{2}{3}$ is $14\frac{2}{3}$.

The correct answer is $\boxed{14\frac{2}{3}}$.

To solve the problem of multiplying the whole number 4 by the mixed fraction $2\frac{2}{7}$, we will follow these steps:

• Step 1: Convert the mixed fraction to an improper fraction.
• Step 2: Multiply the improper fraction by the whole number.
• Step 3: Simplify or convert the result back to a mixed fraction.

Step 1: The mixed fraction $2\frac{2}{7}$ can be converted into an improper fraction as follows:
Calculate: $2 \times 7 + 2 = 14 + 2 = 16$.
So, $2\frac{2}{7} = \frac{16}{7}$.

Step 2: Multiply the whole number 4 by the improper fraction $\frac{16}{7}$:
$4 \times \frac{16}{7} = \frac{4 \times 16}{7} = \frac{64}{7}$.

Step 3: Convert the improper fraction $\frac{64}{7}$ back into a mixed number:
Divide 64 by 7. The quotient is 9 and the remainder is 1, thus $\frac{64}{7} = 9\frac{1}{7}$.

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

To solve the problem, we need to multiply the whole number 3 by the mixed number $3 \frac{3}{8}$. We will proceed with the following steps:

• Step 1: Convert $3 \frac{3}{8}$ into an improper fraction.

• Step 2: Multiply the converted improper fraction by 3.

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

Now, let's work through each step.
Step 1: Convert $3 \frac{3}{8}$ to an improper fraction:
The mixed number $3 \frac{3}{8}$ can be expressed as $\frac{8 \times 3 + 3}{8} = \frac{24 + 3}{8} = \frac{27}{8}$.

Step 2: Multiply by the whole number 3:
$\frac{27}{8} \times 3 = \frac{27 \times 3}{8} = \frac{81}{8}$

Step 3: Convert the improper fraction back into a mixed number:
Divide 81 by 8. The quotient is 10, and the remainder is 1.
Thus, $\frac{81}{8} = 10 \frac{1}{8}$.

Therefore, the solution to the problem is $10\frac{1}{8}$, corresponding to choice 3.

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

• Step 1: Convert the mixed number $3\frac{7}{9}$ to an improper fraction.
• Step 2: Multiply the improper fraction by the whole number 4.
• Step 3: Simplify the result back to a mixed number.

Now, let's work through each step:
Step 1: Convert $3\frac{7}{9}$ to an improper fraction:
The denominator is 9. Multiply the whole number 3 by 9 and add the numerator 7:
$3 \times 9 + 7 = 27 + 7 = 34$.
So, $3\frac{7}{9} = \frac{34}{9}$.

Step 2: Multiply $\frac{34}{9}$ by 4:
$4 \times \frac{34}{9} = \frac{4 \times 34}{9} = \frac{136}{9}$.

Step 3: Simplify $\frac{136}{9}$ to a mixed number:
Divide 136 by 9. The quotient is 15, and the remainder is 1. Thus, $\frac{136}{9} = 15\frac{1}{9}$.

Therefore, the solution to the problem is $15\frac{1}{9}$.

To solve the multiplication of a whole number by a mixed fraction, follow these steps:

• Step 1: Convert the mixed number $6\frac{5}{9}$ to an improper fraction.
• Step 2: Multiply the improper fraction by the whole number $2$.
• Step 3: Simplify the resulting improper fraction back to a mixed number, if necessary.

Now, let's perform these steps:

Step 1: Convert $6\frac{5}{9}$ to an improper fraction.
The mixed number $6\frac{5}{9}$ can be expressed as $\frac{6 \times 9 + 5}{9} = \frac{54 + 5}{9} = \frac{59}{9}$.

Step 2: Multiply the whole number $2$ by the improper fraction $\frac{59}{9}$.
This results in the multiplication $2 \times \frac{59}{9} = \frac{2 \times 59}{9} = \frac{118}{9}$.

Step 3: Convert back to the mixed fraction.
To convert the improper fraction $\frac{118}{9}$ back to a mixed number, divide 118 by 9:

• 118 divided by 9 is 13 with a remainder of 1.
• Thus, the mixed number is $13\frac{1}{9}$.

Finally, the solution to the problem is $13\frac{1}{9}$.