Multiplication of Integers by a Fraction and a Mixed number - Examples, Exercises and Solutions

To solve the problem $6 \times \frac{3}{4}$, we follow these steps:

• Step 1: Express the integer 6 as a fraction: $\frac{6}{1}$.
• Step 2: Multiply the fractions: $\frac{6}{1} \times \frac{3}{4}$.
• Step 3: Multiply the numerators: $6 \times 3 = 18$.
• Step 4: Multiply the denominators: $1 \times 4 = 4$.
• Step 5: Form the resulting fraction: $\frac{18}{4}$.
• Step 6: Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2: $\frac{18}{4} \div \frac{2}{2} = \frac{9}{2}$.
• Step 7: Convert $\frac{9}{2}$ to a mixed number: Divide 9 by 2 to get 4 with a remainder of 1, thus $\frac{9}{2} = 4\frac{1}{2}$.

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

To solve this problem, we'll multiply the whole number 2 by the fraction $\frac{5}{7}$:

• Step 1: Multiply the numerator 5 by the whole number 2:

$2 \times 5 = 10$

• Step 2: Write the result over the original denominator 7:

$\frac{10}{7}$

• Step 3: Convert $\frac{10}{7}$ to a mixed number:

Since 10 divided by 7 is 1 with a remainder of 3, we can express this as:

$1\frac{3}{7}$

Therefore, the solution to the problem is $\textbf{1}\frac{\textbf{3}}{\textbf{7}}$.

To solve this problem, we'll multiply the whole number 4 by the fraction $\frac{2}{3}$ as follows:

• Step 1: Convert the whole number 4 into a fraction. This can be written as $\frac{4}{1}$.
• Step 2: Use fraction multiplication rules: multiply the numerators together and the denominators together.
• Step 3: So, multiply the numerators: $4 \times 2 = 8$.
• Step 4: Multiply the denominators: $1 \times 3 = 3$.
• Step 5: The result is $\frac{8}{3}$.
• Step 6: Since $\frac{8}{3}$ is an improper fraction, convert it to a mixed number.
      $8 \div 3 = 2$ with a remainder of 2.
      Thus, $\frac{8}{3} = 2\frac{2}{3}$.

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

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

• Multiply the numerator of the fraction by the integer.
• Keep the denominator unchanged.
• Convert the resulting improper fraction to a mixed number, if necessary.

Now, let's work through each step:
Step 1: Multiply the numerator of $\frac{1}{2}$, which is $1$, by $3$:
$1 \times 3 = 3$.

Step 2: Write the result over the original denominator:
$\frac{3}{2}$.

Step 3: Convert the improper fraction $\frac{3}{2}$ to a mixed number:
Divide $3$ by $2$. This gives $1$ as the quotient and $1$ as the remainder, so:
$\frac{3}{2} = 1\frac{1}{2}$.

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

To solve this problem, we will start by multiplying the whole number 7 by the fraction $\frac{3}{8}$ using the rule for multiplying a whole number by a fraction.

Calculate the product:

• $7 \times \frac{3}{8} = \frac{7 \times 3}{8}$
• $= \frac{21}{8}$

The fraction $\frac{21}{8}$ is an improper fraction, meaning the numerator is greater than the denominator. To convert it to a mixed number, we divide 21 by 8:

• 21 divided by 8 equals 2 with a remainder of 5.
• This gives us the mixed number: $2\frac{5}{8}$

The remainder becomes the numerator of the fraction part, and the denominator remains the same as in the original fraction.

Therefore, the solution to the problem is $2\frac{5}{8}$.

To solve this mathematical problem, we need to multiply a whole number, 8, with the fraction, $\frac{1}{2}$.

Here are the steps:

• Step 1: Identify the given values. The integer is 8 and the fraction is $\frac{1}{2}$.
• Step 2: Apply the multiplication formula for an integer and a fraction: $a \times \frac{b}{c} = \frac{a \times b}{c}$.
• Step 3: Multiply the integer by the numerator of the fraction: $8 \times 1 = 8$.
• Step 4: Divide the result by the denominator of the fraction: $\frac{8}{2} = 4$.
• Step 5: Simplify the fraction if necessary. In this case, $\frac{8}{2}$ simplifies directly to 4.

Therefore, the multiplication of 8 by $\frac{1}{2}$ is $4$.

In the context of the multiple-choice options provided, the correct answer is choice (4): $4$.

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

• Convert the whole number into a fraction.
• Multiply the fractions.
• Simplify the result.

Now, let's work through each step:

Step 1: Convert the whole number 3 into a fraction:
3 becomes $\frac{3}{1}$.

Step 2: Multiply the fraction $\frac{3}{1}$ by $\frac{6}{7}$:
The numerators are $3 \times 6 = 18$.
The denominators are $1 \times 7 = 7$.
The result is $\frac{18}{7}$.

Step 3: Convert $\frac{18}{7}$ to a mixed number:
Divide the numerator by the denominator: 18 divided by 7 is 2 with a remainder of 4.
Thus, $\frac{18}{7} = 2\frac{4}{7}$.

Therefore, the solution to the problem is $2\frac{4}{7}$.

To solve the problem $7 \times \frac{2}{5}$, we will follow a structured approach:

• Step 1: Multiply the whole number by the numerator of the fraction.
• Step 2: Retain the denominator of the fraction.
• Step 3: Simplify the resulting fraction, if possible.

Let's work through each step:

Step 1: Multiply the whole number by the numerator.
We have $7 \times 2 = 14$.

Step 2: Keep the denominator the same.
The resulting fraction is $\frac{14}{5}$.

Step 3: Convert the improper fraction to a mixed number if possible.
Divide the numerator by the denominator: $14 \div 5 = 2$ with a remainder of $4$.
This results in the mixed number $2\frac{4}{5}$.

Therefore, the solution to the problem $7 \times \frac{2}{5}$ is $2\frac{4}{5}$, which corresponds to choice 3 in the provided options.

To solve the problem of multiplying $8$ by $\frac{5}{9}$, we can follow these steps:

• Step 1: Convert the whole number $8$ into a fraction by expressing it as $\frac{8}{1}$.
• Step 2: Multiply the numerators together: $8 \times 5 = 40$.
• Step 3: Multiply the denominators together: $1 \times 9 = 9$.
• Step 4: Write the product as a fraction: $\frac{40}{9}$.
• Step 5: Convert the improper fraction $\frac{40}{9}$ into a mixed number:
• Divide 40 by 9, which gives 4 (quotient) with a remainder of 4.
• Write the mixed number as $4\frac{4}{9}$.

Therefore, the solution to the multiplication problem $8 \times \frac{5}{9}$ is $4\frac{4}{9}$.

To solve this problem, we'll pursue a step-by-step method:

• Step 1: Simplify the fraction $\frac{8}{12}$. The greatest common divisor of 8 and 12 is 4, so dividing both the numerator and denominator by 4 gives $\frac{2}{3}$.
• Step 2: Multiply the integer 3 by the simplified fraction $\frac{2}{3}$.

Let's proceed with these steps:
Step 1: Simplify $\frac{8}{12}$:
$\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$.

Step 2: Multiply the integer by the fraction:
$3 \times \frac{2}{3} = \frac{3 \times 2}{3} = \frac{6}{3} = 2$.

Thus, the result of the multiplication is $\boxed{2}$.

To solve the problem $10 \times \frac{7}{9}$, we follow these steps:

• Step 1: Multiply the whole number by the numerator. Calculate $10 \times 7 = 70$.
• Step 2: Divide this product by the denominator. Calculate $\frac{70}{9}$.
• Step 3: If the result is an improper fraction, convert it to a mixed number. Divide 70 by 9, which goes 7 times with a remainder of 7. This can be expressed as the mixed number $7\frac{7}{9}$.

Let's work through each step:
Step 1: Multiply 10 by 7 to get 70.
Step 2: Divide 70 by 9 to get 7 remainder 7.
Step 3: The proper whole number from the division is 7, with the remainder over the original fraction denominator giving us the final fraction $\frac{7}{9}$.

Thus, the product $10 \times \frac{7}{9}$ is $7\frac{7}{9}$.

To solve the multiplication of an integer with a fraction, we need to follow these steps:

• Step 1: Multiply the integer 7 by the numerator of the fraction, which is 6.
• Step 2: Keep 8 as the denominator.
• Step 3: Simplify the resulting fraction.
• Step 4: Convert to a mixed number if needed.

Now, let's work through each step:

Step 1: Multiply 7 by 6, which gives us $42$ as the numerator.

Step 2: The denominator remains 8, so we have the fraction $\frac{42}{8}$.

Step 3: Simplify $\frac{42}{8}$ by finding the greatest common divisor (GCD) of 42 and 8. The GCD is 2.

We divide the numerator and the denominator by 2: $\frac{42 \div 2}{8 \div 2} = \frac{21}{4}$.

Step 4: Convert $\frac{21}{4}$ into a mixed number:

Divide 21 by 4, which equals 5 with a remainder of 1. Thus, $\frac{21}{4}$ is equivalent to the mixed number $5\frac{1}{4}$.

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

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: $6\frac{1}{4}$.