Mixed Fractions - Examples, Exercises and Solutions

To divide the whole number 3 by the fraction $\frac{5}{7}$, we follow these steps:

• Step 1: Identify the reciprocal of the fraction. The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$.
• Step 2: Multiply the whole number 3 by this reciprocal.
• Step 3: Perform the multiplication to find the result.

Let's calculate this:
Step 1: The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$.
Step 2: Multiply: $3 \times \frac{7}{5} = \frac{3 \times 7}{5} = \frac{21}{5}$.
Step 3: Convert the improper fraction $\frac{21}{5}$ to a mixed number:

• Divide 21 by 5. It goes 4 times with a remainder of 1.
• The quotient is 4, and the remainder is 1. Therefore, $\frac{21}{5} = 4\frac{1}{5}$.

Thus, the solution to $3 : \frac{5}{7}$ is $4\frac{1}{5}$.

The correct choice among the given answers is: $4\frac{1}{5}$.

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

• Step 1: Express the division as a fraction operation: $1 \div \frac{1}{4}$.
• Step 2: Use the invert-and-multiply rule. Find the reciprocal of $\frac{1}{4}$, which is $4$.
• Step 3: Multiply the whole number by the reciprocal: $1 \times 4 = 4$.

Thus, after performing these operations, we find that the result of the division $1 : \frac{1}{4}$ is $4$.

To solve the expression $2:\frac{2}{3}$, follow these steps:

• Step 1: Rewrite the expression as a division problem:
  This means $2 \div \frac{2}{3}$.
• Step 2: Convert the division to a multiplication by using the reciprocal:
  The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$.
• Step 3: Multiply by the reciprocal:
  $2 \times \frac{3}{2} = \frac{2 \cdot 3}{2 \cdot 1} = \frac{6}{2} = 3$.

Therefore, the solution to the problem is $3$.

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 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 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 problem $3:\frac{3}{4}$, we must perform division of the whole number 3 by the fraction $\frac{3}{4}$. Here are the steps:

• Step 1: Recall the rule for dividing by a fraction. Dividing by $\frac{3}{4}$ is the same as multiplying by its reciprocal, $\frac{4}{3}$.
• Step 2: Rewrite the expression as a multiplication problem: $3 \times \frac{4}{3}$.
• Step 3: Perform the multiplication: $3 \times \frac{4}{3} = \frac{3 \times 4}{3} = \frac{12}{3}$.
• Step 4: Simplify the fraction: $\frac{12}{3} = 4$.

The solution to the division $3:\frac{3}{4}$ is $4$.

To solve the problem $4:\frac{3}{5}$, follow these steps:

• Step 1: Convert the division into a multiplication by using the reciprocal of $\frac{3}{5}$. The reciprocal is $\frac{5}{3}$.
• Step 2: Multiply the whole number 4 by $\frac{5}{3}$:
  $4 \times \frac{5}{3} = \frac{4 \times 5}{3} = \frac{20}{3}$
• Step 3: Simplify $\frac{20}{3}$ to a mixed number:
  Perform the division $20 \div 3$, which gives 6 as a whole number and leaves a remainder of 2.
• Thus, $\frac{20}{3}$ as a mixed number is $6\frac{2}{3}$.

Therefore, the solution to the problem is $6\frac{2}{3}$.

To solve the problem $4 : \frac{4}{7}$, we'll apply the concept of dividing by a fraction:

Step 1: Convert the division to multiplication by the reciprocal of the fraction. The reciprocal of $\frac{4}{7}$ is $\frac{7}{4}$.

Thus, the expression $4 : \frac{4}{7}$ is equivalent to $4 \times \frac{7}{4}$.

Step 2: Perform the multiplication:

$4 \times \frac{7}{4} = \frac{4}{1} \times \frac{7}{4}$

Step 3: Multiply the numerators and the denominators:

$\frac{4 \times 7}{1 \times 4} = \frac{28}{4}$

Step 4: Simplify the fraction:

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

Therefore, the result of dividing $4$ by $\frac{4}{7}$ is $7$.

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 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 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, let's carry out the following steps:

• Step 1: Recognize that the expression $3 : \frac{5}{6}$ represents division, so it becomes $3 \div \frac{5}{6}$.
• Step 2: Use the rule that dividing by a fraction is the same as multiplying by its reciprocal. Thus, convert this to $3 \times \frac{6}{5}$.
• Step 3: Multiply 3 (which can be written as $\frac{3}{1}$) by $\frac{6}{5}$:
  $\frac{3}{1} \times \frac{6}{5} = \frac{3 \times 6}{1 \times 5} = \frac{18}{5}$.
• Step 4: Convert $\frac{18}{5}$ to a mixed number. Divide 18 by 5:
  - 5 goes into 18 three times with a remainder of 3.
  - Therefore, $\frac{18}{5} = 3\frac{3}{5}$.

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

To solve this problem, let's divide $1$ by $\frac{3}{4}$. The solution involves converting the division into a multiplication:

• Step 1: Recognize $\,1:\frac{3}{4}\,$ as the division $\frac{1}{\frac{3}{4}}$.

• Step 2: Convert division into multiplication: $\frac{1}{\frac{3}{4}} = 1 \times \frac{4}{3}$.

• Step 3: Compute the multiplication: $1 \times \frac{4}{3} = \frac{4}{3}$.

• Step 4: Convert $\frac{4}{3}$ into a mixed number: $1\frac{1}{3}$.

Therefore, the solution to the division $1 : \frac{3}{4}$ is $1\frac{1}{3}$

The correct answer is $(1 \frac{1}{3})$.

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 $2\frac{4}{7}$.