Mixed Numbers

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 $4\frac{4}{9}$.