Division of Fractions: Using decimal fractions

To solve the problem $\frac{99.9}{33.3}$, we will simplify it by transforming each decimal into a whole number to make calculation straightforward.

• Step 1: Multiply both the numerator and the denominator by $10$ to eliminate decimals. This gives us:
  $\frac{99.9 \times 10}{33.3 \times 10} = \frac{999}{333}$.
• Step 2: Notice that both $999$ and $333$ can be divided by $3$ (a common factor). Divide each by $3$:
  $\frac{999 \div 3}{333 \div 3} = \frac{333}{111}$.
• Step 3: Notice again that $333$ and $111$ can be simplified further by dividing each by $111$ (another common factor).
  $\frac{333 \div 111}{111 \div 111} = \frac{3}{1} = 3$.

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

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

• Step 1: Understand the rule of division by 10

• Step 2: Calculate by shifting the decimal point to the left

Now, let's work through each step:
Step 1: When dividing a number by 10, we shift the decimal point one position to the left.
Step 2: In the given fraction $\frac{69.5}{10}$, we move the decimal point in 69.5 one place to the left, resulting in 6.95.

Therefore, the solution to the problem is $6.95$, which corresponds to choice 2.

To solve the division $\frac{97.2}{48.6}$, we will simplify by removing the decimals:

• Step 1: Multiply both the numerator and denominator by 10 to eliminate the decimals since 97.2 and 48.6 each have one decimal place.
• Step 2: The problem becomes equivalent to $\frac{972}{486}$.
• Step 3: Simplifying $\frac{972}{486}$ involves performing the division 972 ÷ 486.

By performing the division directly, we note that:

• 972 divided by 486 equals 2 because 972 is exactly double 486.

Therefore, the solution to the problem is $2$.

To solve the expression $\frac{6}{0.75} - 2 \times 3$, we need to carefully follow the order of operations. The order of operations is often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Here, there are no parentheses or exponents, so we focus on multiplication, division, subtraction in the given order.

• Step 1: Division
  First, perform the division: $\frac{6}{0.75}$. To divide by a decimal, convert it to a fraction or adjust the dividend and divisor by a power of 10 to make the divisor a whole number. Here, $\frac{6}{0.75}$ becomes $\frac{6 \times 100}{0.75 \times 100} = \frac{600}{75}$.
  Next, simplify $\frac{600}{75}$. Both numbers are divisible by 15:
  $600 \div 15 = 40$ and $75 \div 15 = 5$, so $\frac{600}{75} = 40 \div 5 = 8$.
• Step 2: Multiplication
  Next, perform the multiplication: $2 \times 3$ which equals $6$.
• Step 3: Subtraction
  Finally, subtract the results of the previous operations: $8 - 6$. This gives $2$.

By carefully following the order of operations, the final answer to the expression $\frac{6}{0.75} - 2 \times 3$ is $2$, which matches the correct answer provided.

To solve the problem $\frac{11.3}{3.8}$, follow these steps:

• Step 1: Convert 11.3 and 3.8 into fractions.
• Step 2: Conduct the division using these fractions.
• Step 3: Simplify and convert to a mixed number if appropriate.

Step 1: Convert 11.3 and 3.8 to fractions:
$11.3 = \frac{113}{10}$ since the decimal point signifies tenths place.
$3.8 = \frac{38}{10}$

Step 2: Use the rule for fraction division, which is to multiply by the reciprocal:
$\frac{113}{10} \div \frac{38}{10} = \frac{113}{10} \times \frac{10}{38} = \frac{113}{38}$.

Step 3: Simplify $\frac{113}{38}$ by performing the division:
113 divided by 38 gives a quotient of 2 and a remainder of 37.
Thus, $\frac{113}{38} = 2\frac{37}{38}$.

Therefore, the division of $\frac{11.3}{3.8}$ results in $2\frac{37}{38}$.

Therefore, the solution to $\frac{11.3}{3.8}$ is $\mathbf{2\frac{37}{38}}$, which matches the second choice provided.

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

• Step 1: Convert each decimal number into a fraction
• Step 2: Perform the division of these fractions using multiplication by the reciprocal
• Step 3: Simplify the resulting fraction

Now, let's work through each step:

Step 1: Convert the decimals to fractions.

For the decimal 2.4, notice it is equal to 24 divided by 10, so it can be written as the fraction $\frac{24}{10}$.

For the decimal 5.1, it is equal to 51 divided by 10, so it can be written as the fraction $\frac{51}{10}$.

Step 2: Divide the fractions by multiplying by the reciprocal.

We have the division: $\frac{24}{10} \div \frac{51}{10}$.

Instead of dividing, we multiply by the reciprocal: $\frac{24}{10} \times \frac{10}{51}$.

When multiplying, we multiply the numerators and the denominators:

$\frac{24 \times 10}{10 \times 51} = \frac{240}{510}$.

Step 3: Simplify the resulting fraction.

Both 240 and 510 are divisible by 30:

$\frac{240 \div 30}{510 \div 30} = \frac{8}{17}$.

Thus, the fraction simplifies to $\frac{8}{17}$.

Therefore, the solution to the problem is $\frac{8}{17}$.