Multiplying and Dividing Decimal Fractions by 10, 100, etc.: Division by numbers greater than 100

To solve the problem of dividing $0.4$ by $200$, we can follow the steps outlined below:

• Step 1: Convert the decimal to a fraction.
  $0.4$ is equivalent to $\frac{4}{10}$.
• Step 2: Set up the division as a fraction.
  The division $0.4 \div 200$ can be rewritten as $\frac{4}{10} \div 200$. This is the same as multiplying by the reciprocal of $200$, thus $\frac{4}{10} \times \frac{1}{200}$.
• Step 3: Simplify the fraction.
  Multiply the fractions: $\frac{4 \times 1}{10 \times 200} = \frac{4}{2000}$.
• Step 4: Simplify further if possible.
  Divide the numerator and denominator by the greatest common factor. The greatest common factor of 4 and 2000 is 4.
  Thus, $\frac{4}{2000} = \frac{1}{500}$.
• Step 5: Convert the simplified fraction back to a decimal.
  The fraction $\frac{1}{500}$ in decimal form is $0.002$.

Therefore, the solution to the problem is $0.002$.

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

• Step 1: Convert $0.5$ to a fraction if necessary.
• Step 2: Divide $0.5$ by $500$ using decimal division or treat $0.5$ as $\frac{1}{2}$.
• Step 3: Simplify the result to a standard decimal number.

Now, let's work through each step:
Step 1: The number $0.5$ can be considered as $\frac{1}{2}$. However, dividing directly $0.5 \div 500$ is suited here.
Step 2: Express this division as a multiplication: $0.5 \times \frac{1}{500}$.
Simplifying, we have: $0.5 \times 0.002 = 0.001$ Hence, the computation as division directly: $\frac{0.5}{500} = 0.001$

Step 3: Utilize division to achieve a simpler form, yielding the decimal $0.001$.

Thus, the solution to the problem is clearly established as $0.001$.

To solve the problem of dividing $0.99$ by $330$, we follow these steps:

• Step 1: Recognize that the operation is division, specifically dividing a decimal by a whole number.
• Step 2: Express the decimal division $0.99 \div 330$ as a fraction $\frac{0.99}{330}$.
• Step 3: To facilitate division, multiply both the numerator and the denominator by 100 to eliminate the decimal. Thus, $\frac{0.99}{330} = \frac{99}{33000}$.
• Step 4: Simplify the fraction $\frac{99}{33000}$. Both numbers can be divided by 3:
• Divide 99 by 3 to get 33.
• Divide 33000 by 3 to get 11000.
• Thus, the simplified fraction is $\frac{33}{11000}$.
• Step 5: Perform division on the simplified fraction: 33 divided by 11000 is a direct calculation that results in $0.003$.

Therefore, the quotient of $0.99$ divided by $330$ is $0.003$.

To solve this problem, we need to divide $10.1$ by $101$. The division can be represented as follows:

First, consider the expression $10.1 \div 101$. To simplify this division, express it as a fraction: $\frac{10.1}{101}$.

To deal with the decimal, convert $10.1$ to an equivalent fraction by multiplying both the numerator and the denominator by 10 to eliminate the decimal point:

At this point, you can simplify the fraction. Divide both the numerator and the denominator by 101. Thus:

Evaluating $\frac{1}{10}$ gives us the decimal form of $0.1$.

Therefore, the solution to the problem $10.1 \div 101$ is $0.1$.

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

• Step 1: Understand the division involved by recognizing the division problem $\frac{10.2}{200}$.
• Step 2: Simplify the division by recognizing how division by 200 affects the decimal point.
• Step 3: Perform the calculation step-by-step by treating the division as a decimal operation.

Let's address each step:

Step 1: We need to divide 10.2 by 200. This can be viewed as scaling down 10.2 by a factor of 200.

Step 2: When dividing by a number like 200, one approach is to consider how the decimal point will shift. Each "zero" in the divisor typically implies moving the decimal point in the dividend to the left. Since 200 is effectively 2 followed by two zeros, dividing by 200 means moving the decimal point two places to the left in the number 10.2.

Step-by-step:

1. Take 10.2 as the original number.

2. Move the decimal point two places to the left because you are dividing by 200 (equivalent to dividing by 100 then 2, which means two decimal places move).

3. As you move the decimal point in 10.2 to the left, you start by moving from after the "0" in "10.2", resulting in 0.102, and then move it one more spot to obtain 0.051.

Step 3: Therefore, the solution to the problem is:

$0.051$

To solve this problem, we will follow these steps:

• Understand the context and arithmetic implication of dividing by $300$.
• Apply the division rule for decimal fractions to manage decimal point movements effectively.
• Verify calculated results with given choices.

Step 1: We know that we need to divide $30.3$ by $300$. To do this, observe that:

Step 2: The number $300$ can be written as $3 \times 100$. When we divide by $300$, it's equivalent to:

• First dividing $30.3$ by $3$.
• Then dividing the result by $100$ (which is moving the decimal point two places to the left).

Step 3: Perform the calculation:
- $30.3 \div 3 = 10.1$.
- Now divide $10.1$ by $100$, which effectively moves the decimal two places left, giving $0.101$.

Therefore, the solution to the problem is $0.101$.

To solve the given problem, we need to divide the decimal number $1.5$ by $200$.

• Step 1: Understand the problem setup as $1.5 \div 200$.
• Step 2: Perform the division.

Let's interpret $1.5$ as the fraction $\frac{15}{10}$ to assist with division:

Rewriting the division, we have:

$\frac{1.5}{200} = \frac{\frac{15}{10}}{200} = \frac{15}{2000}$

Step 3: Now we divide $15$ by $2000$.

First, set up the division: $15 \div 2000$. This is equivalent to finding: $\frac{15}{2000}$.

Perform the division plant normally or using a calculator:

1. Start by noting the decimal point position. $15.000 \div 2000$ essentially. Perform one digit at a time if manual.

2. You calculate $\frac{15}{2000} = 0.0075$ accurately. Be critical with decimal placement!

Therefore, the quotient of dividing $1.5$ by $200$ is $0.0075$.

Hence, the correct answer is $0.0075$, which corresponds to choice $2$.

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

• Step 1: Convert the given decimal number to a fraction.
• Step 2: Divide the fraction by the whole number.
• Step 3: Simplify the result and convert it back to a decimal.

Now, let's work through each step:

Step 1: Convert the decimal $1.66$ into a fraction. We can express $1.66$ as $\frac{166}{100}$ since it can be read as "one point sixty-six," which means 166 hundredths.

Step 2: Perform the division $\frac{166}{100}$ divided by $166$. Mathematically, this is expressed as: $\frac{166}{100} \div 166 = \frac{166}{100} \times \frac{1}{166} = \frac{166 \times 1}{100 \times 166}$.

Step 3: Simplify the expression $\frac{166}{166}$ to 1. Thus, we have: $\frac{1}{100} = 0.01$.

Therefore, the solution to the problem is $0.01$.

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

• Step 1: Examine the given division $20.30:203$.

• Step 2: Adjust the division by removing the decimal.

• Step 3: Divide and obtain the quotient.

Now, let's work through each step:

Step 1: We have $20.30 \div 203$, which we need to simplify.

Step 2: Consider $20.30$ as $2030$ because we can shift the decimal to transform it, and similarly divide $2030$ by $20300$ (adding sufficient zeros).

Step 3: Perform the Division:

Calculate how many times $203$ fits into $2030$:

$203 \times~ 10~ gives~ 2030$. This results in $203$ fitting in $10$ times into our transformed division $20300$, indicating quotient $0.1$.

Therefore, the solution to the division is $\boldsymbol{0.1}$.

To solve this problem, we'll directly perform a division of the given numbers $3.9 \div 150$:

• Step 1: Write out the division: $\frac{3.9}{150}$.
• Step 2: Set up the division using long division. To eliminate the decimal in 3.9, multiply both 3.9 and 150 by 10 to work with whole numbers:
  $\frac{39}{1500}$.
• Step 3: Perform the division:

$39 \div 1500 = 0.026$.

Move the decimal back into the appropriate place considering it was shifted during the adjustment to whole numbers.

Therefore, the result of the division $3.9 \div 150$ is $0.026$.