All Operations in Decimal Fractions: Solving the problem

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

• Align the numbers by their decimal points.
• Add the numbers starting from the rightmost column.
• Ensure each column's sum does not exceed ten; if it does, carry over to the next column.

Now, let's work through each step:
Step 1: Align the decimals: 0.61 + 0.20
Step 2: Add the hundredths place: $1 + 0 = 1$.
Step 3: Add the tenths place: $6 + 2 = 8$.
Step 4: Add the units (whole number) place: There are no whole numbers to add here, so the sum remains as is.

Therefore, the solution to the problem is $0.81$.

To solve this problem, we will add the decimal numbers 0.75 and 0.35 by aligning them according to their decimal points:

• Step 1: Align the numbers vertically by their decimal points:

  $\begin{array}{c} \phantom{0}\\ \phantom{0} \quad 0.75 \\ + \quad 0.35 \\ \hline \end{array}$

• Step 2: Add the digits starting from the rightmost column (hundredths place):

  - Hundredths place: $5 + 5 = 10$. Write 0 and carry over 1 to the tenths place.
  - Tenths place: $7 + 3 + 1 = 11$. Write 1 and carry over 1 to the units place.
  - Units place: $0 + 0 + 1 = 1$.

• Step 3: Write the final sum by placing the decimal point correctly below the decimal points of the addends:

  $\begin{array}{c} 0.75\\ + 0.35\\ \hline 1.10\\ \end{array}$

The sum of 0.75 and 0.35 is therefore $\textbf{1.10}$, which can be simplified to $\textbf{1.1}$ by removing the trailing zero after the decimal point.

The correct answer is, therefore, $\textbf{1.1}$, corresponding to choice $3$.

To avoid confusion in solving the exercise, we will first add the digit 0 to the number 12.0 as follows:

$12.00\\+29.31\\$

Now let's solve the exercise in order:

Let's add the hundreds after the decimal point:

$0+1=1$

Let's add the tenths after the decimal point:

$0+3=3$

Finally, let's add the whole numbers before the decimal point:

$2+9=11$

We'll add ten to the ones digit before the decimal point and get:

$1+1+2=4$

And we obtain the following:

$12.00\\+29.31\\41.31$

To solve the division problem $2.6 \div 0.1$, we can simplify the division by eliminating the decimal in the divisor:

• Multiply both the dividend $2.6$ and the divisor $0.1$ by 10 to get rid of the decimal in the divisor. This gives $2.6 \times 10 = 26$ and $0.1 \times 10 = 1$.
• Now, the division simplifies to $26 \div 1$.
• Dividing $26$ by $1$ gives us $26$.

Thus, the solution to the division problem $2.6 \div 0.1$ is $26$.

Let's solve the following exercise in the correct order.

First, we'll proceed to subtract the tenths after the decimal point:

$1-0=1$

Then, we'll subtract the whole numbers before the decimal point accordingly:

$2-2=0$

$2-1=1$

We obtain the following result:

$22.1\\-12.0\\10.1$

Let's solve the exercise in order:

We'll first subtract the hundreds after the decimal point:

$0-0=0$

Next, we'll subtract the tenths after the decimal point:

$1-1=0$

Finally, we'll subtract the whole numbers before the decimal point accordingly:

$3-1=2$

$3-1=2$

Therefore:

$33.10\\-11.10\\22.00$

To avoid confusion in solving the exercise, we will first add two zeros to the number 210.1 as follows:

$321.301\\+210.100\\$

Now let's solve the exercise in order:

Let's connect the thousandths after the decimal point:

$1+0=1$

Let's connect the hundredths after the decimal point:

$0+0=0$

Let's connect the tenths after the decimal point:

$3+1=4$

Finally, let's connect the whole numbers before the decimal point accordingly:

$1+0=1$

$2+1=3$

$3+2=5$

And we obtain the following:

$321.301\\+210.100\\531.401$

To avoid confusion in solving the exercise, we'll first add two zeros to the number 32.1 as follows:

$32.100\\+37.032\\$

Now let's solve the exercise in order:

Let's multiply the thousandths after the decimal point:

$0+2=2$

Let's multiply the hundredths after the decimal point:

$0+3=3$

Let's multiply the tenths after the decimal point:

$1+0=1$

Finally, let's multiply the whole numbers before the decimal point accordingly:

$2+7=9$

$3+3=6$

And we obtain the following:

$32.100\\+37.032\\69.132$

To solve the problem $0.1 \div 0.02$, let's follow these steps:

• Step 1: Eliminate the decimals by multiplying both the dividend and divisor by 100 to convert them to whole numbers.
• Step 2: Convert $0.1$ to $10$ and $0.02$ to $2$ by multiplication.
• Step 3: Perform the division $10 \div 2$.

Now, let's proceed with the calculations:
- When we multiply $0.1$ by 100, we get $10$. Similarly, multiplying $0.02$ by 100 gives us $2$.
- Next, we divide $10$ by $2$:
$10 \div 2 = 5$.

Therefore, the solution to the problem is $5$.

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

• Step 1: Convert $0.3$ (the divisor) into a whole number by multiplying both $0.18$ and $0.3$ by 10.
• Step 2: Perform the division with whole numbers.

Let's perform the steps:

Step 1: Multiply both $0.18$ and $0.3$ by 10 to get:

$(0.18 \times 10) = 1.8$

$(0.3 \times 10) = 3$

Step 2: Now divide the whole numbers:

$\frac{1.8}{3} = 0.6$

Thus, the division of $0.18$ by $0.3$ results in $0.6$.

To solve the division problem $0.36 \div 0.06$, we'll follow these steps:

• Step 1: Eliminate the decimals:

Both $0.36$ and $0.06$ are decimal numbers. In order to make the division easier, multiply both numbers by 100 to remove the decimals:

• $0.36 \times 100 = 36$
• $0.06 \times 100 = 6$

Converting the problem, we now have $36 \div 6$.

• Step 2: Perform the division:

Now divide $36$ by $6$:

$36 \div 6 = 6$

Therefore, the solution to the problem $0.36 \div 0.06$ is $6$.

To solve the problem $0.4 \div 0.2$, we can follow these steps:

• Step 1: Convert the division of decimals into a division of whole numbers by multiplying both the dividend (0.4) and the divisor (0.2) by 10.
• Step 2: This operation results in dividing 4 by 2 (since $0.4 \times 10 = 4$ and $0.2 \times 10 = 2$).
• Step 3: Perform the division $4 \div 2$, which equals 2.

Thus, the result of dividing $0.4$ by $0.2$ is $\boxed{2}$.

To solve the division of two decimal numbers $0.49$ and $0.7$, we will follow these steps:

• Step 1: Convert the decimals to fractions. Since $0.49$ has two decimal places, it can be written as $\frac{49}{100}$. Similarly, $0.7$, which has one decimal place, can be written as $\frac{7}{10}$.
• Step 2: Set up the division of fractions: $\frac{49}{100} \div \frac{7}{10}$.
• Step 3: Recall that division by a fraction is the same as multiplication by its reciprocal. Thus, we rewrite the division as multiplication by the reciprocal of $\frac{7}{10}$:

$\frac{49}{100} \times \frac{10}{7}$.

• Step 4: Multiply the numerators and the denominators:

$\frac{49 \times 10}{100 \times 7} = \frac{490}{700}$.

• Step 5: Simplify the fraction by finding the greatest common divisor (GCD) of $490$ and $700$. The GCD is $70$.

After dividing both the numerator and the denominator by $70$, we have:

$\frac{490 \div 70}{700 \div 70} = \frac{7}{10}$.

Thus, the result of $0.49 \div 0.7$ is $\frac{7}{10}$, which is $0.7$.

Therefore, the solution to the problem is $0.7$.

To solve the problem of dividing $0.64$ by $0.8$, we need to convert the decimals to whole numbers to simplify the process. Here's a step-by-step solution:

• Step 1: Notice both numbers are decimals. Our goal is to make them whole numbers to simplify the division.
• Step 2: The number $0.64$ can be written as $\frac{64}{100}$ and $0.8$ as $\frac{8}{10}$. To eliminate decimals, multiply both numbers by 100 (the larger factor of 10 to ensure both numbers become integers): $0.64 \times 100 = 64$ and $0.8 \times 100 = 80$.
• Step 3: Perform the division using these whole numbers: $\frac{64}{80}$.
• Step 4: Simplify $\frac{64}{80}$. The greatest common divisor of 64 and 80 is 16. Thus, dividing both the numerator and the denominator by 16, we get $\frac{64 \div 16}{80 \div 16} = \frac{4}{5}$.
• Step 5: Convert $\frac{4}{5}$ to decimal form. To do this, divide 4 by 5, which gives us $0.8$.

Therefore, the solution to the problem $0.64 \div 0.8$ is $0.8$.

To solve this problem, we will follow these steps:

• Step 1: Remove the decimals by multiplying both the dividend and the divisor by the same power of 10.
• Step 2: Perform the division on the resulting whole numbers.
• Step 3: Simplify and verify the result against the multiple-choice options.

Now, let's work through the solution:

Step 1: Multiply both 15.5 and 0.5 by 10 to eliminate the decimals.
$15.5 \times 10 = 155$
$0.5 \times 10 = 5$

Step 2: Divide the resulting whole numbers:
$155 \div 5 = 31$

Step 3: We check the result against the choices. The result $31$ corresponds to choice 1.

Therefore, the solution to the problem is $31$.

To solve the problem $1.6 \div 0.2$, we will eliminate the decimals by converting the numbers into whole numbers and performing the division.

• Step 1: Convert the decimals to whole numbers. Multiply both 1.6 and 0.2 by 10 to remove the decimal places.
  $1.6 \times 10 = 16$ and $0.2 \times 10 = 2$.
• Step 2: Perform integer division. Now divide the two whole numbers obtained:
  $16 \div 2 = 8$.
• Step 3: Verify the result. Verify by direct division using decimals:
  Perform $1.6 \div 0.2$ which is equivalent to $\frac{16}{2} = 8$.

The calculation confirms that our approach is correct, ensuring accuracy both in removing decimals and dividing.

Therefore, the solution to $1.6 \div 0.2$ is $8$.

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

• Step 1: Eliminate decimals by multiplying both numbers by 10.
• Step 2: Perform integer division.
• Step 3: Evaluate and verify the final answer.

Now, let's work through each step:
Step 1: Multiply both numbers $2.4$ and $1.2$ by 10 to eliminate the decimal points. This gives us $24$ and $12$, respectively.
Step 2: Divide the integers: $24 \div 12 = 2$.
Step 3: Verify: Multiplying the result $2$ by $1.2$ should give us $2.4$, confirming the division is correct.

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

Let's solve the following exercise in the correct order:

Subtract the hundredths after the decimal point:

$9-3=6$

Subtract the tenths after the decimal point:

$2-1=1$

Finally, we'll subtract the whole numbers before the decimal point accordingly:

$3-2=1$

$1-1=0$

We'll obtain the following result:

$13.29\\-12.13\\01.16$

Let's solve the following exercise in the correct order:

Add up the hundredths after the decimal point:

$8+1=9$

Add up the tenths after the decimal point:

$3+4=7$

Finally, we'll add up the whole numbers before the decimal point accordingly:

$1+3=4$

$2+1=3$

We obtain the following result:

$21.38\\+13.41\\34.79$

Let's solve the exercise in order:

Let's subtract the hundreds after the decimal point

$9-5=4$

Let's subtract the tenths after the decimal point:

$2-3=$

Since we cannot subtract, we'll borrow ten from the tens digit of the whole number and get:

$12-3=9$

Finally, let's subtract the whole numbers before the decimal point accordingly.

Since we borrowed ten from the tens digit of the whole number we get:

$6-1-3=2$

$1-1=0$

And we get:

$16.29\\-13.35\\02.94$

We can ignore the 0 before the 2, so we got the number: 2.94