Addition and Subtraction of Decimal Fractions: Vertical operations

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

• Step 1: Align the decimal numbers by their decimal points.
• Step 2: Check if each digit is correctly aligned with its corresponding digit based on place value.
• Step 3: Ensure the operation symbol is properly placed.

Now, let's work through each step:
Step 1: Start with the number 13.45 and align 3.21 directly below it such that the decimal points are vertically aligned. This ensures that the tenths, hundredths, and whole numbers are in the correct columns.
Step 2: Verify that: - The '1' in 13.45 is in the tens place, and the '3' in 3.21 is in the ones place, both aligned left of the decimal. - The '3' in 13.45 and '2' in 3.21 are aligned in the tenths column. - The '4' in 13.45 and '1' in 3.21 are in the hundredths column.
Step 3: Place the '+' sign outside and to the left, in line with the numbers, ensuring it is clearly indicating addition.

Therefore, the correct alignment for the addition of these decimal numbers is:

To correctly set up a vertical addition of decimal fractions, it is crucial to align the numbers by their decimal points. Let's break down the choices:

• Choice 1 presents the numbers in the proper format. The decimal points of $15.5604$ and $4.32$ align vertically, ensuring accurate addition.

• Choice 2 misaligns the decimal points, incorrectly starting $4.32$ further to the left than necessary.

• Choice 3 also misaligns the decimals, wanting to begin $4.32$ even further than choice 2.

Thus, the correct choice is Choice 1, since it correctly aligns the numbers by their decimal points, facilitating accurate addition of the values involved.

Therefore, the correct format choice is Choice 1.

To solve this problem, we need to verify if the numbers are properly aligned for subtraction:

• Step 1: Identify the given numbers. We have the numbers 16.234 and 5.35 written in a vertical format.
• Step 2: Check if decimal points are aligned. For proper subtraction of decimal numbers, the decimal points in both numbers should line up vertically.
• Step 3: Align digits relative to their place values. We need to compare the positioning of digits before and after the decimal point to confirm correct alignment.

Now, let's apply these steps:
Step 1: We have the numbers $16.234$ and $5.35$.
Step 2: Inspect the vertical alignment of the decimal points. We notice that $16.234$ has three digits after the decimal, while $5.35$ has only two digits.
Step 3: Evaluate the alignment. The decimal point in $5.35$ is not properly aligned because it does not have the same number of decimal places as $16.234$.

Therefore, the format is not correct for vertical subtraction, as the alignment of decimal points and digits is incorrect. Hence, the correct answer is No.

When setting up a subtraction problem with decimals, it is crucial that the decimal points in both numbers directly align. This ensures that each digit is correctly placed in its respective place value column (units, tenths, hundredths, etc.) to facilitate accurate subtraction.

In this problem, the number $25.754$ is aligned vertically with $3.13$. Upon inspection, both decimals are indeed arranged vertically so that the decimal points are in the same column. The digits following the decimal in $3.13$ are tenths and hundredths, leaving the thousandths position empty, which is acceptable as we can treat the missing place value as zero for alignment purposes (i.e., treat $3.13$ as $3.130$).

Therefore, the numbers are set up correctly for vertical subtraction, where:

This alignment confirms that the subtraction operation can reliably proceed as the format is correct.

Therefore, the correct answer to the problem is Yes.

To determine if the addition of decimals is set up correctly, follow these steps:

• Step 1: Check the alignment of the decimal points in both numbers.
• Step 2: Ensure each corresponding place value (units, tenths, hundredths, etc.) is aligned vertically.
• Step 3: Look at the layout provided:
  $\begin{array}{c} 7.462 \\ +\phantom{.}2.13 \\ \hline \end{array}$

The first number, $7.462$, has three decimal places, whereas the second number, $2.13$, has two decimal places. The decimal point in $2.13$ should be directly below the decimal point in $7.462$. However, it appears that the digits in the tenths and hundredths place of $2.13$ are not properly aligned with $7.462$. Hence, the addition is not aligned correctly as the decimal points are not vertically aligned.

Therefore, the addition layout is incorrect, and the solution to the problem is:

No

To solve this problem, let's perform the addition of the two decimal numbers $0.1$ and $0.2$ step-by-step.

Step 1: Align the decimal points of the numbers.

We write the numbers as:

$0.1$

$+0.2$

Here, the decimal points are already aligned.

Step 2: Add the numbers column by column starting from the rightmost side, which is the decimal fraction part.

$0.1$ consists of 1 in the tenths place, and $0.2$ consists of 2 in the tenths place. By adding these:

• $0.1 + 0.2 = 0.3$

Thus, the sum of $0.1$ and $0.2$ is $0.3$.

Looking at the answer choices, the correct choice is:

• Choice 2: $0.3$

Therefore, the solution to the problem is $0.3$.

To solve the decimal addition $0.4 + 0.3$, we will follow these steps:

• Step 1: Line up the numbers by the decimal point.

• Step 2: Add the decimal fractions directly as both numbers have only one digit after the decimal point.

• Step 3: Sum the values and retain the decimal point in the same position for the result.

Let's begin the calculation:

Write the numbers aligning them by the decimal point:

$0.4$
$+0.3$
_______
$0.7$

Adding the two numbers $0.4$ and $0.3$ gives $0.7$.

Therefore, the solution to the problem is $0.7$, which matches choice 1.

To solve this problem, let's follow these steps to perform decimal addition:

• Step 1: Align the decimal points of numbers $1.3$ and $1.1$.

• Step 2: Add the numbers starting from the right, moving towards the left.

Now, let's perform the calculation:

Align the numbers:
$\begin{aligned} &{1.}3 \\ +&1.1 \\ \hline \end{aligned}$

Step 1: Add the digits in the tenths place: $3 + 1 = 4$.

Step 2: Add the digits in the units place: $1 + 1 = 2$.

Combine these results to find the total sum: $2.4$.

Therefore, the solution to the problem is $2.4$, which corresponds to choice 4.

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

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

• Step 2: Add the numbers starting from the rightmost digit, moving left.

• Step 3: Handle any carryovers properly.

Now, let's work through each step:

Step 1: Align the numbers:

  10.03
+    5.51

Step 2: Start adding from the rightmost digit (hundredths):
$\begin{array}{c} & 10.03\\ + & \,5.51\\ \hline & 15.54\\ \end{array}$

Step 3: Add the digits from right to left:
- Hundredths place: $3 + 1 = 4$
- Tenths place: $0 + 5 = 5$
- Whole numbers: $0 + 5 = 5$, $1 + 0 = 1$

Thus, the sum of $10.03$ and $5.51$ is $\textbf{15.54}$.

After evaluating the answer choices, the correct answer is choice 3: $\textbf{15.54}$.

To solve the addition problem $11.32 + 10.36$, we follow a straightforward procedure:

• Step 1: Align the Numbers
  Write the numbers one on top of the other, aligning by the decimal point:
  $\begin{array}{c} 11.32 \\ + 10.36 \\ \hline \end{array}$
• Step 2: Add Each Column
  Start adding from the rightmost column and move left to the next columns, carrying over if necessary:
  • Add the hundredths place: $2 + 6 = 8$
  • Add the tenths place: $3 + 3 = 6$
  • Add the units place: $1 + 0 = 1$
  • Add the tens place: $1 + 1 = 2$
• Step 3: Write the Result
  After adding all the columns, write down the result with the decimal point aligned:
  $\begin{array}{c} 21.68 \\ \hline \end{array}$

Therefore, the solution to the problem is $21.68$.

To solve this problem, follow these steps:

• Identify the given numbers: $5.3$ and $2.45$.
• Align them by the decimal point:

  5.30
+ 2.45
------

Note that adding a zero to the right of $5.3$ makes it $5.30$, which does not change its value but ensures alignment.

Next, perform the addition starting from the rightmost digit:

• Hundredths place: $0 + 5 = 5$
• Tenths place: $3 + 4 = 7$
• Units place: $5 + 2 = 7$

Thus, the sum is:

  5.30
+ 2.45
------
  7.75

The sum of $5.3$ and $2.45$ is $7.75$.

Therefore, the correct choice corresponding to our solution is choice 1: $7.75$.

To find the sum of $2.16$ and $3.214$, follow these steps:

• First, write the numbers vertically, aligning them by their decimal points:

$\phantom{00}2.160$
$+3.214$
$\_\_\_\_\_\_$

• Add the numbers from right to left, adding zeros if necessary to balance the number of decimal places:

• Start from the rightmost digit: $0 + 4 = 4$.

• Next column to the left: $6 + 1 = 7$.

• Next column to the left: $1 + 2 = 3$.

• Finally, add the whole number part: $2 + 3 = 5$.

• Combine the results to get the final sum.

Therefore, the sum of $2.16$ and $3.214$ is $\textbf{5.374}$.

To solve this problem, we'll add the numbers 100 and 10.3.

Let's break down the steps as follows:

• Step 1: Align the numbers vertically by the decimal point. While $100$ is a whole number, it can be written as $100.0$ to facilitate alignment. Therefore, write the numbers as: $\begin{array}{r} 100.0 \\ + 10.3 \\ \hline \end{array}$
• Step 2: Proceed with addition, starting from the rightmost digits. The setup allows addition digit-by-digit:
• Add the tenths place: $0 + 3 = 3$.
• Add the ones place: $0 + 0 = 0$.
• Add the tens place: $0 + 1 = 1$.
• Add the hundreds place: since there's nothing above the 1, it remains $1$.
• Step 3: Carefully maintain the decimal position. The result of this addition will be $110.3$.

Therefore, the sum of $100 + 10.3$ is $110.3$.

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

• Step 1: Align the numbers based on the decimal point.
• Step 2: Perform addition starting from the rightmost digit, carrying over as necessary.
• Step 3: Ensure the decimal point is placed correctly in the final answer.

Now, let's work through each step:

Step 1: Align the numbers vertically:

  6.40
+ 2.56
------

Step 2: Start with the hundredths place (rightmost) and move left.

• Add the hundredths: $0 + 6 = 6$.
• Add the tenths: $4 + 5 = 9$.
• Add the units: $6 + 2 = 8$.

Step 3: Place the decimal point directly below the other decimal points in the final answer.

  6.40
+ 2.56
------
  8.96

Therefore, the solution to the problem is $8.96$.

To solve this problem, follow these steps:

• Step 1: Recognize the numbers to be added are $3$ and $4.57$.

• Step 2: Write $3$ as $3.00$ for easier alignment by the decimal point.

• Step 3: Align the numbers:
        $\phantom{0}3.00$
  $+4.57$

• Step 4: Add the numbers column by column starting from the rightmost column.

Let's perform the addition:

• Rightmost column (hundredths): $0 + 7 = 7$.

• Next column (tenths): $0 + 5 = 5$.

• Next column (units): $3 + 4 = 7$.

The sum of $3$ and $4.57$ is calculated as follows:

First, verify the alignment of the decimals, then add each column:

$3.00 + 4.57 = 7.57$

The calculated sum is $7.57$.

The correct answer from the given choices is: $7.57$, which matches choice 3.

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

• Step 1: Align the decimal numbers vertically, ensuring that the decimal points are perfectly aligned.
• Step 2: Subtract the numbers starting from the rightmost digit (the tenths place in this case).
• Step 3: Write the result directly below, keeping the decimal point aligned in the final answer.

Now, let's work through each step:

Step 1: We have the numbers $15.6$ and $0.3$. Align these two numbers:

  15.6
-  0.3
------

Step 2: Start the subtraction from the rightmost digit. Subtract $0.3$ from $15.6$. The tenths place: $6 - 3 = 3$.

In the units place, we then directly bring down the 15, as there is no need to subtract anything further.

  15.6
-  0.3
------
  15.3

Step 3: Ensure the result is aligned with a decimal point directly below the original decimal points: the result is $15.3$.

Therefore, the solution to the problem is $15.3$.

To solve the problem of $0.98 - 0.36$, we will perform the following steps, ensuring clarity in how decimal subtraction operates:

• Step 1: Align the numbers by their decimal points:
     0.98
  -  0.36
       ------

• Step 2: Subtract from the rightmost column (hundredths place):
  We have 8 minus 6 in the hundredths place, which gives us 2.

• Step 3: Move to the tenths place:
  In the tenths place, we subtract 3 from 9, which gives us 6.

• Step 4: Since there is no borrowing across the decimal, move to the units place. Nothing to subtract from left of the decimals, 0 remains unaffected.
      0.98
  -  0.36
      ------
      0.62

Therefore, the final result of the subtraction $0.98 - 0.36$ is $0.62$.

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

• Align the decimal numbers vertically by placing the decimal points in the same column.
• Subtract each digit from right to left, ensuring to borrow from the next column if necessary.

Let's apply these steps to the problem:

1. Write the numbers in a vertical format:
    101.23
- 100.10
---------

2. Start subtracting from the rightmost column.
- In the hundredths place, subtract $3 - 0$, which equals $3$.
- Move to the tenths place: subtract $2 - 1$, which equals $1$.
- Move to the ones place: $1 - 0$ equals $1$.
- Finally, in the tens place, $0 - 0$ equals $0$, and in the hundreds place, $1 - 1$ equals $0$.

3. Combine the digits we calculated by restoring the decimal point placement to get the result:
    1.13

Therefore, the solution to the problem is $1.13$.

To solve this problem, we will subtract $22.1$ from $56.2$ using the subtraction of decimal fractions method:

Step-by-step solution:

• Step 1: Align the numbers by the decimal point: $\begin{array}{c} \quad ~~~56.2 \\ - \quad 22.1 \\ \hline \end{array}$

• Step 2: Subtract the digits in each column, starting from the right (tenths place): - First, subtract the tenths: $2 - 1 = 1$. - Then, subtract the units: $6 - 2 = 4$. - Finally, subtract the tens: $5 - 2 = 3$.

This gives us the result of $34.1$ after performing the subtraction.

Thus, the solution to the problem $56.2 - 22.1$ is $34.1$.

To solve the problem of finding $47.5 - 3.5$, we will follow these steps:

• Step 1: Write the numbers vertically, aligning them by the decimal point.
• Step 2: Subtract each digit starting from the right (tenths place) and moving left, borrowing where necessary.

Let's perform these steps:

Step 1: Write the numbers aligned by the decimal point:
  $47.5$
-   $3.5$
————

Step 2: Subtract the digits starting from the tenths place:
- Tenths place: $5 - 5 = 0$
- Units place: $7 - 3 = 4$
- Tens place: $4 - 0 = 4$
This results in $44.0$. Since decimal points don't alter when .0 is at the end, $44.0$ is equal to $44$.

Therefore, the solution to the problem is $44$.