Addition and Subtraction of Decimal Fractions: Addition and subtraction with regrouping

To solve the problem $2.2 - 1.5$, follow these steps:

• Step 1: Write down the numbers vertically, ensuring the decimal points are aligned:

$2.2$

$-1.5$

• Step 2: Begin subtraction from the tenths place (rightmost decimal place).

• Step 3: Subtract $5$ from $2$ in the tenths place, which necessitates borrowing:

  • Change $2.2$ to $2.1$ (borrowing 1 from the units place, making it $12$ tenths).

• Step 4: Perform the subtraction:

  • $12 - 5 = 7$ in the tenths place, with a carry back to units (effectively $1$ in the units becomes $0$).

  • $1 - 1 = 0$ in the units place.

Therefore, the result of $2.2 - 1.5$ is $0.7$.

To solve this problem, let's add the decimal numbers $1.6$ and $0.5$ as follows:

• Step 1: Write the numbers $1.6$ and $0.5$ in a column, ensuring that the decimal points are aligned:
  $\begin{array}{c} 1.6 \\ +0.5 \\ \hline \end{array}$
• Step 2: Begin with the rightmost digits (tenths place). Add $6$ (from $1.6$) and $5$ (from $0.5$):
  $6 + 5 = 11$. Write down the $1$ in the tenths place and carry over the other $1$ to the units place.
• Step 3: Move to the units place. Add $1$ (carried over) to $1$ (from $1.6$) and $0$ (from $0.5$):
  $1 + 1 + 0 = 2$.
• Step 4: Combine the sums from Steps 2 and 3 to get the final result.

Thus, the sum of $1.6$ and $0.5$ is $2.1$.

The correct answer is $2.1$.

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

• Step 1: Identify the given numbers 1.5 and 0.7.

• Step 2: Align the decimal points of the numbers.

• Step 3: Subtract the numbers digit by digit from right to left.

Now, let's work through each step:
Step 1: The problem gives us the numbers 1.5 and 0.7, which need to be subtracted from each other.
Step 2: Write them vertically one on top of the other, aligning the decimal points:
$1.5$
$- 0.7$
Step 3: Subtract from right to left starting from the tenths place. In the tenths place, subtract $7$ from $5$, which requires borrowing since $5 < 7$.
- Borrow 1 from the units place of $1.5$, making it $0.5 + 1.0 = 10.5$, and giving the tenths place $15$
- Now we have:
$(1.0 - 0.1)$ in the units, and $15 - 7 = 8$ in the tenths.
our expression becomes 0.8 after subtraction.

Therefore, the solution to the problem is $0.8$.

To solve this problem, we'll perform the following steps:

• Align the numbers by their decimal points:

$1.0$
$- 0.2$
$----$

• Perform the subtraction, starting from the rightmost digit (the tenths place since decimals are aligned):

$0.0$ in the tenths place subtract $0.2$ results in $0.8$.

• In the ones place, there is no further subtraction, simply keeping the placed zero.

Overall, the subtraction $1 - 0.2$ results in $0.8$.

Therefore, the solution to the problem is $0.8$.

To solve this problem, we'll perform the following steps:

• Step 1: Align the decimal points of $20.2$ and $10.4$.

• Step 2: Subtract starting from the rightmost digits (tenths place).

Now, let's work through each step:

Step 1: We align the two numbers with their decimal points:
$20.2$
$-10.4$
_____

Step 2: Subtract the digits from right to left:

• Tenths place: $2 - 4$ can't be done directly, so we need to look at the whole number.

• Units place: We need to borrow from the $20$, making the $0$ a $10$ and reducing the hundreds digit: this gives us $12.0 - 10.4$.

• Now, tenths place calculation $2 - 4 = -2$, but with borrowing from next digit $(10) + (2 - 4) = 8$.

• Units place results into $20 - 10 = 10$. Including borrows, results into $9$.

The final subtraction shows that $20.2 - 10.4 = 9.8$.

Therefore, the solution to the problem is $\mathbf{9.8}$

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

• Step 1: Align the numbers by their decimal points:
  $0.8$
  $+0.4$

• Step 2: Add each corresponding position from right to left:

Adding the tenths place, we have $8 + 4 = 12$. Write down the $2$ and carry over the $1$ to the ones place.

Add the ones place, which includes the carried-over $1$:
$0 + 0 + 1 = 1$.

Combining this, the sum of $0.8$ and $0.4$ is $1.2$.

Therefore, the solution to the problem is $1.2$. This corresponds to choice 2.

To solve this problem, we will add the decimal numbers 20.2 and 8.8. Here is how we proceed:

• Step 1: Align the decimal numbers:

• $\phantom{1}20.2$
  $+$ $\phantom{1}8.8$
  $\overline{\phantom{99.99}}$
  

• Step 2: Add column by column, starting from the right.

  • First, add the tenths: $2 + 8 = 10$. Since 10 is more than 9, write 0 and carry over 1 to the units column.

  • Then, add the units: $0 + 8 = 8$, plus the carry-over 1 makes it 9.

  • Finally, add the tens: $20 + 0 = 20$.

• When combining the results, we get:

  $29$.
  

  Therefore, the sum of $20.2$ and $8.8$ is \( \textbf{$29$} \).

To solve the problem $16.1 - 4.7$, follow these steps:

• Step 1: Align the numbers by their decimal points:
  $\quad 16.1$
  $-\ 4.7$
  ---------------------
  
• Step 2: Subtract from right to left: Start with the decimal part:
  The problem is $16.1 - 4.7$. Since 1 (tenths place of 16.1) is less than 7 (tenths place of 4.7), we need to borrow.
• Step 3: Borrow from the whole number part: Modify the 16 to become 15, and the decimal part becomes 11 ten-thousandths. After getting 15 from 16, the subtraction becomes:
  $\quad 15.11$
  $-\ 4.7$
  ---------------------
  
• Subtract the tenths place: $11 - 7 = 4$. Now subtract the integer part: $15 - 4 = 11$.
• Finalize the result:
  $\quad 11.4$.

Thus, the solution to $16.1 - 4.7$ is $11.4$.

To solve the problem $24.2 - 3.7$, follow these detailed steps:

• Step 1: Align the numbers by their decimal points:
  $\begin{array}{r} 24.2 \\ -3.7 \\ \hline \end{array}$
• Step 2: Subtract the digits from the rightmost side. Starting with the tenths place, $2 - 7$ cannot be done without borrowing. Borrow 1 from the number in the units place (4 becomes 3), making it $12 - 7 = 5$.
• Step 3: Proceed to the whole number place. After borrowing, subtract $3 - 3 = 0$.
• Step 4: Move to the tens place, subtract $2 - 0 = 2$.

Therefore, the result of the subtraction $24.2 - 3.7$ is $20.5$.

To solve the subtraction problem $30.4 - 12.6$, follow these steps:

Step 1: Write the numbers vertically, ensuring that the decimal points are aligned:

$\begin{array}{c} 30.4 \\ - 12.6 \\ \hline \end{array}$

Step 2: Begin subtracting from the rightmost column, which is the tenths place in this case:

• The tenths column: $4 - 6$. Since 4 is less than 6, we need to borrow from the whole numbers to the left. The 0 in the units column of 30 becomes 10 (because we borrow 1, which makes it 10 in decimal terms), and the 3 becomes 2.
• Now, 14 - 6 = 8. Write 8 in the tenths place. The result so far is 0.8.

Step 3: Move to the units place (whole numbers column):

• Subtract $2 - 2 = 0$.
• Write 0 in the units place. The result so far is 0.8.

Step 4: Move to the tens place:

• Subtract $3 - 1 = 1$ (remember we borrowed 1 earlier, hence it is now 2 in place of originally 3, subtracting 1 gives 1).
• Write 1 in the tens place. The final result is $17.8$.

Therefore, the solution to the problem $30.4 - 12.6$ is $17.8$.

To solve this problem, we'll add two decimal numbers $15.6$ and $7.8$.

• Step 1: Align the decimal points of the numbers, writing them one under the other:

  $\begin{array}{c} 15.6 \\ + \ 7.8 \\ \hline \end{array}$

• Step 2: Add the numbers starting from the rightmost digit, which are the tenths after the decimal point:

  $6 + 8 = 14$. Place $4$ under the tenths column and carry over $1$ to the next column (the units column).

• Step 3: Add the numbers in the units column, considering the carry:

  $5 + 7 = 12$; with the carry $1$, it becomes $12 + 1 = 13$. Place $3$ under the units column and carry over $1$ to the next column (the tens column).

• Step 4: Add the numbers in the tens column, considering the carry:

  $1 + 0 + 1 = 2$ (as $15.6$ can be seen as $015.6$, for alignment purpose). Place $2$ under the tens column.

• Step 5: After adding all decimal digits, position the decimal point in the result directly below where the other decimal points are aligned in the original numbers.

Thus, the sum of $15.6 + 7.8$ is $23.4$.

Finally, checking against the answer choices, we find that $\boxed{23.4}$ matches the correct answer choice.

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 right-most column.
• Step 3: Address any carry-over if it occurs.

Now, let's work through each step:
Step 1: Write the numbers $10.6$ and $4.5$ one on top of the other, aligning their decimal points.
Step 2: Start adding from the right-most digit (tenths place in this case): $\begin{array}{c} 10.6 \\ + 4.5 \\ \hline \end{array}$

Add the digits in the tenths column: $6 + 5 = 11$. Write $1$ in the tenths place and carry over $1$ to the ones place.

Continue with the ones column: $0 + 4 + 1$ (carry over) equals $5$.

In the tens column, $1 + 0$ equals $1$.

Putting it all together, the sum is: $\begin{array}{c} 15.1 \\ \end{array}$

Therefore, the solution to the problem is $15.1$.

To solve this problem, we will perform the subtraction of two decimal numbers: $12.3$ and $5.4$.

Here are the steps to follow:

• Align the decimals: Write the numbers $12.3$ and $5.4$ vertically with their decimal points aligned.
• Subtract the numbers digit by digit: Start from the rightmost digit (tenths place).
  $\phantom{x} \, \, 12.3$
  $- \, \, \, 5.4$
  $\rule{1cm}{0.4pt}$
• From the tenths place, $3 - 4$ requires borrowing. Borrow $1$ from the units place of the top number, turning $12.3$ into $12.2$ and making the tenths digit $3+10=13$. Subtract: $13 - 4 = 9$.
• In the units place, $2 - 5$ needs borrowing as well. Borrow from the tens place: change $12$ into $11$ and $2$ becomes $12$. Then subtract: $12 - 5 = 7$.
• Since no borrowing is needed in the tens place, retain $1$ as is.
• Therefore, the final result is $6.9$.

Thus, the result of $12.3 - 5.4$ is $6.9$.

To solve this subtraction problem involving decimals, follow these steps:

• Step 1: Write the numbers $8.4$ and $5.6$, aligning the decimal points.
• Step 2: Begin subtracting from the rightmost digit (tenths).
• Step 3: Perform the subtraction in the tenths column: $4 - 6$. This requires borrowing. Borrow 1 from the whole number 8, leaving it as 7 and converting the 4 tenths into 14 tenths.
• Step 4: Subtract the tenths: $14 - 6 = 8$.
• Step 5: Move to the whole number column. Subtract: $7 - 5 = 2$.

The result of subtracting $5.6$ from $8.4$ is $2.8$.

Therefore, the correct answer is $\boxed{2.8}$.

Let's solve the subtraction problem $27.4 - 10.5$ step-by-step:

Step 1: Align the numbers vertically by the decimal point:

$\begin{aligned} 27.4 \\ -10.5 \\ \hline \end{aligned}$

Step 2: Start subtracting from the rightmost digit, moving to the left.

• In the tenths column (4 - 5), we need to borrow from the ones column.

• Borrowing 1 from the ones column changes 27.4 to 26.14:

$\begin{aligned} 26.14 \\ -10.50 \\ \hline \end{aligned}$

• Now subtract the tenths: 14 - 5 = 9.

• Next, subtract the ones column: 6 - 0 = 6.

• Finally, subtract the tens column: 2 - 1 = 1.

Therefore, the result of $27.4 - 10.5$ is $16.9$.

The correct answer is $\boxed{16.9}$.

To solve the addition problem of $1.54 + 0.27$, we start by aligning the numbers by their decimal points:

• Step 1: Write the numbers vertically, one under the other, ensuring that the decimal points are aligned:
  $\begin{array}{c} ~~~1.54 \\ +0.27 \\ \hline \end{array}$

• Step 2: Add the digits starting from the rightmost column (hundredths place):
  - Hundredths place: $4 + 7 = 11$. Write down $1$ and carry over $1$ to the tenths place.
  - Tenths place: $5 + 2 = 7$ plus the carried over $1$ gives us $8$. Write down $8$.
  - Ones place: $1 + 0 = 1$. Write down $1$.

• Step 3: Combine the results:
  $\begin{array}{c} ~~~1.54 \\ + 0.27 \\ \hline ~~~1.81 \\ \end{array}$

Therefore, the sum of $1.54$ and $0.27$ is $1.81$.

From the given answer choices, the correct answer is choice 1: $1.81$.

To solve the problem of adding $103.8$ and $96.4$, follow these steps:

• Step 1: Write the numbers vertically, aligning the decimal points:

• 103.8
  + 96.4

• Step 2: Start from the rightmost digit (the tenths place) and add:

• $8 + 4 = 12$. Write down 2 in the tenths place and carry over 1 to the units place.

• Step 3: Add the units place:

• $3 + 6 = 9$, and add the carry-over of 1: $9 + 1 = 10$. Write down 0 in the units place and carry over 1 to the tens place.

• Step 4: Add the tens place:

• $0 + 9 = 9$, and add the carry-over of 1: $9 + 1 = 10$. Write down 0 in the tens place and carry over 1 to the hundreds place.

• Step 5: Add the hundreds place:

• $1 + 0 = 1$, and add the carry-over of 1: $1 + 1 = 2$. Write down 2 in the hundreds place.

• Step 6: Write the final sum:

• The resulting sum is $200.2$.

Thus, the calculated sum of $103.8$ and $96.4$ is $200.2$.

The answer choice corresponding to this result is: 200.2

To solve the problem of subtracting $46$ from $574.86$, follow these steps:

• Step 1: Write the numbers in a vertical format aligned by the decimal point:
  574.86
  - 46.00
  ------------

• Step 2: Subtract starting from the rightmost digits and move towards the left. Begin with the tenths place:

  • The tenths place: $6 - 0 = 6$.

  • The units place: $4 - 0 = 4$.

  • In the tens place, $7 - 6 = 1$.

  • The hundreds place: $5 - 0 = 5$.

• Write down the result: 528.86

Thus, the result of subtracting $46$ from $574.86$ is $528.86$.

To solve the subtraction $70 - 14.4$, we need to perform the following steps:

• Step 1: Convert $70$ to a decimal number for easier alignment, writing it as $70.0$.

• Step 2: Write the two numbers vertically, aligning to the right and with the decimal points:

$\begin{aligned} 70.0 \\ - \ 14.4 \\ \hline \end{aligned}$

• Step 3: Perform the subtraction starting from the rightmost digit.

• Step 4: Subtract tenths: $0$ (borrow if necessary) - $4$ becomes $10 - 4 = 6$, since we needed to borrow 1 from $7$, making it $6$.

• Step 5: Subtract whole numbers: $6 - 1 = 5$ and $7 - 1 = 5$ (remember we borrowed to balance the tenths).

Therefore, after the subtraction, we find that the solution is $55.6$.

To solve this problem, follow these steps:

• Step 1: Align the numbers $0.77$ and $30.6$ by their decimal points:
  $\begin{aligned} 30&.6 \\ + ~~ 0&.77 \\ \hline \end{aligned}$

• Step 2: Add these numbers starting from the rightmost digit, considering each place value.

• Adding hundredths place:
  $0 + 7 = 7$

• Adding tenths place:
  $6 + 7 = 13$, write down $3$ and carry over 1 to the next column.

• Adding whole number and carry over:
  $0 + 1 + 30 = 31$

• Thus, the sum is $31.37$.

Therefore, the solution to the problem is $31.37$.