Question

Find each sum or difference without using a calculator. \(54.32+45.68\)

Short answer

100.00

Step-by-step solution

- Align the numbers

Write the numbers vertically and align the decimal points. This will help in adding the numbers column by column.\[\begin{array}{r} 54.32 \+45.68 \hline\end{array}\]

- Add the hundredths place

Begin by adding the digits in the hundredths place (2 and 8).\[ 2 + 8 = 10 \] Write down 0 and carry over 1 to the tenths place.

- Add the tenths place

Add the digits in the tenths place (3 and 6) along with the carried over 1.\[ 3 + 6 + 1 = 10 \] Write down 0 and carry over 1 to the units place.

- Add the units place

Add the digits in the units place (4 and 5) and the carry over 1.\[ 4 + 5 + 1 = 10 \] Write down 0 and carry over 1 to the tens place.

- Add the tens place

Add the digits in the tens place (5 and 4) and the carry over 1.\[ 5 + 4 + 1 = 10 \] Write down 10.

- Combine the results

Now combine all the results, placing the decimal point in the correct position: \[ 100.00 \]

Key concepts

Aligning Decimal Points

When adding decimal numbers, it is crucial to align the decimal points. This means writing one number directly above the other, ensuring their decimal points are in a vertical line.
This alignment allows you to add each column of digits correctly.

For example, to add 54.32 and 45.68, write them as:

54.32
+45.68
------

By aligning the decimal points, we ensure each digit is in its proper place value: units under units, tenths under tenths, and hundredths under hundredths.

Carrying Over in Addition

Carrying over is an important part of addition, especially when the sum of a column is 10 or more. To illustrate, let’s look at the hundredths place in the addition of 54.32 and 45.68.

When you add the hundredths digits (2 + 8), you get 10. Here, you write down the 0 and carry over the 1 to the next column, which is the tenths place.
This process repeats in higher places if needed:
• In the tenths place: 3 + 6 + the carried over 1 = 10
• In the units place: 4 + 5 + the carried over 1 = 10
• In the tens place: 5 + 4 + the carried over 1 = 10

Each time the result of adding exceeds 9, the leftmost digit is written in the next higher place value.

Step-by-Step Calculation

Let's summarize the calculation of 54.32 + 45.68 using the step-by-step approach.

1. **Align the numbers** with their decimal points lined up:

54.32
+45.68
------

2. **Add the hundredths place** (2 + 8 = 10):Write down 0 and carry over 1.

3. **Add the tenths place** (3 + 6 + carried over 1 = 10):Write down 0 and carry over 1.

4. **Add the units place** (4 + 5 + carried over 1 = 10):Write down 0 and carry over 1.

5. **Add the tens place** (5 + 4 + carried over 1 = 10):Write down 10.

Combining these results and placing the decimal point back in its correct position, we get the final sum:54.32 + 45.68 = 100.00