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Addition and Subtraction of Decimals
Addition and subtraction of decimals are basic arithmetic operations performed on decimal numbers. Decimals are a way to represent numbers that have fractional parts, and they are expressed using a decimal point. The decimal point separates the whole number part from the fractional part.
Addition of Decimals:
To add decimals, follow these steps:
Step 1: Align the decimal points of the numbers.
Step 2: Add the digits in each column, starting from the rightmost column (the units column) and moving to the left.
Step 3: If the sum of any column is greater than 9, carry over the excess to the next left column.
Step 4: Once all columns are added, place the decimal point in the sum exactly below the decimal points of the numbers being added.
Example:
12.34
+ 56.78
________
69.12
Subtraction of Decimals:
To subtract decimals, follow these steps:
Step 1: Align the decimal points of the numbers.
Step 2: Subtract the digits in each column, starting from the rightmost column (the units column) and moving to the left.
Step 3: If the digit in the minuend (the number being subtracted from) is smaller than the digit in the same column of the subtrahend (the number being subtracted), borrow 1 from the next left column.
Step 4: Once all columns are subtracted, place the decimal point in the result exactly below the decimal points of the numbers.
Example:
56.78
– 12.34
_______
44.44
It’s essential to maintain proper alignment of decimal points to ensure accurate addition and subtraction of decimals. These operations are commonly used in various real-life situations, such as dealing with money, measurements, and calculations involving fractions of a whole.
What is a Decimal Number?
A decimal number is a base-10 numeral system that uses ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent numeric values. Each digit in a decimal number has a place value based on its position in the number, with the rightmost digit representing the units place, the next digit to the left representing the tens place, the next representing the hundreds place, and so on.
For example, the number “123.45” is a decimal number. In this number, “1” is in the hundreds place, “2” is in the tens place, “3” is in the units place, “4” is in the tenths place, and “5” is in the hundredth place.
Decimal numbers are commonly used in everyday life for counting, measuring, and performing arithmetic operations like addition, subtraction, multiplication, and division. They are the most familiar number system to people, and many financial and scientific calculations are performed using decimal numbers.
What is Addition and Subtraction of Decimals?
Addition and subtraction of decimals is a way of combining or taking away numbers with decimal points. It’s similar to adding and subtracting whole numbers, but now we include numbers with parts after the decimal point.
Addition: To add decimals, line up the decimal points, and then add the digits just like you would with whole numbers. The key is to make sure the decimals are aligned.
Subtraction: To subtract decimals, again, line up the decimal points, and then subtract the digits just like you would with whole numbers. The key is to make sure the decimals are aligned.
How to Add and Subtract Decimals?
Adding and subtracting decimals in mathematics can be done by following these steps:
Adding Decimals:
- Line up the decimal points of the numbers you want to add.
- If one number has more decimal places than the other, add zeros to the shorter decimal so that they have the same number of decimal places.
- Start adding the digits from right to left, just like you do with whole numbers. Add the digits in the ones place, then in the tenths place, then in the hundredths place, and so on.
- Carry over any values that exceed 9 in each column, just like you would with whole numbers.
- Once you’ve added all the digits in each column, write the sum with the decimal point aligned with the original numbers.
Subtracting Decimals:
- Line up the decimal points of the numbers you want to subtract.
- If one number has more decimal places than the other, add zeros to the shorter decimal so that they have the same number of decimal places.
- Start subtracting the digits from right to left, just like you do with whole numbers. Subtract the digits in the ones place, then in the tenths place, then in the hundredths place, and so on.
- Borrow from the next higher place value, just like you would with whole numbers if the top digit is smaller than the bottom digit in a given column.
- Once you’ve subtracted all the digits in each column, write the result with the decimal point aligned with the original numbers.
- It’s essential to be careful with the decimal alignment, as errors in this can lead to incorrect results. Always double-check your work to ensure accuracy.
Remember that practicing with different examples will help reinforce your understanding and improve your skills in adding and subtracting decimals.
Decimals in Real Life
Decimals play a significant role in real life and are used in various contexts across many different fields. Here are some examples of how decimals are used in everyday life:
- Money and Finance: Decimals are crucial in handling money and financial transactions. Currency is often represented in decimal form, with dollars and cents (e.g., $10.50). Financial calculations, such as taxes, interest rates, and investments, also involve decimals.
- Measurement and Units: Decimals are used extensively in measurements. For instance, length, weight, volume, and temperature are often expressed with decimal values. For example, 2.5 meters, 1.75 liters, or 37.8 degrees Celsius.
- Cooking and Recipes: Cooking often requires precise measurements, and decimals are commonly used in recipes to indicate ingredient quantities. For example, 1.5 cups of flour or 0.25 teaspoons of salt.
- Science and Engineering: In scientific and engineering disciplines, decimals are used for precise measurements and calculations. Scientists might measure the pH of a solution as 6.8, or engineers may specify a tolerance of 0.01mm for a mechanical part.
- Medicine and Healthcare: In the medical field, decimals are used for various purposes, such as measuring medication dosages, blood pressure readings (e.g., 120/80 mmHg), or the concentration of substances in the body.
- Sports and Athletics: In sports, decimals play a role in recording and comparing scores or timing. For instance, a runner might complete a race in 10.45 seconds.
- GPA and Grading Systems: In education, grade point averages (GPAs) are often calculated using decimals to represent the average of a student’s grades.
- GPS and Navigation: GPS coordinates use decimals to indicate precise locations on the Earth’s surface, such as latitude and longitude values.
- Statistics and Data Analysis: Decimals are used in data analysis to represent percentages, probabilities, or numeric values with greater precision.
- Time: Time can be represented using decimals in various contexts. For example, a half-hour can be represented as 0.5 hours, or 3.5 hours.
These examples demonstrate how decimals are a fundamental part of our daily lives, enabling us to make precise measurements, calculations, and comparisons across various fields and activities.
Solved Examples on Addition and Subtraction
Example 1: Decimal Addition
Compute the sum of 3.25 and 1.75.
Solution:
Step 1: Align the decimal points.
3.25
1.75
Step 2: Add the numbers without the decimal first.
325
175
Step 3: Add the numbers with the decimal.
500 (5 carried over from the previous step)
Step 4: Place the decimal point in the result.
5.00
So, 3.25 + 1.75 = 5.00
Example 2: Decimal Subtraction
Find the difference between 6.84 and 2.15.
Solution:
Step 1: Align the decimal points.
6.84
2.15
Step 2: Subtract the numbers without the decimal.
684
215
Step 3: Subtract the numbers with the decimal.
469
So, 6.84 – 2.15 = 4.69
Example 3: Decimal Addition and Subtraction
Evaluate the expression: 9.75 + 2.13 – 1.29
Solution:
Step 1: Perform the addition first.
9.75
2.13
1175 (1 carried over from the previous step)
Step 2: Perform the subtraction.
11.75
1.29
1046 (1 carried over from the previous step)
So, 9.75 + 2.13 – 1.29 = 10.46
Remember to always align the decimal points when performing addition or subtraction of decimals.
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