Step-by-Step Guide to Subtraction with Borrowing and Regrouping
Learning the Subtraction Of Numbers Using The Decomposition Method is a crucial arithmetic skill, especially for students in upper primary and middle school classes. This method, also called the borrowing or expanded subtraction method, is used to handle more complex subtractions, particularly when the digits in the minuend are smaller than those in the subtrahend. Mastering this technique helps students solve subtraction problems with confidence and accuracy, both in exams and everyday life.
Understanding Subtraction Using the Decomposition Method
Subtraction using the decomposition method involves breaking apart numbers into their place values and borrowing from higher place values when needed. This technique is especially useful when the digits in the top number are smaller than those directly below in the subtraction columns. By decomposing or 'borrowing', we reorganize numbers to make the subtraction possible, reinforcing our place value understanding.
How Does the Decomposition Method Work?
Let's break down how this method works, step-by-step:
- Write both numbers in columns, aligning digits by place value (ones, tens, hundreds, etc.).
- Start subtracting from the rightmost column (ones).
- If the digit above is smaller than the digit below, borrow 1 from the next left column. Increase the top digit by 10 and decrease the digit you borrowed from by 1.
- Continue this process for each column, borrowing as needed.
- Write the answer for each column after subtraction.
Step-by-Step Example
Example: Subtract 562 from 285 using the decomposition method.
| Hundreds | Tens | Ones | |
|---|---|---|---|
| Number | 5 | 6 | 2 |
| Subtract | 2 | 8 | 5 |
| Let's subtract column by column: | |||
| Ones | 2 - 5 (can't do, so borrow 1 from tens) | ||
| After borrow | 6 → 5 | 2 → 12 | |
| Subtract Ones | 12 - 5 = 7 | ||
| Subtract Tens | 5 - 8 (can't do, so borrow 1 from hundreds) | ||
| After borrow | 5 → 4 | 5 → 15 | |
| Subtract Tens | 15 - 8 = 7 | ||
| Subtract Hundreds | 4 - 2 = 2 | ||
| Answer | 277 | ||
So, 562 - 285 = 277 using the decomposition method.
Worked Examples
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Example 1: Subtract 406 from 258.
- Ones: 8 - 6 = 2
- Tens: 5 - 0 = 5
- Hundreds: 2 - 4 (can't do, borrow 1 from the thousands place if given or adjust problem)
Since we can't borrow and no higher place value, this indicates the answer would be negative in this case. For school-level problems, numbers are usually chosen so borrowing is possible.
-
Example 2: Subtract 158 from 407.
- Ones: 7 - 8 (can't do; borrow 1 from tens, tens becomes -1, ones becomes 17)
- 17 - 8 = 9
- Tens: 0 - 5, after borrow is -1 - 5 = -6 (can't do; borrow 1 from hundreds, hundreds becomes 3, tens becomes 9)
- 9 - 5 = 4
- Hundreds: 3 - 1 = 2
- Final Answer: 407 - 158 = 249
-
Example 3: Subtract 347 from 602.
- Ones: 2 - 7 (can't do; borrow 1 from tens, tens becomes -1, ones becomes 12)
- 12 - 7 = 5
- Tens: 0 - 4 (after borrow, -1 - 4 = -5; borrow 1 from hundreds, hundreds becomes 5, tens becomes 9)
- 9 - 4 = 5
- Hundreds: 5 - 3 = 2
- Final Answer: 602 - 347 = 255
Practice Problems
- Subtract 274 from 635 using the decomposition method.
- Subtract 489 from 903 using borrowing.
- Subtract 765 from 832 using decomposition.
- Solve 501 - 288 step by step.
- Subtract 550 from 800 using the decomposition method.
Common Mistakes to Avoid
- Forgetting to decrease the digit after borrowing from it – always subtract 1 from the next higher place value.
- Mixing up place values while aligning digits in columns.
- Not adjusting every digit from which you've borrowed, which can give wrong answers.
- Forgetting zeros can be borrowed from; treat zeros as valid "lenders" if higher digits are available.
Real-World Applications
The decomposition method is practical whenever dealing with money, time, or measurements. For example, if you have ₹500 and spend ₹263, decomposing the numbers makes it easier to calculate your change. Businesses, banks, and retailers use similar strategies for cash calculations. At home, you might use this method to measure ingredient differences, count change, or calculate the time left between events.
At Vedantu, we break down concepts like Subtraction Of Numbers Using The Decomposition Method in a simple and easy-to-follow way, so students can master math skills for exams and everyday life. Explore more on Borrowing Subtraction and Regrouping for deeper practice and understanding.
Page Summary
In this topic, you learned to use the decomposition or borrowing method for subtraction by breaking numbers into place values, borrowing when needed, and subtracting column by column. Practicing these steps makes subtracting large and complex numbers easy and error-free—essential for school math success and real-world problem-solving.
FAQs on How to Subtract Numbers Using the Decomposition Method
1. What is the decomposition method in subtraction?
The decomposition method, also known as the borrowing method, breaks down numbers into their place values (ones, tens, hundreds, etc.) to make subtraction easier. It's especially helpful when subtracting larger numbers where a digit in the top number is smaller than the corresponding digit in the bottom number.
2. How to subtract using decomposition?
To subtract using the decomposition method: 1. Break down both numbers into their place values (ones, tens, hundreds). 2. If a digit in the top number is smaller than the corresponding digit below, 'borrow' from the next higher place value. 3. Subtract each place value column separately. 4. Combine the results to get the final answer. This is especially useful for solving problems involving subtraction with regrouping.
3. How can you decompose one number to subtract 696 − 275?
To solve 696 - 275 using decomposition: 1. Decompose 696 into 600 + 90 + 6. 2. Decompose 275 into 200 + 70 + 5. 3. Subtract the place values: 600 - 200 = 400; 90 - 70 = 20; 6 - 5 = 1. 4. Add the results: 400 + 20 + 1 = 421. Therefore, 696 - 275 = 421. This demonstrates decomposing numbers for subtraction.
4. How to decompose 37?
You can decompose 37 in several ways depending on the need. Some examples include: 30 + 7 (tens and ones); 20 + 17 (adjusting place values for subtraction); or 10 + 10 + 10 + 7 (breaking it into individual tens and ones).
5. How to decompose 345 into 4 parts?
There are multiple ways to decompose 345 into four parts. Here are a few examples: 100 + 100 + 100 + 45; 100 + 100 + 120 + 25; 50 + 100 + 100 + 95. The key is that the sum of the four parts equals 345.
6. How to subtract with borrowing?
Borrowing is a key part of the decomposition method. When a digit in the top number is smaller than the digit below, you 'borrow' from the next higher place value. For example, in 32 - 18, you borrow 1 ten from the 30, making it 20 + 12. Then subtract 18 from 20 + 12, which is 14.
7. How do you regroup in subtraction?
Regrouping is another term for borrowing in subtraction. It involves changing the place value of a digit to make subtraction possible when a digit in the top number is smaller. This is a core component of decomposition in subtraction.
8. What if I can't borrow?
If you can't borrow from the next place value (e.g., there's a zero), you need to borrow from a higher place value. For example, in 502 - 375, you must borrow from the hundreds place to perform the subtraction in the tens and ones columns. Mastering this aspect of subtraction with regrouping is crucial.
9. What if numbers have zeros?
Zeros in subtraction require careful application of the decomposition method. You might need to borrow from multiple place values. For instance, in 400 - 156, you'll borrow from the hundreds to help with the tens, and then from the tens to help with the ones. Understanding place value is critical for solving such problems.
10. Can the decomposition method be used for decimal numbers?
Yes, the decomposition method can be applied to decimal numbers. You simply extend the place value system to include tenths, hundredths, etc., and follow the same principles of borrowing and regrouping.
11. Why is the decomposition method important?
The decomposition method provides a structured and easily understood approach to subtraction, particularly for larger numbers requiring regrouping. Mastering this technique is essential for building a strong foundation in arithmetic and improving problem-solving skills. It directly addresses subtraction with regrouping.
