Mastering Decimal Division: A Comprehensive Guide with Step-by-Step Instructions
Dividing decimals might seem daunting at first, but with a clear understanding of the process and some practice, it becomes a manageable and essential mathematical skill. This comprehensive guide will break down the steps involved in dividing decimals, providing detailed instructions and examples to help you master this important concept. Whether you’re a student struggling with homework or an adult looking to brush up on your math skills, this article will equip you with the knowledge and confidence to tackle any decimal division problem.
Understanding the Basics of Decimal Division
Before diving into the process, let’s quickly review what decimals are and why they matter. Decimals represent numbers that are not whole, and they are composed of two parts: the whole number part to the left of the decimal point and the fractional part to the right. The decimal point separates these two parts. For example, in the number 3.14, ‘3’ is the whole number part, and ’14’ is the fractional part.
Division, in its essence, is the process of splitting a quantity into equal parts. Decimal division simply extends this concept to include numbers with fractional components. When dividing decimals, we’re essentially asking, “How many times does one decimal number fit into another decimal number?”
The Standard Division Algorithm: A Foundation for Decimal Division
The standard algorithm for long division serves as the foundation for dividing decimals. This algorithm involves several key steps:
- Set up the problem: Write the dividend (the number being divided) inside the division symbol and the divisor (the number you’re dividing by) outside the symbol.
- Divide: Divide the first digit (or group of digits) of the dividend by the divisor. Write the quotient (the result of the division) above the corresponding digit(s) of the dividend.
- Multiply: Multiply the quotient by the divisor and write the product below the corresponding digits of the dividend.
- Subtract: Subtract the product from the corresponding digits of the dividend.
- Bring down: Bring down the next digit of the dividend and append it to the remainder from the subtraction.
- Repeat: Repeat steps 2 through 5 until there are no more digits to bring down.
While this algorithm works for whole numbers, it needs a few adjustments when dealing with decimals. The primary adjustment involves the decimal point and how it’s treated during the process.
Step-by-Step Guide to Dividing Decimals
Now, let’s explore the step-by-step process for dividing decimals, incorporating the necessary adjustments:
Step 1: Setting up the Decimal Division Problem
The first step is the same as for whole number division: write the dividend inside the division symbol and the divisor outside. The crucial difference now is to take note of any decimal places in both the dividend and the divisor. For example, if you’re dividing 12.6 by 2.1, the setup would be:
______
2.1 | 12.6
Step 2: Dealing with the Decimal in the Divisor
The most important adjustment in decimal division is that we cannot directly divide if the divisor has a decimal. To resolve this, we need to transform the divisor into a whole number. We do this by moving the decimal point to the right until the divisor becomes a whole number. The number of places we move the decimal point in the divisor must also be the exact same number of places we move it in the dividend.
In our example of 12.6 ÷ 2.1, we need to move the decimal one place to the right in 2.1 to get 21, which is a whole number. Correspondingly, we must move the decimal one place to the right in 12.6 to get 126. Our adjusted problem now looks like this:
______
21 | 126
If the divisor is a whole number to start with, such as in the case of 5.4 ÷ 3, then this step can be skipped, and we proceed with our division.
Example 1:
Divide 15.75 by 0.5
Move decimal in 0.5 one position to right to get 5, Move decimal one position to the right in 15.75 to get 157.5, Thus the division becomes 157.5 divided by 5
Example 2:
Divide 4.5 by 0.09
Move the decimal two positions to right in 0.09 to get 9, Move decimal 2 positions to right in 4.5 to get 450 (We add the zero), Thus the division becomes 450 divided by 9
Step 3: Perform the Division
Now that you’ve adjusted the decimal points, you can perform the division as you normally would with whole numbers. Follow the standard division algorithm, but take the decimal point in the dividend into account when writing the quotient. In our example of 126 ÷ 21, we perform regular long division.
Step 3a: Divide
Divide the first digit (or group of digits) of the dividend by the divisor. In the case of 126 ÷ 21, we need to determine how many times 21 can go into 126. We find that 21 goes into 126 exactly 6 times. Write ‘6’ above the ‘6’ in the dividend.
_6____
21 | 126
Step 3b: Multiply
Multiply the quotient (6) by the divisor (21): 6 * 21 = 126. Write 126 below 126 in the dividend.
_6____
21 | 126
126
Step 3c: Subtract
Subtract the product (126) from the corresponding digits of the dividend (126): 126 – 126 = 0.
_6____
21 | 126
126
---
0
Since the remainder is zero and we have no further digits to bring down, we are finished. The quotient is 6.
In our original example of 12.6 ÷ 2.1, the answer is 6
Important Note Regarding Decimal Placement in the Quotient: When placing the decimal point in the quotient, place it directly above the decimal point in the adjusted dividend. The decimal point of the dividend affects the decimal point placement in the quotient.
Step 4: Handling Remainders and Repeating Decimals
Sometimes, division doesn’t result in a clean whole number; there may be a remainder. In such cases, we can continue the division by adding zeros to the right of the decimal point in the dividend and continuing with steps 2 through 5 from the standard algorithm. This allows us to find a decimal quotient (often to a certain level of precision). It is also possible that when we add zeroes we might end up with repeating decimals. In that case, you have to either round it to certain decimal places or indicate the repeating part using an overline.
Example 3: Division with Remainders
Divide 17.5 by 4:
Step 1 & 2: Set up, since the divisor does not have a decimal, there is no decimal movement required
______
4 | 17.5
Step 3: Divide normally:
4 goes into 17 4 times (4 * 4 = 16), Write 4 in the quotient and subtract 16 from 17
_4.____
4 | 17.5
16
---
1
Bring down the 5 in dividend to get 15
_4.____
4 | 17.5
16
---
15
4 goes into 15 3 times (3 * 4 = 12). Write 3 in the quotient, and subtract 12 from 15
_4.3____
4 | 17.5
16
---
15
12
---
3
Now there is a remainder, add a zero to the end of the 17.5 and bring it down
_4.3____
4 | 17.50
16
---
15
12
---
30
4 goes into 30 7 times(7 * 4 = 28) write 7 in quotient and subtract 28 from 30
_4.37____
4 | 17.50
16
---
15
12
---
30
28
--
2
We continue, again add a zero and bring it down
_4.37____
4 | 17.500
16
---
15
12
---
30
28
--
20
4 goes into 20 5 times and now we have reached a zero remainder. The final quotient is 4.375
_4.375____
4 | 17.500
16
---
15
12
---
30
28
--
20
20
--
0
Example 4: Repeating Decimals
Let’s divide 10 by 3:
3 goes into 10 3 times, and then we have 1 as remainder:
3.___
3 | 10
9
--
1
Add a zero, 3 goes into 10 3 times again, remainder 1:
3.3__
3 | 10.0
9
--
10
9
--
1
This will repeat so we get 3.333….
In such case we can write it as 3.3(with a line above the last 3 indicating repeating decimal)
Step 5: Check Your Answer
A great practice after each division problem is to check your answer. You can do this by multiplying the quotient you’ve obtained by the original divisor. If done correctly, the result should be equal to the original dividend. This serves as a final verification of the accuracy of your calculation. For example if you divide 12.6 by 2.1 and get 6, then 2.1 * 6 = 12.6 which means your answer is correct. If you divided 17.5 by 4 and got 4.375, then 4 * 4.375 = 17.5, again verifying the correct answer
Practical Tips for Decimal Division
Here are some additional tips to help you perform decimal division effectively:
- Estimation: Before dividing, try to estimate the result. This will help you catch major errors.
- Organization: Keep your work neat and well-organized. This will reduce the chance of making careless errors.
- Practice: The more you practice, the easier decimal division will become.
- Technology: Don’t hesitate to use calculators to check your answers, but strive to understand the underlying process.
- Understand the importance of each step: Each step in the decimal division algorithm is important, understanding the purpose of each step makes the whole process simpler.
Examples with Detailed Solutions
Let’s work through a few more examples together:
Example 5: Divide 3.45 by 1.5
Solution:
- Set up: Write 3.45 inside the division symbol and 1.5 outside.
- Adjust decimal: Move the decimal one place to the right in both divisor and dividend to get 34.5 divided by 15.
- Divide: 15 goes into 34 twice (2 * 15 = 30), subtract 30 from 34 getting a remainder of 4, bring down the 5 to get 45, 15 goes into 45 three times. So, 2.3 is the quotient.
- Check: 1.5 * 2.3 = 3.45, the answer is correct.
Example 6: Divide 1.044 by 0.12
Solution:
- Set up: Write 1.044 inside the division symbol and 0.12 outside.
- Adjust decimal: Move the decimal two places to the right in both divisor and dividend to get 104.4 divided by 12.
- Divide: 12 goes into 104 eight times with remainder of 8, bringing down 4 we get 84, 12 goes into 84 seven times and the remainder is 0. So, 8.7 is the quotient.
- Check: 0.12 * 8.7 = 1.044, the answer is correct.
Example 7: Divide 25.5 by 4
Solution:
- Set up: Write 25.5 inside the division symbol and 4 outside.
- Adjust decimal: No adjustment required
- Divide: 4 goes into 25 6 times (6*4=24), subtract 24 from 25 remainder is 1. Bring down 5. 4 goes into 15 3 times, remainder 3. Add zero, 4 goes into 30 7 times, remainder 2. Add zero, 4 goes into 20 5 times, and remainder is 0. So, quotient is 6.375
- Check: 4 * 6.375 = 25.5
Conclusion
Dividing decimals is an essential skill that builds upon your understanding of basic division and decimal concepts. By following these step-by-step instructions and practicing regularly, you can confidently tackle any decimal division problem. Remember to adjust your decimal places properly, pay attention to remainders, and always check your answers. With consistent effort, you will master this important mathematical skill.