Performing division without a calculator primarily involves using the long division method, often referred to as the "bus stop" method due to its visual setup. This systematic approach allows you to break down larger division problems into manageable steps.
Understanding the Long Division Method
Long division is a step-by-step process that systematically divides a larger number (the dividend) by a smaller number (the divisor) to find the quotient and, if applicable, a remainder.
Steps for Manual Division
To perform division manually, follow these core steps:
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Set up the problem: Write the division operation in the "bus stop" form. The divisor (the number you are dividing by) goes outside the bracket, and the dividend (the number being divided) goes inside.
- Example: For 735 ÷ 5, 5 would be outside, and 735 inside.
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Divide the first digit (or group of digits): Start with the leftmost digit of the dividend. Determine how many times the divisor can go into this digit (or the first few digits if the first digit is smaller than the divisor). Write the quotient digit directly above the corresponding digit in the dividend.
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Multiply and Subtract:
- Multiply the quotient digit you just wrote by the divisor.
- Write this product directly below the part of the dividend you just divided.
- Subtract this product from that part of the dividend to find the remainder.
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Bring down the next digit and repeat: Bring down the next digit from the dividend next to your remainder. This creates a new number. Now, repeat steps 2 and 3 with this new number. If the divisor does not go into the current number, write a zero in the quotient, carry the entire number over, and bring down the next digit.
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Continue until finished: Keep repeating the process until all digits of the dividend have been used. Any number left over at the end is the remainder.
Practical Example: Dividing 735 by 5
Let's walk through an example to illustrate the process: 735 ÷ 5
| Step | Action | Calculation | Explanation |
|---|---|---|---|
| 1 | Setup | 5 | 735 |
Place the divisor (5) outside and the dividend (735) inside the division bracket. |
| 2 | Divide 7 by 5 | 1 <br> 5 | 735 |
How many times does 5 go into 7? Once (1 x 5 = 5). Write '1' above the '7'. |
| 3 | Multiply & Subtract | 1 <br> 5 | 735 <br> - 5 <br> --- <br> 2 |
Multiply 1 by 5 (result is 5). Write 5 below 7. Subtract 5 from 7, leaving 2. |
| 4 | Bring Down 3 | 1 <br> 5 | 735 <br> - 5 <br> --- <br> 23 |
Bring down the next digit, '3', next to the remainder '2', forming '23'. |
| 5 | Divide 23 by 5 | 14 <br> 5 | 735 <br> - 5 <br> --- <br> 23 |
How many times does 5 go into 23? Four times (4 x 5 = 20). Write '4' above the '3'. |
| 6 | Multiply & Subtract | 14 <br> 5 | 735 <br> - 5 <br> --- <br> 23 <br> - 20 <br> --- <br> 3 |
Multiply 4 by 5 (result is 20). Write 20 below 23. Subtract 20 from 23, leaving 3. |
| 7 | Bring Down 5 | 14 <br> 5 | 735 <br> - 5 <br> --- <br> 23 <br> - 20 <br> --- <br> 35 |
Bring down the next digit, '5', next to the remainder '3', forming '35'. |
| 8 | Divide 35 by 5 | 147 <br> 5 | 735 <br> - 5 <br> --- <br> 23 <br> - 20 <br> --- <br> 35 |
How many times does 5 go into 35? Seven times (7 x 5 = 35). Write '7' above the '5'. |
| 9 | Multiply & Subtract | 147 <br> 5 | 735 <br> - 5 <br> --- <br> 23 <br> - 20 <br> --- <br> 35 <br> - 35 <br> --- <br> 0 |
Multiply 7 by 5 (result is 35). Write 35 below 35. Subtract 35 from 35, leaving 0. |
The quotient is 147 with no remainder.
Handling Remainders and Decimals
- Remainders: If you have a number left over after processing all digits and it's not zero, that's your remainder. You can express the answer as "quotient remainder R" (e.g., 147 R 0, or just 147 in this case), or as a mixed number (quotient + remainder/divisor).
- Decimal Answers: To get a decimal answer, add a decimal point and zeros to the dividend (e.g., 735.000). Continue the long division process, placing the decimal point in the quotient directly above the decimal point in the dividend.
Essential Skills for Manual Division
Mastering long division without a calculator relies on a few fundamental arithmetic skills:
- Multiplication Tables: A strong recall of multiplication facts is crucial for quickly determining how many times the divisor goes into a number.
- Subtraction: Accurate subtraction is necessary for finding the remainder at each step.
- Place Value Understanding: Knowing the value of each digit helps in setting up the problem and carrying over numbers correctly.
- Practice: Consistent practice with various division problems will build speed and confidence.
For more detailed explanations and practice, you can explore resources like How to divide without a calculator.
