Long division, which is a part of basic arithmetic, is a method of solving and finding the remainder in division problems involving numbers with at least two digits. Learning the basic steps of long division will allow you to divide numbers of any size, including integrals and decimals. This process is easy to learn, and the ability to do long divisions will help you hone your understanding of mathematics in ways that will be beneficial both in school and in other areas of your life.
Steps
Part 1 of 4: Divide
Step 1. Formulate the equation
On a piece of paper, write the dividend (the number that you will divide) on the right, below the division symbol, and the divisor (the number that will perform the division) on the left on the outside.
- The quotient (the answer) will go to the top, just above the dividend.
- Leave plenty of room under the equation to perform multiple subtraction operations.
- Here's an example: if there are six mushrooms in a 250 gram package, how much does each one weigh approximately? In this case, we must divide 250 by 6. The 6 goes on the outside and the 250 on the inside.
Step 2. Divide the first digit
By dividing from left to right, determine the number of times the divisor fits into the first digit of the dividend without exceeding it.
In our example, you will need to determine how many times 6 goes into 2. Since six is greater than two, the answer is zero. If you want, you could write a 0 directly above 2 as an indicator and delete it later. You can also leave that space blank and move on to the next step
Step 3. Divide the first two digits
If the divisor is a number greater than the first digit, determine the number of times the divisor fits into the first two digits of the dividend without exceeding it.
- If your answer to the previous step was 0, as in the example, expand the number by one digit. In this case, the question is how many times does 6 go into 25?
- If the divisor has more than two digits, you will need to expand it even further to the third or even the fourth digit of the dividend in order to get a number that will fit the divisor.
- Work in terms of whole numbers. If you use a calculator, you will find that 6 goes into 25 a total of 4, 167 times. In long division, it always rounds to the nearest whole number, so in this case our answer would be 4.
Step 4. Write the first digit of the quotient
Place the number of times the divisor fits into the first digit (or digits) of the dividend above the appropriate digit (s).
- In a long division, it is important to ensure that the columns of the numbers remain correctly aligned. Be careful or you could make a mistake that leads to the wrong answer.
- In the example, you must place a 4 on top of the 5, since it is the number of times that 6 goes into 25.
Part 2 of 4: Multiply
Step 1. Multiply the divisor
You must multiply the divisor by the number you just wrote above the dividend. In our example, this is the first digit of the quotient.
Step 2. Write the product
Place the result of your multiplication in step 1 below the dividend.
In the example, 6 times 4 equals 24. After you have written a 4 in the quotient, write the number 24 below the 25, always being careful to keep the numbers aligned
Step 3. Draw a line
You must put a line under the product of your multiplication, in this case, 24.
Part 3 of 4: Subtract and Carry a Digit
Step 1. Subtract the product
Subtract the number you just wrote below the dividend from the digits directly above it in the dividend area. Write the result below the line you just made.
- In the example, we will subtract 24 from 25, which gives us 1.
- Do not subtract the entire dividend, but only those digits that you worked with in the previous sections. In the example, you should not subtract 24 from 250.
Step 2. Move down the next digit
Write the next digit of the dividend after obtaining the result of the subtraction.
In the example, since 6 does not fit into 1 without exceeding it, you need to lower another digit. In this case, you will take the 0 out of 250 and place it next to the 1, making it 10 and allowing 6 to fit in it
Step 3. Repeat the whole process
Divide the new number by the divisor and write the result above the dividend as the next digit of the quotient.
- In the example, determine the number of times 6 can go into 10. Write that number (1) in the quotient above the dividend. Then multiply 6 by 1 and subtract the result from 10. In the end, you should have 4.
- If the dividend has more than three digits, keep repeating this process until you have done the operation on all the numbers. For example, if we had started with 2506 grams of mushrooms, we would lower the 6 and place it next to the four.
Part 4 of 4: Finding the remainder or decimal
Step 1. Write the remainder
Depending on what you use for this division, you may want to end up with a quotient that is a whole number with a remainder, that is, an indicator of how much is left after the division has been completed.
- In the example, the remainder would be 4, since 6 cannot fit into 4 and there are no more digits to go down.
- Put the remainder after the quotient along with a letter "r." In the example, the answer would be expressed as follows: “41 r4”.
- You will have to stop at this point if you want to calculate something that could not be expressed in partial units, such as if you are trying to determine the number of cars that are needed to move a certain number of people. In a case like this, it wouldn't be helpful to think in terms of cars or biased people.
- If you want to calculate a decimal, you can skip this step.
Step 2. Add a decimal point
If you want to calculate a precise answer instead of one with a remainder, you will need to go beyond whole numbers. When you have reached a point where you are left with a number smaller than your divisor, add a decimal point to both the quotient and the dividend.
In the example, since 250 is an integer, all digits after the decimal point will be 0, making it 250,000
Step 3. Repeat the same process
Now you have more digits you can lower (all of them zero). Lower a zero and continue as you did before, determining the number of times the divisor can fit into the new number.
In the example, determine the number of times 6 can go into 40. Add that number (6) in the quotient above the dividend and after the decimal point. Then multiply 6 by 6 and subtract the result from 40. Again, you'll end up with a 4
Step 4. Stop at this point and round
In some cases, you will find that when you begin to solve the decimal, the answer is repeated over and over again. At this point, it's time to stop and round your answer up (if the repeating number is 5 or greater) or down (if it is 4 or less).
- In the example, you could keep getting 4 as a remainder and adding 6 to the quotient indefinitely. Instead, stop and round that quotient. Since 6 is greater than (or equal to) 5, you can round your answer to 41.67.
- You can also indicate a repeating decimal by placing a small horizontal line over the repeating digit. In this case, since the quotient is 41.6, you must draw a line over the 6.
Step 5. Add the unit back to your answer
If you are working with units such as kilograms, liters, or degrees, you should add them at the end once you have obtained your answer.
- If you added a leading zero as an indicator, you should delete it at this point.
- In the example, because your question was how much each mushroom weighs in a 250 gram package, you will need to put the answer in grams. Therefore, your final answer will be 41.67 grams.
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Advice
- If you have time, it is a good idea to perform calculations on paper first and then verify them with a calculator or computer. Remember that machines sometimes give wrong answers for a number of reasons. If there is an error, you can check a third time using logarithms. Doing the division by hand rather than relying on machines is beneficial for your math skills and your conceptual understanding.
- One way to remember the steps for a long division is with the names: “Diana, María, Rosa and Beatriz”, where D is to divide the number, M to multiply, R to subtract and B to lower a digit.
- Look for practical examples in your everyday life. This will help you learn the process, because you can see how useful it is in the real world.
- Start by doing simple calculations. This will give you confidence and help you develop the skills necessary to move on to more advanced calculations.