Division is one of the four main operations in arithmetic, along with multiplication, addition and subtraction. In addition to whole numbers, it is also possible to divide exponents, fractions and decimal numbers. Normally long division is used, but be aware that there is also short division, which can be used when one of the numbers has only one digit. However, start by mastering the long division as it contains all the elements of the operation.
Method 1 of 5: Making a Long Division
Step 1. Write the problem using a long division bar
The split bar (厂) looks like a parenthesis connected to a horizontal line and sits on top of the numbers. Place the divider (the number you are going to divide) outside the divider bar. The dividend (the number that will be divided) goes inside the bar.
- Example problem #1 (for beginners): 65 ÷ 5. Place the 5 outside the split bar and the 65 inside it. you must get 5厂65, with 65 under the horizontal line.
- Example problem #2 (intermediate difficulty): 136 ÷ 3. Put the 3 outside the bar and 136 inside it. you must get 3厂136, with the 136 below the horizontal line.
Step 2. Divide the first digit of the dividend by the divisor
In other words, find out how many times the divisor (the number outside the slash) fits into the first digit of the dividend. Place the result on the division line, just above the first digit of the divisor.
- In example #1 (5厂65), 5 is the divisor and 6 is the first digit of the dividend (65). 5 fits within 6 once, so place 1 at the top of the bar, just above 6.
- In example #2 (3厂136), 3 (the divisor) does not fit within 1 (the first digit of the dividend) in full. In that case, write a 0 above the split bar, aligned above the 1.
Step 3. Multiply the digit above the divider bar by the divisor
Take the number you just wrote on the slash and multiply it by the divisor (the number to the left of the slash). Write the result in a new row below the dividend, in line with its first digit.
- In example problem #1 (5厂65), multiply the number above the bar (1) by the divisor (5), resulting in 1 x 5 = 5. Put the answer (5) below the 6 within 65.
- In example problem #2 (3厂136), there is a zero above the division bar, so when you multiply it by the divisor (3), the result will be 0. Put the answer (0) below the 1 within 136.
Step 4. Subtract the result of multiplying the first digit of the dividend
In other words, subtract the number you just typed in the bottom row from the digit just above it. Write the result on a new line, in line with the subtraction digits.
- In example problem #1 (5厂65), subtract 5 (the result of multiplication) from the 6 above it (the first digit of the dividend): 6 - 5 = 1. Put the result (1) in a new row, below 5.
- In example problem #2 (3厂136), subtract 0 (the result of multiplication) from the 1 above it (the first digit of the dividend): 1 – 0 = 1. Put the result (1) in a new row, below 0.
Step 5. Pass the second digit of the dividend down
Drop it down to the row below, to the right of the result of the subtraction you just made.
- In example problem #1 (5厂65), take the 5 down from 65, putting it next to the 1 you got from subtracting 6 - 5. So you get 15.
- In example problem #2 (3厂136), drop the 3 of 136 and place it next to the 1, resulting in 13.
Step 6. Repeat the long division process (problem example #1)
Now use the dividend (the number to the left of the division bar) and the new number in the bottom row (the result of the first calculation and the digit that was descended). As before, divide, multiply and subtract to get the final result.
- To continue 5厂65, divide 5 (the dividend) into the new number (15) and write the result (3, given that 15 ÷ 5 = 3) above the division bar, to the right of the 1. Then multiply the 3 above the bar by the 5 (the dividend) and write the result (15, given that 3 x 5 = 15) below 15 under the split bar. Finally, subtract 15 from 15, getting 0. Write the result on a new row under everything.
- Example problem #1 is resolved, as there are no more digits in the divisor to pass down. The answer (130 will be above the split bar.
Step 7. Repeat the long division process (problem example #2)
As before, start dividing and multiplying. Finish by subtracting the results.
- For 3厂136: Find out how many times 3 fits into 13 and write answer (4) above the divider bar to the right of 0. Then multiply 4 by 3 and write answer (12) below 13. Finally, subtract 12 of 13 and write down the answer (1) under 12.
Step 8. Do one more long division and get the rest (example problem #2)
When you finish the problem, note that there is a remainder (a number left over from the calculations), which should be placed next to the answer.
- In case of 3厂136: Continue the splitting process. Drop the 6 of 136, making 16 on the bottom row. Divide 16 by 3 and note the result (5) above the division line. Multiply 5 by 3 and note the result (15) in the bottom row. Subtract 15 from 16, writing the result (1) in the bottom row.
- Since there are no more digits to pass down in the dividend, the problem is over, and the 1 that is left is the remainder of the division. Write it above the split bar with an "r." ahead. Therefore, the final result is "45 r.1".
Method 2 of 5: Doing a Short Division
Step 1. Write the problem with a divider bar
Place the divider (the number you are going to divide) on the outside, to the left of the bar. Place the dividend (the number that will be divided) inside the division bar on the right.
- For a short division, the divisor cannot be more than one digit.
- Example problem: 518 ÷ 4. In this case, 4 will be outside the split bar, with 518 inside it.
Step 2. Divide the divisor by the first digit of the dividend
In other words, find out how many times the number outside the division fits within the first digit of the number inside the division bar. Write the result above the division bar, putting the remainder (the remainder of the division) superscript next to the first digit of the dividend.
- In the example, 4 (the divisor) fits within 5 (the first digit of the dividend) only 1 time, leaving 1 (5 ÷ 4 = 1 r.1). Place the quotient (1) above the division bar and place a 1 next to the 5, remembering that 1 is left.
- The 518 under the bar should now look like this: 5118.
Step 3. Divide the divisor by the remainder and the second digit of the dividend
The idea is to match the superscript number with the right dividend digit. Find out how many times the divisor fits into this new two-digit number and write the whole number and the rest as you did before.
- In the problem used as an example, the number formed by the remainder and the second digit of the dividend is 11. The divisor (4) fits 2 times inside the dividend (11), leaving 3 (11 ÷ 4 = 2 r.3). Write the 2 above the division line (resulting in 12) and write the 3 next to the 1 in 518.
- The original dividend, 518, should now read: 51138.
Step 4. Repeat the process until the dividend is finalized
Continue evaluating how many times each divisor fits within the number formed by the digit of the dividend and the superscript to the left of it. When you finish all the digits, you will find the answer to the problem.
- In the same example, the last number of the dividend is 38 - the 3 left over from the previous Step and the original 8 of 518. The divisor (4) fits 9 times into the dividend (38), leaving 2 (38 ÷ 4 = 9 r.2), as 4 x 9 = 36. Write the final remainder (2) above the division bar to complete the answer.
- Therefore, the final answer above the split bar is 129 r.2.
Method 3 of 5: Dividing Fractions
Step 1. Write the equation with the two fractions side by side
To divide fractions, write them side by side, with the division symbol (÷) between the two.
- For example, the problem might be 3/4 ÷ 5/8. To make your life easier, use horizontal lines instead of diagonals to separate the numerator (the top number) from the denominator (the bottom number) of each of the fractions.
Step 2. Invert the numerator and denominator of the second fraction
This inverse fraction is what we call reciprocal.
In the example problem, invert 5/8, putting the 8 on top and the 5 on the bottom
Step 3. Replace the division sign with a multiplication sign
To divide fractions, multiply the first by the reciprocal of the second.
- For example: 3/4 x 8/5.
Step 4. Multiply the fraction numerators
Follow the same procedures as you would when multiplying two fractions.
- In this case, the numerators are 3 and 8. The result would be 3 x 8 = 24.
Step 5. Multiply the denominators of the fractions in the same way
Again, the process is the same as for ordinary fraction multiplication.
- The denominators are 4 and 5, so 4 x 5 = 20.
Step 6. Place the product of numerators over that of denominators
Now that you've multiplied the two fractions, you can form their product.
- In the same problem, it would be 3/4 x 8/5 = 24/20.
Step 7. Reduce the fraction if necessary
To do this, find the greatest common divisor, the largest number capable of dividing the two numbers evenly. Then divide the numerator and denominator by it.
- In the case of the 24/20 fraction, 4 is the largest number that fits equally within 24 and 20. To confirm this, 'factor the numbers and choose the largest number capable of factoring both:
- 24: 1, 2, 3,
Step 4., 6, 8, 12, 24.
- 20: 1, 2,
Step 4., 5, 10, 20.
- 24: 1, 2, 3,
- Since 4 is the highest denominator of 20 and 24, divide the two numbers by it to reduce the fraction.
- 24/4 = 6
- 20/4 = 5
- 24/20 = 6/5. Therefore: 3/4 ÷ 5/8 = 6/5.
Step 8. Rewrite the fraction as mixed numbers if necessary
To do this, divide the denominator by the numerator and write the answer as an integer. The rest, the number on the left, will be the numerator of the new fraction. The denominator will remain the same.
- In the example, 5 fits into 6 with a remainder of 1. So the new integer is 1, the new numerator is 1, and the denominator is still 5.
- As a result: 6/5 = 1 1/5.
Method 4 of 5: Dividing Exponents
Step 1. Check that the exponents have the same base
You can only split numbers with exponents when they share the same base. Otherwise, you have to manipulate them until it becomes a reality - if possible, obviously.
- To practice, practice with a calculus in which the two exponent numbers have the same base - for example, 38 ÷ 35.
Step 2. Subtract the exponents
Subtract the second exponent from the first, without worrying about the base for now.
- In the same problem: 8 - 5 = 3.
Step 3. Place the new exponent on the original base
Just write the new number on the base, and you're done!
- Therefore: 38 ÷ 35 = 33.
Method 5 of 5: Dividing Decimals
Step 1. Write the problem using a division bar
Place the divider (the number to be divided) outside to the left of the divider bar. The dividend (the number that will serve as the basis for the division) must be within the bar. To divide decimals, the first step is to convert them to whole numbers.
- For example 65, 5 ÷ 0, 5, 0, 5 is outside the bar and 65, 5 is inside.
Step 2. Move the decimal places equally to create two whole numbers
Shift decimal places to the right until they reach the end of each number. It is important to move them the same number of places for the two numbers. For example, if you have to move two places on the divisor, do the same on the dividend.
- In the example problem, it is enough to move the square once to the right, both in the divisor and in the dividend. Therefore, 0, 5 becomes 5 and 65, 5 becomes 655.
- Another example: 0, 5, and 65, 55. In this case, you would need to move two decimal places into 65, 55 making it 6555. As a result, you would also need to move two decimal places into 0, 5. To do this, add one 0 at the end, getting 50.
Step 3. Align the decimal points on the split bar
Place a decimal point on the long part of the division bar, just above the decimal point of the dividend.
In the example problem, the decimal point of 655 would appear above the last 5 (such as 655, 0). So write the other decimal point above the division line, just above the 655 point
Step 4. Solve the problem as a long division
To divide 5 into 655, do the following:
- Divide 5 into the hundred 6. You'll get 1 as a result, leaving 1. Put the 1 in the hundredth place on the divide bar and subtract 5 from 6, putting the result at the bottom.
- The 1 that's left is on top. Pass the first 5 of 655 down, creating the number 15. Then divide 5 into 15, getting 3 as a result. Place the 3 on the split bar, next to the 1.
- Pass the last 5 down. Divide 5 by 5, getting 1, and place it on top of the split bar. In this case there is nothing left, as 5 is divided by 5 equally.
- The answer is the number above the dividing bar (131). That is, 655 ÷ 5 = 131. If you pick up a calculator, you'll see that this is the answer to the original problem, 65, 5 ÷ 0, 5.