Content

• To step
• Method 1 of 5: Long division
• Method 2 of 5: Short division
• Method 3 of 5: Dividing fractions
• Method 4 of 5: Share exponents
• Method 5 of 5: Dividing decimal numbers

Division is one of the four major arithmetic operations, in addition to addition, subtraction, and multiplication. In addition to whole numbers, you can also divide decimals, fractions or exponents. You can do long division or, if one of the numbers is a single digit, do short division. Start by mastering long division, though, because that's the key to the whole operation.

To step

Method 1 of 5: Long division

1. Write out the problem using a long division sign. The long division sign ( 厂 ) looks like an "end bracket" with a number below it. Place the denominator, the number you're dividing by, outside the long division sign, and the numerator, the number you're dividing, inside the long division sign.
   • Sample exercise # 1 (beginner): 65 ÷ 5. Place the 5 outside the division sign, and the 65 inside. It should look like 5厂65, but with the 65 below the horizontal.
   • Sample exercise # 2 (advanced): 136 ÷ 3. Place the 3 outside the division sign, and the 136 inside. It should look like 3厂136, but with the 136 below the horizontal.
2. Divide the first digit of the numerator by the denominator. In other words, find out how many times the denominator (the number outside the division sign) goes into the first digit of the numerator. Place the integer result above the division sign, right above the first digit of the denominator.
   • In exercise # 1 (5厂65), 5 is the denominator and 6 is the first digit of the numerator (65). 5 goes into 6 once, so put a 1 on the division sign, above 6.
   • In exercise # 2 (3厂136), 3 (the divisor) does not fit entirely into 1 (the first digit of the numerator). In this case, write a 0 above the division sign, above the 1.
3. Multiply the number above the division sign by the denominator. Take the number you wrote just above the division sign and multiply it by the denominator (the number to the left of the division sign). Write the result in a new row below the counter, aligned with the first digit of the counter.
   • In exercise # 1 (5厂65), multiply the number above the bar (1) by the denominator (5), which results in 1 x 5 = 5, and place the answer (5) just below 6 of 65.
   • In exercise # 2 ("3厂136) there is a zero above the division sign, so if you multiply this by 3 (the denominator), the result is zero. Write a zero on a new line just below 1 of 136.
4. Subtract the product (result of the multiplication) from the first digit of the numerator. In other words, subtract the number you just wrote in the new line below the counter from the number in the counter immediately above it. Write the result in a new row, aligned below the digits of the subtraction sum.
   • In exercise # 1 (5厂65), subtract the 5 (the product in the new row) from the 6 above it (the first digit of the numerator): 6 - 5 = 1. Place the result (1) in another new row directly below the 5.
   • In exercise # 2 (3厂136) subtract the 0 (the product in the new row) from the 1 at the top right (the first digit in the numerator). Place the result (1) in another new row directly below the 0.
5. Bring down the second digit of the counter. Bring the second digit of the numerator down to the new bottom row, just to the right of the result of the subtraction you just got.
   • In exercise # 1 (5厂65), bring the 5 down from 65 so that it is next to the 1 you obtained by subtracting 5 from 6. There is now 15 in this row.
   • In exercise # 2 (3厂136), bring the 3 down from 136 and place it next to the 1, giving you 13.
6. Repeat long division (exercise # 1). This time, use the numerator (the number to the left of the division sign) and the new number on the bottom row (the result of your first round of math and the number you carried down). Just like before, divide, multiply and subtract numbers to get the result.
   • To continue with 5厂65, divide the new number (15) by 5 (the denominator), and write the result (3, because 15 ÷ 5 = 3) to the right of the 1 above the division sign. Then multiply these 3 above the division sign by 5 (the denominator) and write the result (15, because 3 x 5 = 15) below 15 under the division sign. Finally, subtract 15 from 15 and write 0 in a new bottom row.
   • Sample exercise # 1 is now complete, as there are no more digits to lower in the denominator. The answer (13) is above the division sign.
7. Repeat long division (exercise # 2). As before, you start by dividing, multiplying and then subtracting.
   • In front of 3厂136: Determine how many times 3 goes completely into 13, and write the answer (4) to the right of the 0 above the division sign. Then multiply 4 by 3 and write the answer (12) below 13. Finally, subtract 12 from 13 and write the answer (1) below 12.
8. Do another long division round and get the rest (problem # 2). When you are done with this problem, make sure there is a remainder (that is, a number that remains at the end of your calculation). You place this remainder next to your entire answer.
   • In front of 3厂136: Continue the process for another round. Bring the 6 down from 136, leaving 16 in the bottom row. Divide 16 by 3 and write the result (5) above the division sign. Multiply 5 by 3 and write the result (15) in a new bottom row. Subtract 15 from 16 and write the result (1) in a new bottom row.
   • Since there are no more digits to include in the counter, you are done with the problem and the 1 on the bottom line is the remainder (the number that remains). Write it above the division sign, optionally with an "r." In front of it, so that your final answer becomes "45 r.1".

Method 2 of 5: Short division

1. Use a dash to write the problem. Place the denominator, the number by which you are going to divide, outside (and to the left of) the division line. Place the numerator, the number you are going to divide, inside (to the right of and below) the division line.
   • For quick division, the denominator can only be one digit.
   • Statement: 518 ÷ 4. In this case, the 4 will be outside the dash and 518 will be inside.
2. Divide the first digit of the numerator by the denominator. In other words, determine how many times the number outside the dash fits into the first digit of the number inside the dash. Write the integer of the result above the dash, and write any remainder in superscript next to the first digit of the numerator.
   • In this problem, 4 (the denominator) fits once into 5 (the first digit of the numerator), with a remainder of 1 (5 ÷ 4 = 1 r.1). Place the quotient, 1, above the long division line. Place a small superscript 1 next to the 5 to remind yourself that you had a remainder of 1.
   • The 518 below the dash should now look like this: 518.
3. Divide the remainder and the second digit of the numerator by the denominator. Treat the superscript number that indicates the rest as a full digit, and combine it with the digit of the numerator to the immediate right of it. Determine how many times the denominator goes completely into this new 2-digit number, and write down the whole number and any remainder as you did before.
   • In the problem, the number formed by the remainder and the second number of the numerator is 11.the denominator (4), goes into 11 twice, leaving a remainder of 3 (11 ÷ 4 = 2 r.3) remains. Write the 2 above the dash (giving you 12) and the 3 as a superscript number next to the 1 in 518.
   • The original counter, 518, should now look like this: 518.
4. Repeat this until you have gone through the entire counter. Keep determining how many times the denominator goes in the number formed by the next digit of the numerator and the remainder in superscript to the immediate left of it. Once you have gone through all the digits of the counter, you have your answer.
   • In the problem, 38 is the next (and last) number of the counter - the remainder 3 from the previous step, and the number 8 is the last term of the counter. The denominator (4) goes into 38 nine times with a remainder of 2 (38 ÷ 4 = 9 r.2), because 4 x 9 = 36, which is two less than 38. Write this last remainder (2) above the dash to complete your answer.
   • Your last answer above the division line is therefore 129 r.2 ..

Method 3 of 5: Dividing fractions

1. Write the division sum so that the two fractions are next to each other. To divide fractions, write the first fraction followed by the division symbol (÷), then the second fraction.
   • For example, the statement could be something like: 3/4 ÷ 5/8. For convenience, use horizontal instead of diagonal lines to separate the numerator (the top number) and the denominator (the bottom number) of each fraction.
2. Reverse the numerator and denominator of the second fraction. The second fraction becomes its own inverse.
   • In this example problem, we'll flip 5/8 so that the 8 is at the top and the 5 at the bottom.
3. Change the dash to a multiplication sign. To divide fractions, multiply the first fraction by the reciprocal of the second.
   • For instance: 3/4 x 8/5.
4. Multiply the numerators of the fractions. Follow the same procedure as for multiplying two fractions.
   • In this case, the counters are 3 and 8, and 3 x 8 = 24.
5. Multiply the denominators of the fractions in the same way. Again, this is exactly what you would do to multiply two fractions.
   • The denominators are 4 and 5 in the problem, and 4 x 5 = 20.
6. Place the product of the numerators above the product of the denominators. Now that you've multiplied the numerators and denominators of both fractions, you can form the product of the two fractions.
   • In the statement: 3/4 x 8/5 = 24/20.
7. Simplify the fraction, if necessary. To simplify the fraction, find the greatest common divisor, or the greatest number that fits in both numbers in its entirety, and then divide both the numerator and denominator by that number.
   • In the case of 24/20, 4 is the largest number that evenly goes into both 24 and 20. You can confirm this by writing out all divisors of both numbers and choosing the largest number that is a divisor of both:
     • 24: 1, 2, 3, 4, 6, 8, 12, 24
     • 20: 1, 2, 4, 5, 10, 20
   • Since 4 is the greatest common divisor of 24 and 20, divide both numbers by 4 to simplify the fraction.
     • 24/4 = 6
     • 20/4 = 5
     • 24/20 = 6/5. So: 3/4 ÷ 5/8 = 6/5
8. Rewrite the fraction as a mixed number, if necessary. To do this, divide the numerator by the denominator and write the answer as an integer. The remainder (the number that remains) is the numerator of the new fraction. The denominator of the fraction remains the same.
   • In the problem, 5 goes into 6 once with a remainder of 1. So the new integer is 1, the new numerator is 1, and the denominator remains 5.
   • The result: 6/5 = 1 1/5.

Method 4 of 5: Share exponents

1. Make sure the exponents have the same base. You can divide exponents if they have the same base. If they don't have the same base, you will have to manipulate them until they do, if possible.
   • If you are just starting with this, first do a problem where both exponents already have the same base. For instance: 3 ÷ 3.
2. Subtract the exponents. Just subtract the second exponent from the first. Don't worry about the base for now.
   • In the statement: 8 - 5 = 3.
3. Place the new exponent above the original base. Just write the new exponent above the original base. That's all!
   • Thus: 3 ÷ 3 = 3.

Method 5 of 5: Dividing decimal numbers

1. Write down the problem with a dash. Place the denominator, the number by which you are going to divide, outside (and to the left of) the long division bar, and the numerator, the number that you are going to divide, inside the long division bar. To divide decimals, first convert the decimals to whole numbers.
   • In the example 65,5 ÷ 0,5 0.5 is placed outside the division line and 65.5 inside it.
2. Move the decimal points by the same amount to create two integers. Just slide the decimal points to the right until they are at the end of each number. Make sure you move them the same number of positions for each number - if you need to move the decimal point two places in the denominator, do the same for the numerator.
   • In the problem, all you have to do is move the decimal point one position for both the denominator and the numerator. So 0.5 becomes 5 and 65.5 becomes 655.
   • However, if the numbers in the problem were 0.5 and 65.55, then you have to move the decimal point two places in 65.55, making it 6555. As a result, you should also shift the decimal point two places in 0.5. To do this, add a zero to the end and make it 50.
3. Place the decimal point directly above the division line. Place a decimal point on the long division sign directly above the decimal in the numerator.
   • In the problem, the decimal in 655 comes after the last 5 (as 655.0). So write the decimal point above the division line directly above the decimal point in 655.
4. Solve the problem by doing long division. To divide 655 by 5, do the following:
   • Divide the hundredth (6) by 5. You get 1, with remainder 1. Place 1 in place of the hundredth on top of the long division line, and subtract 5 from 6 below the number six.
   • The rest, 1, remains. Bring down the first five in 655 and you get the number 15. Divide 15 by 5 and you get 3.Place the three above the long division sign, next to the 1.
   • Bring down the last 5. Divide 5 by 5 and you get 1 - place the 1 above the long division sign. There is no remainder as 5 goes into 5 once.
   • The answer is the number above the long division sign (131), so 655 ÷ 5 = 131. If you bring in a calculator, you will see that this is also the answer to the original division: 65,5 ÷ 0,5.