- Div_1
Numbers: Getting to grips with division
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978 1 47300 034 6 (.kdl)

978 1 47300 129 9 (.epub)
After studying this course, you should be able to:
divide one number by another
divide using decimals
practise division skills learnt.
IntroductionDo you want to improve your ability to divide one number by another without having to rely on a calculator? This course will help you get to grips with division and give you some practice in dividing numbers.You don’t need to complete the whole course if only certain sections are relevant to you. I start with the basics, where you’ll have the opportunity to get some practice in dividing small numbers in your head. Then I deal with dividing bigger numbers and decimals. If you are confident you already have the skills covered by the early sections, you can move through them quickly until you get to the information you really need.On the other hand, if you find you do need to start with the basics, it may be best to take your time and work on this course over several visits.Find out more about studying with The Open University by visiting our online prospectus1 Ways of expressing divisionDivision is the opposite of multiplication. If you say:What do I get if I divide 20 by 4?It is equivalent to saying:What do I have to multiply 4 by to get 20?Division can be expressed in various ways. You can say:divided by – as in ‘8 divided by 4’over – as in ‘8 over 4’divided between – as in ‘8 apples divided equally between 4 people’.It can also be expressed as how many times one number ‘goes into’ another. As you will see below, this is similar to wording the division as if it were a multiplication.So the following expressions all mean the same thing:What do I get if I divide 20 by 4?What is 20 over 4?What do I have to multiply 4 by to get 20?How many times does 4 go into 20?Division calculations can also be represented in several different ways. All of the following mean 20 divided by 4:2 Dividing in your headTo divide one number into another without using a calculator you need to know basic multiplications up to 10. This means that you need to know, by heart, what result you get if you multiply (times) any number up to 10 by any other number up to 10. For example you have to remember that 9 multiplied by 6 is 54, or 4 multiplied by 5 is 20, and so on.If you are confident that you know the basic multiplications up to 10, just carry on with the rest of this course. If you are unsure, or would like some practice to help you remember them, go now to the last section of this course (Appendix: multiplication tables) by following the link on the left. Then return here to continue.<b>3 Division rules – order matters</b>An important thing to remember about division is that it has different rules from multiplication.For example when you multiply two numbers together, it doesn’t matter what order you multiply them in. So 8 x 4 is exactly the same as 4 x 8. The result is 32 in both cases.But in division, order matters. So 8 ÷ 4 is different from 4 ÷ 8.The answer to the first is 2. If you multiply 4 by 2 you get 8.But the answer to the second is 0.5 (a half). If you multiply 8 by 0.5, you get 4 (in other words eight halves are equivalent to 4 wholes).4 Whole numbers, fractions and decimals<b>4.1 Whole numbers</b>If you add, subtract or multiply whole numbers, you always end up with a whole number. But division is different. One whole number may not ‘go into’ another exactly. There may be something left over.For example if you have 6 biscuits to divide between 4 people, each person can have one biscuit each and 2 are left over. In division, the 2 left over are called the remainder. This is expressed as:6 ÷ 4 = 1, remainder 24.2 FractionsInstead of putting the two left over biscuits back in the tin, you might instead decide to break them into halves and then give one of the resulting four halves to each person so that everyone receives a total of one and a half biscuits. In this case, the answer has to be expressed as a fraction or the equivalent decimal number:6 ÷ 4 = 1½6 ÷ 4 = 1.5A fraction is really just another way of expressing a division because ½ means ‘1 over 2’, which is another way of saying 1 divided by 2. Each of the following fractions is the same because in each case you have to multiply the top number by 2 to get the bottom number. 1 is half of 2, 2 is half of 4, 6 is half of 12, and 25 is half of 50.It’s a convention when using fractions to ‘reduce’ them so that they have the smallest possible numbers at the top and bottom. So you usually write instead of, for example, .4.3 DecimalsA decimal number is a different way of representing numbers smaller than one. You put them after a full stop (the decimal point), for instance 0.5. The first digit after the decimal point represents tenths. If you sliced a cake into 10 slices, each slice would be a tenth of the cake. So 0.5 is the same as saying 5 tenths, and can be written . This is the same as because can be reduced to .Further digits after the decimal point represent hundredths, then thousandths, and so on. For example the number 2.25 represents 2 whole numbers, 2 tenths and 5 hundredths. But it’s easier if you think of the digits after the decimal point as being a single fraction. Work out what the last digit after the decimal point represents. For example in 2.25, the 5 is in the hundredths position. Then take all the digits after the decimal point and put them over whatever the last digit represents. So .25 becomes 25 hundredths, or . 2.25 as whole numbers and a fraction would be .Sometimes numbers need a lot of digits after the decimal point to show the exact number. In most cases, however, it is sufficient to show two decimal places. For example you would usually represent as 0.33, even though the number actually has a lot more digits after the decimal point.5 Practice dividing in your headBefore you go any further, you might want to practise doing some divisions in your head using basic multiplications of numbers up to 10. For example if you know:8 multiplied by 6 is 48then you also know:48 divided by 6 is 8To practise, go to one of the following websites:The Practice sums page of the Numbers website. Select **Division** from the pull-down list next to **Type of sum**. Choose which number you want to practise dividing by from the pull-down list next to **Times table**. Then follow the instructions on the page.The Basic Math page of the math.com website. Select **Divide** under **1. Choose an operation**. Under **2. Choose numbers from 0 to 12**, choose the highest and lowest numbers you want to divide by, then click on **Go**.When you have finished practising, return here and carry on with this course.6 Dividing on paperIf the numbers you want to divide are too large for you to do the calculation in your head, you can use a calculator. Alternatively, you can do the calculation on paper. In the example below, click on each step in turn to see how to divide 126 by 6.
Activity 1Click on ‘Reveal discussion’ to see all the individual steps at once.This division was straightforward because the dividing number (6) was less than 10 and the calculation at each stage worked out exactly, with no remainder. But what do you do if, at a particular stage, you end up with a remainder? This requires an extra step called carrying, which is explained in the next section.7 Dividing when you have to carryIf the number you are dividing by does not go exactly (with no remainder) into the digit you are dividing into, you need to do something called carrying.Say you want to divide 952 by 7. The process is basically the same as in the previous section. First write it down on paper. Then, to do the calculation, you take each digit from the number being divided in turn, starting with the one on the far left, and see how many times the dividing number, 7 in this case, goes into it. The calculation is shown below. Click on each step in turn to see how carrying is done.
Activity 2Click on ‘Reveal discussion’ to see all the individual steps at once.In your head, as you do this calculation, you’ll probably be thinking something like ‘Seven into 9 goes once, leaving 2 over. Seven into 25 goes 3 times (because 3 sevens are 21), leaving 4 over. Seven into 42 goes 6 times (because 6 sevens are 42)’.8 Dealing with remaindersHow do you deal with divisions where there is a remainder and there are no more digits you can carry to? In most cases you will need to express your answer as a decimal number rather than as a whole number plus a remainder.Take the example of 518 divided by 8.This calculation leaves a remainder of 6 but there seem to be no more digits to carry the remainder over to. But think of 518 as actually being 518.0. The decimal point is always there in any number, it’s just that, normally, it’s not written in if it only has zeros after it. So 518 can be written as 518.0 or 518.00, or as 518 with any number of zeros after the decimal point.So if you insert the decimal point and a zero after 518, you can carry the remainder 6 over and put it in front of the zero, making it 60. Click on each step in turn to see how to finish the calculation.
Activity 3Click on ‘Reveal discussion’ to see all the individual steps at once.As long as there is a remainder, you can keep on adding zeros after the decimal point, continuing either until the division works out exactly or you have as many decimal places as you need.<b>9 Summary of what you’ve learned so far</b>When you divide by a number up to 10, the steps are as follows:Take each digit of the number under the line in turn, starting from the left.Work out how many times the dividing number goes into it.Write the answer to the division above the lineIf there is a remainder, carry it by putting it in front of the next digit on the right.Work out how many times the dividing number goes into the next digit, including any carried remainder.Continue, repeating the cycle, and adding zeros after the decimal point if necessary.Stop when you get an exact division with no remainder or your answer has as many decimal places as you want.10 Dividing by big numbers – long divisionIn the previous sections you saw how to divide a big number by a small number up to 10. Things get harder if you want to do a division where both the numbers are big. This kind of calculation is called long division, probably because you write the steps of the calculation out on paper in a long sequence.The principle of doing long division is the same as when you divide by a number up to 10. The only difference is that, because the numbers involved in long division are usually too big to work out in your head, you need to write more steps down on paper. The next section looks at how you do it.11 Example of long divisionThe example of 25546 divided by 53 is suitable for long division. First write the calculation down on paper in the same way you did before.Look at the first digit of the number you are dividing into. This is 2, but 53 doesn’t go into 2. So you take the next digit as well, which makes it 25, but 53 doesn’t go into 25 either. So you take the next digit as well, which makes it 255.Now you have to work out how many times 53 goes into 255. This step usually involves trying out multiples of 53. First make an estimate in your head. For example there are two 50s in a hundred, so there will be four 50s in 200. But 255 probably isn’t big enough for 53 multiplied by 5. So assume for the moment that the answer is 4 and work out what 53 multiplied by 4 is.The result is 212 as shown below. This is less than 255, which it needs to be. It is also close enough to 255 to suggest that 53 multiplied by 5 would be bigger than 255, so 4 is probably right. Click on each step below in turn to see how to continue with long division.
Activity 4Click on ‘Reveal discussion’ to see all the individual steps at once.At this point you can check whether your answer for this column is correct or not. You can’t have a remainder that is bigger than the number you are dividing by. So if the result of your subtraction is smaller than the number you are dividing by, then your answer for this column is correct. If it isn’t, then you need to multiply 53 again. In this case, the result of the subtraction is 43. This is smaller than 53, which is the number you are dividing by, so the answer of 4 is correct.Next you need to carry the remainder and put it in front of the next digit, turning the 4 into 434. But if you have a big number to carry (and in divisions with big numbers the remainder could be several digits long), trying to write it into the small space between the 5 and the 4 is difficult. So instead, write the 4 after the remainder of 43, as shown below, to make it 434.Now work out how many times 53 goes into 434. Do an estimate in your head then multiply it out to see if it is close enough to the total for any remainder to be less than the number you are dividing by. Click on each step below in turn to see how the long division is completed.
Activity 5Click on ‘Reveal discussion’ to see all the individual steps at once.So far, all the division calculations I’ve considered have involved one whole number divided by another whole number. What do you do if you want to do a division that involves decimal numbers? This is what the next section will look at.12 Dividing decimal numbers by moving the decimal pointDoing division when decimal numbers are involved is the same as doing divisions involving whole numbers, with a few extra steps to take care of the decimal point.Either the number you are dividing into or the number you are dividing by, or both of them, may be a whole number or a decimal number. So, for example, you might want to do the following divisions:Example 1: 49.26457 ÷ 8Example 2: 2.601 ÷ 1.22Example 3: 678 ÷ 27.356Before starting to do the division, you need to adjust the numbers so that the number you are dividing **by** is a whole number. In Example 1 above, the number you are dividing by (8) is already a whole number so you don’t need to do anything, even though the number you are dividing into (49.26457) is a decimal number.In Examples 2 and 3 above, the number you are dividing by **is** a decimal number, so you do need to make adjustments. You have to:Move the decimal point to the right in the number you are dividing **by**, until the divider becomes a whole number.Count the number of places you have to move the decimal point in order to do this.Move the decimal point in the number you are dividing **into** the same number of places to the right as you did in the number you are dividing **by**.Look at Example 2 above. To make 1.22 into the whole number 122 you have to move the decimal point two places to the right. So you also move the decimal point two places to the right in 2.601, making it 260.1.
Activity 6Click on ‘Reveal discussion’ to see all the individual steps at once.In Example 3 above, you have to move the decimal point three places to the right to make the divider into the whole number 27356. But the number you are dividing into is **already** a whole number, so there is no decimal point to move. Click on each step below to see how the decimal point moves in Example 3.
Activity 7Click on ‘Reveal discussion’ to see all the individual steps at once.Once the number you are dividing **by** is a whole number, you proceed with the division on paper in exactly the same way as when dealing entirely with whole numbers.The only difference is that you ignore the decimal point in the number you are dividing **into**, if it still has one, until the end of the calculation. Then, to complete your answer, you insert the decimal point in the appropriate position. How you do this is shown in the next section.<b>13 Dividing decimal numbers – an example</b>Say you want to divide 39.44 by 2.9. After adjusting the decimal points this becomes 394.4 divided by 29. The division, ignoring the decimal point, is shown below.The final step is to insert the decimal point in the right place in your answer. Click on step 1 below to see how.
Activity 8Click on ‘Reveal discussion’ to see all the individual steps at once.As an extra check to make sure you have the decimal point in the right place, it is always a good idea to do an estimate of the answer you expect. The division above, in its original form, was roughly 40 divided by 3, which works out at approximately 13. Knowing this you can be confident you have put the decimal point where it should be.This example of decimal division worked out, with no remainder. If there **is** a remainder, you simply add extra zeros at the end of the number you are dividing into, just like when dividing with whole numbers. This time of course the decimal point is already in its place, so there is no need to add it.14 Practice dividing on paperNow that you’ve learned the principles of doing division on paper, you may want to practise your new skills. If so, go to the Dividing decimals page of the math.com website and follow the instructions. You will need a pen and paper to carry out each calculation. You can then enter your answer on the website to check if it is correct.The practice calculations on this web page all involve decimal numbers. This is good practice because if you can divide decimals, you can divide anything.15 Appendix: multiplication tablesIf you want to be able to do division without using a calculator, you need to know by heart what you get if you multiply any two numbers up to 10. All the possible combinations can be shown in a multiplication table (also called a times table), like the one below.Say you want to multiply 6 by 8. Look along the top row for the ‘6’ column. Look down the leftmost column for the ‘8’ row. The answer is shown where the ‘8’ row meets the ‘6’ column – so in this case the answer is 48.In other words, the top row shows one of the numbers you want to multiply. The leftmost column shows the number you want to multiply it with. The result of the multiplication is shown where the row and column cross.Notice that it doesn’t make any difference which way round you choose your row and column. Because 6 x 8 is the same as 8 x 6, the intersection between row 6 and column 8 is the same as the intersection between row 8 and column 6.If you want to, you can use this table to practise remembering all the possible combinations. There is also an interactive version of a multiplication table on the Multiply page of the Numbers website, which you might find helpful.To help you remember you can do some practice calculations if you want by going to one of the following websites:The Practice sums page of the Numbers website. Select **Times tables** from the pull-down list next to **Type of sum**. Choose which number you want to practice multiplying by from the pull-down list next to **Times table**. Then follow the instructions on the page.The Basic Math page of the math.com website. Select **Multiply** under **1. Choose an operation**. Under **2. Choose numbers from 0 to 12**, choose the highest and lowest numbers you want to multiply by, then click on **Go**.Both of these websites can also give you practice in addition, subtraction or division if you want – just select the appropriate option from the **Type of sum** or **1. Choose an operation** menu.ConclusionThis free course provided an introduction to studying Mathematics. It took you through a series of exercises designed to develop your approach to study and learning at a distance and helped to improve your confidence as an independent learner.Keep on learning Study another free courseThere are more than **800 courses on OpenLearn** for you to choose from on a range of subjects. Find out more about all our free courses. Take your studies furtherFind out more about studying with The Open University by visiting our online prospectus. If you are new to university study, you may be interested in our Access Courses or Certificates. What’s new from OpenLearn?
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