CONNECT: Decimals OPERATIONS: x and To be able to perform the usual operations (+,, x and ) using decimals, we need to remember what decimals are. To review this, please refer to CONNECT: Fractions Fractions 4 FRACTIONS, DECIMALS, PERCENTAGES how do they relate? You might also like to look at Fractions FRACTIONS 2 OPERATIONS WITH FRACTIONS: x and Before looking at x and with decimal numbers in general, we need to be familiar with x and by 10, 100, 1000, (and other powers of 10). Let s think about 5 x 10. I think you will know that the answer to this problem is 50. This is because if we look at our decimal system of numbers, each column to the left is obtained from the one on its right by multiplying by 10. To show that we have moved over a column, we need to attach a 0 in the units column. x 10 10 millions hundred thousands ten thousands thousands hundreds tens units. tenths hundredths thousandths 5 5 0 If there is a decimal number involved, we still pick up the number and move it across to the left if multiplying by 10, 100, or 1000. This is the same as moving the decimal point in the number to the right the number of times that there are 0s involved (this method is probably the one you learned at school). Example: 5.9 x 100. We pick up 5.9 and move it to the left 2 columns (because we are multiplying by 100) which is equivalent to moving the decimal point to the right 2 places. The result would be 590. (This makes sense because 5 x 100 is 500 and we have a little more, but not as much as 6 x 100, which would be 600). With division, we go the other way. This means that to divide 38 by 10, for example, we need to move the number across to the right by one column, which means we no longer have a whole number, and we need to go into the tenths column. 1
millions hundred thousands ten thousands thousands hundreds tens units. tenths hundredths thousandths 3 8 3. 8 Moving the number to the right one column is the equivalent of moving the decimal point one place to the left. Moving the decimal point is probably easier than drawing up the columns and picking up numbers, however you need to remember that both methods are based on our decimal system of numbers. When multiplying by 10, 100, 1 000, if there is no decimal point, attach the same number of 0s on the end as there are 0s in the 10, 100, or 1 000. If there IS a decimal point, move it to the right the same number of places that there are 0s. When dividing by 10, 100, 1 000 and so on, move the decimal point to the left as many places as there are 0s. So when dividing by 10, move the decimal point one place, by 100 two places, by 1 000 three places and so on. If there is no decimal point in the original number, then imagine it at the end of the number. If there are 0s in the number you are dividing, then remove the same number of 0s as there are in the 10 or 100 or 1 000. Here are some examples: 5.3 x 10 = 53 (the decimal point is moved one place to the right that is, we pick up the number and move it all to the left one place) 23.1 1 000 =.0 231 (the decimal point is moved three places to the left that is, we pick up the number and move it all to the right three places) 78 x 100. There is no decimal point here, so we can just attach two 0s, and our answer is 7 800 593 400 10. Remove a 0 to get 59 340. (We remove just one 0 because we are only dividing by 10.) Here are some for you to try. You can check your answers with the solutions at the end of this resource. 1. 4 100 100 2. 58 x 1 000 3. 4.9 x 100 2
4. 0.05 x 10 5. 0.5 10 6. 1.05 10 7. 23.1 x 100 8. 0.003 x 1 000 9. 58 000 100 10. 0.581 423 x 1 000 000 Multiplying decimal numbers by whole numbers or decimal numbers When we multiply (times) 2 by 3, we get 6, because we think 2 lots of 3. It follows that if we multiply 2.4 by 3, then we must get a little bit more than 6 because now we ve got just a little bit more than 2 lots of 3. If we ignore the decimal point, and just multiply 24 by 3 we get 72. But where does the decimal point go? We know our answer has to be just a bit more than 6, so 72 is too much, 0.72 is too little and 7.2 is just right! But what if we have 0.2 x 0.3? We know 2 x 3 is 6, but where does that decimal point go this time? What we have to realise is that we are actually multiplying two numbers that are both smaller than 1, that is both 0.2 and 0.3 are fractions. We are really finding two tenths of three tenths ( 2 3 10 10 ). Here are our three tenths. We need two tenths of them. So we would get this: 3
We have 100 cells altogether, and 6 of them are shaded because they form the two tenths of the three tenths. This means that the result is 6 100 (6 hundredths) which is the same as 0.06. The quick way to remember what to do with decimals when multiplying is this: Ignore the decimal points and multiply the numbers together (in this case, 2 x 3 = 6). Count up how many digits are after (to the right of) the decimal points in each of the numbers in the question (in this case each number has 1 so a total of 2) and place that many digits after a decimal point in the answer. Our result was 6, so when we insert a decimal point so that there are 2 digits on its right, we must get.06. I like to put a 0 in front of the decimal point as well, just in case it s a little hard to see, so for my answer I would put 0.06. This agrees with the result from the diagram using fractions. Further examples: 1. 4.91 x 0.01 Step 1: 491 x 1 = 491 Step 2: How many digits after (to the right of) the decimal points? A total of 4 Step 3: Insert 4 digits to the right of a decimal point so 0.0 491 Answer? 0.0 491 2. 0.058 x 0.23 Step 1: 58 x 23 58 x 23 174 1 160 1 334 Step 2: How many digits after the decimal points? A total of 5 Step 3: Insert 5 digits to the right of a decimal point so 0.01 334 Answer? 0.01 334 4
Here are some for you to try. You can check your results with the solutions at the end. 1. 14.9 x 3 2. 2.01 x 0.4 3. 0.08 x 0.09 What about 3.8 x 40? Here, we can calculate 3.8 x 4, then move the decimal point in the appropriate direction the appropriate number of places. We work out 38 x 4, which is 152, insert the decimal point for 1 decimal place, gives 15.2, then move the decimal point to the right for the 0 gets us back to 152. Try this one: (check your result with the solutions at the end) 2.03 x 2 000 Dividing decimal numbers by whole numbers or decimal numbers We know that 10 2 = 5 and so can probably realise that 10.6 2 will be a little bit more than 5. In fact, it is 5.3. To do this, we can do a short division like so, making sure the decimal points always line up with each other, because of place value. Once you have put the decimal points in, you can ignore them and continue as if they are not there: 5.3 2)10.6 If there is a remainder, such as with 10.7 2, we can attach as many 0s as we wish to the end of the 10.7: 5.3 5 2)10.7 1 0 We can divide any decimal number by a whole number using this method. Here are some for you to try. You can check your results from the solutions at the end. 5
1. 28.5 3 2. 9.17 7 3. 0.0048 6 4. 1 351.2 16 When dividing decimal numbers by decimal numbers, the trick is to work out where to put the decimal point in the answer. For example, 16 2 is 8. But what about 16 0.2? We need to remember that this means how many 0.2s in 16? There are going to be quite a lot because 0.2 is a small number, less than 1 (in fact, we know it is two tenths). The answer should be 80. To work this out, there is a standard algorithm and this will work no matter what values we are talking about. Because we do NOT know our 0.2 times tables, but we DO know our 2 times tables, let s make this easier, and change the 0.2 into a 2. We do this by multiplying 0.2 by 10, that is we move the decimal point to the right one place. In fact, we move it until the number we are dividing by (in this case 0.2) becomes a whole number. 0.2 This changes the question, however, and to balance this, we also need to multiply the 16 by 10 or by the same number as we multiplied the 0.2 by. So, 16 becomes 160. Our new problem (which has the same value as the old one) becomes 160 2. We can work this out: 80 2)160 so 16 0.2 = 80. Another example: 0.58 0.029 This time we need to move the decimal point three places in 0.029 to make it 29. To agree with this, we need to move the decimal point three places in 0.58. So: 0.029 0.580 and our problem becomes 580 29 20 29)580 and our answer is 20. 6
Again, if there is a remainder, we can add 0s to the decimal in the number we are dividing INTO (not in the one we are dividing by). Last example: 153.013 0.03 This will become 15301.3 3 5100. 4 3 3)15301. 1 3 1 0 The answer will continue indefinitely and can be written as 5 100.43 Here are some for you to try. You can check your results with the solutions at the end. 1. 18.64 0.4 2. 0.005 82 0.03 3. 1 984 0.005 4. 0.003 1.5 If you need help with any of the Maths covered in this resource (or any other Maths topics), you can make an appointment with Learning Development through Reception: phone (02) 4221 3977, or Level 3 (top floor), Building 11, or through your campus. 7
Solutions (Multiplying and Dividing by Powers of 10, page 2) 1. 4 100 100 = 41 (you can remove both 0s) 2. 58 x 1 000 = 58 000 (attach three 0s) 3. 4.9 x 100 = 490 (either pick up the number and move it to the left 2 columns, or move the decimal point 2 places to the right) 4. 0.05 x 10 = 0.5 (either pick up the number and move it to the left by 1 column, or move the decimal point 1 place to the right) 5. 0.5 10 = 0.05 (either pick up the number and move it to the right by 1 column, or move the decimal point 1 place to the left) 6. 1.05 10 = 0.105 (either pick up the number and move it to the right by 1 column, or move the decimal point 1 place to the left. You do not have to put a 0 in front of the decimal point but it does help the viewer see that there is a point there) 7. 23.1 x 100 = 2 310 (either pick up the number and move it to the left 2 columns, or move the decimal point 2 places to the right) 8. 0.003 x 1 000 = 3 (either pick up the number and move it to the left 3 columns, or move the decimal point 3 places to the right) 9. 58 000 100 = 580 (remove 2 0s) 10. 0.581 423 x 1 000 000 = 581 423 (either pick up the number and move it to the left 6 columns, or move the decimal point 6 places to the right) 8
Multiplying decimal numbers (page 4) 1. 14.9 x 3 149 x 3 447 One decimal place is required, so the result is 44.7 2. 2.01 x 0.4 201 x 4 804 Three decimal places are required, so the result is 0.804 3. 0.08 x 0.09 8 x 9 = 72. Four decimal places are required, so the result is 0.0072 2.03 x 2 000 (from page 5) 203 x 2 = 406. 2 decimal places are needed, giving 4.06. Move decimal point 3 places to the right for the 0s, gives 4 060. Dividing decimals by whole numbers or decimals (from page 5) 1. 28.5 3 2. 9.17 7 9. 5 1. 3 1 3)28. 1 5 7)9. 2 17 Answer: 9.5 Answer: 1.31 3. 0.0048 6 0.000 8 6)0.004 4 8 Answer: 0.000 8 9
4. 1 351.2 16 84. 45 16)1351.20 16x 16x 16x 128 8 4 5 71 128 64 80 64 72 64 80 80 0 Dividing decimals by whole numbers or decimals (from page 7) 1. 18.64 0.4 will become 186.4 4 4 6. 6 4)18 2 6. 2 4 Answer is 46.6 2. 0.00 5 82 0.03 will become 0.582 3 0.1 9 4 3)0. 5 2 8 1 2 Answer is 0.194 3. 1 984.000 0.005 will become 1 984 000 5 3 9 6 8 0 0 5)19 4 8 3 4 4 000 Answer is 396 800 4. 0.003 1.5 will become 0.03 15 0.00 2 15)0.03 3 0 Answer is 0.002 10