Decimals

Let’s start with this example, 3.04 plus 17 plus 21.479. First of all set out the decimals as shown. A couple of things to notice here. The first thing is that the 17 has been turned into a decimal, in other words, it’s now 17.000. Why? Because the 21.479 decimal has got three numbers after the decimal point. They’re called decimal places. So the 17 has three decimal places as well. Notice also the 3.04 has got an additional decimal place to make it three decimal places as well. So notice how the columns line up, each of the decimals has now got three decimal places.

The other thing to notice are the decimal points have been lined up in one column, and that’s most important. So let’s go ahead and do the addition. Starting with the right-hand column, zero plus zero plus nine is nine. The next column, four plus seven is 11, which is one and we carry the one shown in red and put it next to the four. The next column, zero plus zero plus four plus one is five. Now moving onto the other side of the decimal points, three plus seven plus one is 11, which is one and we carry the one as we did with the other one. In other words, we now have to add one plus two plus one which gives us four. So the answer to this decimal addition is 41.519.

Now do some questions for yourself. The answers to these questions are on the next slide. Thank you for watching this short movie.

I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on subtracting decimals.

Let’s start with an example. 825.709 take away 272.456. Make sure at the outset that you line up the decimal points. Now let’s carry out the subtraction. Nine take away six is three. The next column isn’t so easy, we’re saying zero take away five. In this case we have to add a one which effectively is a 10 to the top number. So we’re now saying 10 take away five is five. Now we carry the one at the bottom number, in other words the bottom number is four plus one which is five. So the subtraction now is seven take away five which is two. Now moving over to the left hand side of the decimal point, five take away two is three. The next column is two take away seven, which as before with a zero we can’t do, so what we do, we add the one or effectively the 10. So the 10 plus the two on the top becomes the number 12. So we’re saying 12 take away seven which is five. As before make sure you carry the one on the bottom, so the number on the bottom now becomes two plus one which is three. So we say eight take away three is five. So the subtraction to this problem is 553.253.

Now do some questions for yourself. The answers to these problems are on the next slide. Thanks for watching this short movie.

Let’s start by multiplying decimals by 10, 100 and 1000. So let’s start with the decimal 5.2 times 10. When we multiple by 10 notice that we have one zero, and when you multiply by 10 you simply move the decimal point one place to the right. In other words, 5.2 times 10 becomes 52.0, or just 52. In other words, the rule is move the decimal point one place to the right for each zero. Another one, 62.7 times 100. Notice two zeros, so we simply move the decimal point two places to the right, so this answer becomes 6270. Finally, let’s move 3.14 times 1000. Here we move the decimal point three places to the right because we have three zeros, 1000 has got three zeros. So the answer is 3140.

Now let’s look at a multiplication which does not involve multiplying by 10, 100 or 1000. We have 8.5 times four. It’s set out as shown with the four underneath the five on the right hand side. Notice also that there is no decimal point in this multiplication. We’ll come back to that in a moment. So we start by multiplying the right hand columns, four times five is 20, zero under the four, carry the two, which I placed in red above the eight. Now I say four times eight is 32 and now I add the two to the 32 to give us 34. So we now have an answer with 340. Now we have to decide where to place the decimal point, and what you do now is to go up to the question again and count up how many numbers there are after the decimal point. Here we have one number, the five, which is after the decimal point. There are no numbers after the four. In other words we have one decimal place. So what we do is we move back from the right hand side one decimal place also. So the decimal point is now between the four and the zero. So our answer is 34.0.

Now let’s look at a slight more complicated multiplication, 4.9 times 2.8. First of all set it out as shown. Note again there are no decimal points, 49 times 28. First of all we start on the right hand side by multiplying eight times nine which is 72, carry the seven, shown in red at the top. Multiply eight times four which is 32, add the seven and we get 39. So our first row is 392. Move onto the next row. Here we’re multiplying by 2, so first of all we place a zero in the right hand column. Now we multiply two by nine which is 18, carry the one shown at the top. Two times four is eight plus one is nine. Now we have our second row 980. All we need to do now is add up the two rows. Two and zero is two, nine and eight is 17, carry one, nine and three is 12 plus one is 13. Finally we decide where to place the decimal point. Remember the rule, how many decimal places are there in the question. There are two, there’s a nine and there’s an eight, so there are two decimal places, which mean that we must move the decimal point back two places from the right hand side. In other words, the answer to this question is 13.72.

Now try some questions for yourself. The answers to these questions are on the next slide. Thank you for watching this short movie.

Let’s start by dividing decimals by 10, 100 and 1000. Here are some examples. Let’s look at the first example, 53.7 divided by 10. Notice here that 10 has got one zero. So what we do when we divide by 10 is to move the decimal point one place to the left. In other words, the decimal point is now between the five and the three, giving us an answer of 5.37. Second example, 462.9 divided by 100. Here 100 has got two zeros, so we move the decimal point two places to the left. So the decimal point is now between the four and the six, giving us an answer of 4.629. Finally, 3782 divided by 1000. Notice here that 1000 has got three zeros, so we move the decimal point three places to the left, giving us a decimal point between three and seven, so the answer now is 3.782.

Now let’s look at some decimal divisions where we’re not dividing by 10, 100 or 1000. Here’s a fairly straight forward one, 8.4 divided by four. Setting it out as shown, four into 8.4, notice I’ve lined up the decimal points to begin with. Four into eight goes two, and four into four goes one, giving us an answer of 2.1.

What happens if we’re not dividing by a whole number? Here’s an example, 8.48 divided by 0.4. Notice 0.4 is not a whole number. What we do is to turn that 0.4 into a whole number and to do that we have to move the decimal point one place to the right. In other words, instead of the decimal point between zero and four, it’s now after the four, or just four. But what we do to one decimal we have to do the other decimal, the 8.48 has to be changed as well. So moving the decimal point one place to the right give the new decimal 84.8. So we now have 84.8 divided by 4. So now we go ahead and carry out the division, four into 84.8. Four into eight goes two, four into four goes one. Notice the decimal points are lined up, and finally four into eight goes two, giving us an answer of 21.2.

Now let’s extend this decimal division to a slightly harder problem, 145.25 divided by 0.25. Again notice the problem. The problem is 0.25 is not a whole number. So we first of all have to turn that into a whole number. So what we do is move the decimal point two places to the right, giving us 25. Similarly, because we’ve done that to the 0.25, we must move the decimal point two places to the right for our 145.25, giving us an answer of 14525. So what we’re carrying out now is a division 14525 divided by 25. So let’s go ahead and carry out that division 25 into 14525. 25 into one won’t go, 25 into four won’t go, 25 into 145 goes five times and that gives us a remainder of 20. So we place the 20 next to the two. So we’re now dividing 25 into 202, which goes eight times and that gives us a remainder of two, which we place next to the five. So we’re now saying 25 into 25 goes one exactly. So the answer to this question 145.25 divided by 0.25 is 581.

Now do some problems for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.

Let’s start by converting a decimal into a fraction. Here’s the decimal 0.455. The first thing to notice about this decimal is that there are three digits after the decimal point. The first digit is the 10th, the second digit the 100th, and the third digit is the 1000th. It’s the third digit you look at because the 1000th is the number that you place on the bottom of the fraction. In other word, what goes on the bottom of this fraction is 1000. What goes on the top is the decimal without the decimal point, in other words it’s 455. So 0.455 as a fraction is 455 over 1000. Now you look to see if anything cancels top and bottom. And as you can see here, you can divide both top and bottom by five. Five into the top goes 91. Five into 1000 goes 200. So 0.455 as a fraction is 91 over 200.

Now let’s look at a whole number decimal. Here’s the decimal, 3.75. As before look at the digits, the first digit before the decimal point is a one. The digit after the decimal point is a 10th. The second digit is the 100th. Concentrate on that last digit, which is a 100th, in other words, the denominator of your fraction will be 100. The top of the fraction will be the decimal without the decimal point, in other words, 375. So 3.75 is 375 over 100. Now look to see if anything cancels and as you can see fives will divide into both top and bottom, 75 and 20. But also notice that 75 over 20 will cancel again, again five will divide into both top and bottom, giving us 15 over four. Turn that improper fraction into a mixed fraction, in other words, 15 over four is three and three quarters. So 3.75 as a decimal is three and three quarters as a fraction.

Now let’s convert a fraction into a decimal. Here’s the fraction, two and three 8ths. The first thing we do is to convert this mixed fraction into an improper fraction, so two and three 8ths is 19 over eight. 19 over eight means 19 divided by eight, so when we turn it into a decimal we have to use the process as shown. Notice that the eight goes outside of the division and 19 goes inside. So now we carry out the division, eights into 19 goes two times. Notice here we have two times with a remainder of three. Now if there’s a remainder then we have to place a decimal point and zeros after that decimal point. So carrying on, because we remainder three, we place the three next to the zeros, the number becomes 30. So we divide eight into 30, goes three times, three eights are 24, remainder six. Place the six next to the zero. The number is now 60. Eight into 60 goes seven, eight sevens are 56, remainder four. So the new number is 40. Eight into 40 goes five times exactly. Notice we keep going until we do not have a remainder. So our fraction two and three 8ths is 2.375 as a decimal.

Now do some questions for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.