WEBVTT Kind: captions Language: en 00:00:01.576 --> 00:00:05.846 >> Welcome to the Cypress College Math Review on decimal operations. 00:00:07.406 --> 00:00:10.286 Objective 1: Place settings. 00:00:10.736 --> 00:00:15.466 Here we have a table displaying the first few place settings around the decimal point. 00:00:16.186 --> 00:00:20.556 Notice that each digit after a decimal point will represent a different fraction, 00:00:20.586 --> 00:00:27.726 with the denominators of 10, 100, 1,000 etc. Example. 00:00:28.416 --> 00:00:34.596 For the number 48.1268, we want to identify the digit in the hundredth place. 00:00:35.286 --> 00:00:41.006 As we can see from the previous chart, the digit in the hundredth place will be the second digit 00:00:41.006 --> 00:00:42.576 to the right of the decimal point. 00:00:42.886 --> 00:00:46.136 Hence, in our example, we have the number 2. 00:00:46.586 --> 00:00:50.496 So our answer is 2. 00:00:50.496 --> 00:00:54.916 Here, we have another example. 00:00:55.736 --> 00:01:02.286 Write out the decimal 0.517, as a sum of fractions representing each decimal place. 00:01:03.036 --> 00:01:08.616 In this case 5 is in the tenths place, so we have five-tenths. 00:01:12.636 --> 00:01:18.126 One is in the hundredths place, so we have one-hundredth. 00:01:18.546 --> 00:01:26.916 And the 7 is in the one-thousandths place, so we have seven-thousandths. 00:01:31.056 --> 00:01:34.796 And then we want to write it as a sum 00:01:35.080 --> 00:01:38.160 of all these three fractions, and that will be our answer. 00:01:39.940 --> 00:01:42.360 Pause the video and try these problems. 00:01:46.220 --> 00:01:49.940 Objective 2: Adding and subtracting decimals. 00:01:50.606 --> 00:01:54.636 To set up problems that add or subtract decimals, we need to be sure to line 00:01:54.636 --> 00:01:57.956 up the decimals, and add or subtract straight down. 00:01:58.776 --> 00:02:01.716 The reason we do this is to be sure we are adding 00:02:01.716 --> 00:02:05.006 or subtracting numbers that are in the correct place. 00:02:05.636 --> 00:02:12.616 Tenths place, ones place, hundredths place, etc. To understand, let us refer to fractions. 00:02:15.276 --> 00:02:19.656 Example. Add 0.3 plus 0.4. 00:02:20.486 --> 00:02:23.156 In this case, we are going to do that using fractions, 00:02:24.146 --> 00:02:29.436 0.3 is equal to three-tenths, and 0.4 is equal to four-tenths. 00:02:30.256 --> 00:02:43.756 So if we want to add 0.3 to 0.4 that will be the same as adding three-tenths plus four-tenths. 00:02:44.096 --> 00:02:50.006 Since we are adding fractions and the denominators are the same, this case, 00:02:50.166 --> 00:02:54.546 we add the numerators which is 7, and keep the denominator. 00:02:55.176 --> 00:02:58.366 So in this case, we have seven-tenths. 00:02:58.736 --> 00:03:01.676 So now we convert that to decimal. 00:03:01.836 --> 00:03:04.846 That will be 0.7. 00:03:05.246 --> 00:03:07.646 And that will be our answer. 00:03:08.826 --> 00:03:13.076 With fractions, we only add the numerators, and the denominators stay the same. 00:03:13.726 --> 00:03:16.606 This is why we line up decimals when adding or subtracting, 00:03:16.846 --> 00:03:19.836 so that we are adding terms that have the same placeholder. 00:03:20.956 --> 00:03:32.116 And so from a previous example, 0.3 plus 0.4 will then equal to, let's line up the decimals. 00:03:33.516 --> 00:03:42.246 So we add 3 and 4, that will give us 7, and our decimal, and then 0, and so our answer is 0.7, 00:03:42.606 --> 00:03:45.666 the same as if we were to add these two decimals, with fractions. 00:03:49.936 --> 00:03:55.866 Example. Add 1.652, plus 20.91. 00:03:56.026 --> 00:03:59.356 This case, we are going to add the numbers by lining up the decimals. 00:04:00.416 --> 00:04:10.666 So we have 1.652, plus 20.91. 00:04:11.566 --> 00:04:15.016 To this second number, we are going to add a zero at the end, 00:04:15.476 --> 00:04:17.886 so that it will also have three decimal places, 00:04:17.966 --> 00:04:21.126 and then we just add as we would add any two numbers. 00:04:21.326 --> 00:04:30.666 We have 2 plus 0 is 2, 5 plus 1 is 6, 6 plus 9 is 15, and hence, we carry 1, 00:04:31.356 --> 00:04:37.006 1 plus 1 plus 0 will give us 2, and here at the end we have 2 also. 00:04:37.556 --> 00:04:41.146 So a decimal place will go here. 00:04:41.626 --> 00:04:47.236 And our answer will then be 22.562. 00:04:47.846 --> 00:04:54.476 Keep in mind that carrying over and borrowing will work the same way 00:04:54.506 --> 00:04:56.886 with decimals as they do with whole numbers. 00:05:00.726 --> 00:05:05.816 Example. Subtract 2.83, minus 1.206. 00:05:06.226 --> 00:05:20.756 To subtract these two numbers, we are going to line up the decimals, 2.83 minus 1.206, 00:05:21.306 --> 00:05:26.956 since the first number only has two decimal places, we are going to add a third one. 00:05:27.946 --> 00:05:32.376 And then we subtract as we would subtract any two whole numbers. 00:05:32.426 --> 00:05:42.006 So we are going to have to borrow 1 from 3, 10 minus 6, that's 4, 2 minus 0 is 2, 8 minus 2, 00:05:42.006 --> 00:05:47.456 that's 6, bring down the decimal point, 2 minus 1, that's 1. 00:05:47.540 --> 00:05:54.820 So 2.83 minus 1.206 is equal to 1.624. 00:05:57.900 --> 00:06:01.660 Pause the video and try these problems. 00:06:05.500 --> 00:06:09.060 Objective 3: Multiplying and dividing with decimals. 00:06:09.496 --> 00:06:14.616 To set up decimal multiplication problems, lining up the decimals is not necessary. 00:06:15.066 --> 00:06:18.566 Instead, we will treat the numbers as regular integers, 00:06:18.896 --> 00:06:21.416 then account for all the decimal places at the end. 00:06:21.836 --> 00:06:25.346 To understand this, let us refer to fractions. 00:06:25.556 --> 00:06:28.366 So here, we have an example. 00:06:28.706 --> 00:06:31.456 Multiply 0.32 times 0.4. 00:06:31.596 --> 00:06:42.896 Since 0.32 is equal to thirty-two hundredths, and 0.4 is equal to four-tenths, 00:06:43.046 --> 00:06:58.206 we have 0.32 times 0.4 is equal to 32 over 100, times 4 over 10, 00:06:58.206 --> 00:07:03.166 and when we multiply fractions, we multiply the numerators. 00:07:03.296 --> 00:07:13.386 So 32 times 4, that is 128, and we multiply the denominators, 100 times 10, that is 1,000. 00:07:13.546 --> 00:07:20.596 And so 0.32 times 0.4 is 128 thousandths. 00:07:20.896 --> 00:07:28.536 This will then equal to 0.128, that represents 128 thousandths. 00:07:28.946 --> 00:07:32.936 Notice that this times we end up with a different denominator. 00:07:33.066 --> 00:07:37.486 And this is because when we multiply fractions, we also multiply the denominators. 00:07:38.456 --> 00:07:45.396 So when we multiply decimals, that means that we will keep track of all the decimal places, 00:07:45.466 --> 00:07:49.046 and move our decimal point over that many places in the end. 00:07:52.296 --> 00:07:57.726 To show the same idea we see with fractions, we can line up the numbers 00:07:57.726 --> 00:07:59.816 like regular integer multiplication. 00:08:00.696 --> 00:08:05.166 But just note that there will be three shifts of the decimal in the final answer. 00:08:05.496 --> 00:08:16.426 From our previous example, 0.32 times 0.4, is equal to-so now let's line up the numbers 00:08:16.426 --> 00:08:27.606 like regular integer multiplication, 0.32 times 0.4, and then we multiply 00:08:27.606 --> 00:08:39.076 as we would multiply any two integers, 4 times 2 is 8, 4 times 3 is 12, 4 times 0 is 0, 00:08:39.236 --> 00:08:41.306 and then we carried 1, so that will be 1. 00:08:42.486 --> 00:08:46.026 The first number has two decimal places. 00:08:46.296 --> 00:08:48.796 The second number has one decimal place. 00:08:49.086 --> 00:08:52.606 So we have a total of three decimal places in our answer. 00:08:53.236 --> 00:08:55.796 So we move the decimal place three times. 00:08:56.136 --> 00:09:02.136 And so our answer is 0.128. 00:09:06.816 --> 00:09:08.266 Here we have another example. 00:09:08.566 --> 00:09:12.656 Multiply 819 times 7.756. 00:09:13.016 --> 00:09:17.176 This case, to multiply, we are going to put the largest number of digits on top, 00:09:17.946 --> 00:09:20.006 regardless of which number is actually larger. 00:09:22.336 --> 00:09:28.256 The number 7.756 is the number with the largest number of digits. 00:09:28.296 --> 00:09:30.796 That is going to go on top. 00:09:30.796 --> 00:09:33.216 Times 819. 00:09:33.966 --> 00:09:42.146 And just multiply as we would multiply any two integers, 9 times 6 is 54, 4, carry 5, 00:09:42.726 --> 00:09:53.336 9 times 5 is 45, plus 5, that is 50, 9 times 7 is 63, plus 5, that is 68. 00:09:55.556 --> 00:09:58.956 Again 9 times 7 is 63, plus 6, is 69. 00:10:02.406 --> 00:10:07.736 Now, let's multiply by 1, 1 times 6 is 6, 1 times 5 is 5, 00:10:08.326 --> 00:10:11.096 1 times 7 is 7, and 1 times 7 is 7. 00:10:13.706 --> 00:10:24.526 Now let's multiply by 8, 8 times 6, that's 48, 8 and carry 4, 8 times 5 is 40, plus 4, that's 44, 00:10:25.196 --> 00:10:32.076 4 and carry 4, 8 times 7 is 56, and 4, that's 60. 00:10:33.426 --> 00:10:37.656 Again, 8 times 7, that's 56, and 6, that's 62. 00:10:40.676 --> 00:10:44.586 And now, let's go ahead and add. 00:10:44.586 --> 00:10:57.486 So we have 4, 6, 8, plus 5, that's 13, plus 8, that's 21, 2 plus 9, that's 11, plus 7, 00:10:57.486 --> 00:11:07.436 that's 18, plus 4, that's 22, 2 plus 6 is 8, plus 7, that's 15, 1 plus 2, 00:11:07.436 --> 00:11:11.756 that's 3, and we have 6 here at the end. 00:11:11.856 --> 00:11:15.116 Now, the first number has three decimal places. 00:11:15.386 --> 00:11:19.536 The second number has 0 decimal places, therefore, 00:11:19.536 --> 00:11:23.176 we have a total of three decimal places in our answer. 00:11:23.796 --> 00:11:30.846 Let's move the decimal place three times to the left then, 1, 2, 3. 00:11:33.736 --> 00:11:43.156 Therefore, our answer is going to be 6,352.164. 00:11:49.626 --> 00:11:57.036 Example. Set up division for the problem, 72.8 divided by 0.26. 00:11:57.916 --> 00:12:02.456 In this case, to set up a long division problem with decimals, we are going to move the decimal 00:12:02.456 --> 00:12:07.106 of the divisor, the number on the outside, to the right, until it is gone. 00:12:07.436 --> 00:12:12.406 However far you must move it, do the same to the dividend, the number on the inside. 00:12:13.166 --> 00:12:16.646 The decimal in your new dividend will line up with the answer. 00:12:17.616 --> 00:12:27.676 So in this case, we are dividing, 72.8 by 0.26. 00:12:27.866 --> 00:12:31.426 So 0.26 is our divisor. 00:12:31.846 --> 00:12:35.946 So we are going to move the decimal place of this number to the right until it's gone. 00:12:36.206 --> 00:12:42.826 So 1, 2, so we move the decimal place twice to the right. 00:12:43.306 --> 00:12:45.896 So we must do the same to the dividend. 00:12:46.286 --> 00:12:53.216 So we'll move the decimal place twice, 1, 2. 00:12:53.326 --> 00:13:02.586 The final setup of this problem will then be 26 dividing 7280. 00:13:06.696 --> 00:13:10.396 Example. Divide 2.814 by 2.1. 00:13:10.396 --> 00:13:19.716 So we are dividing 2.814 by 2.1. 00:13:20.026 --> 00:13:25.496 This divisor is 2.1, so we are going to move its decimal place to the right once. 00:13:25.696 --> 00:13:30.776 And we are going to do the same to the number on the inside, the dividend. 00:13:30.776 --> 00:13:35.436 So its new place is going to be between 8 and 1. 00:13:35.776 --> 00:13:48.426 Our set up will then be 21 dividing 28.14, then now we can go ahead and divide. 00:13:48.426 --> 00:14:00.816 So 21 goes into 28 once, 1 times 21, is 21, subtract, this will give us 7, move the 1 down, 00:14:01.596 --> 00:14:08.776 so 21 goes into 71 three times, 3 times 1, that's 3, 00:14:09.246 --> 00:14:16.486 3 times 2, that's 6, 71 minus 63 will be 8. 00:14:17.636 --> 00:14:32.026 Now we move the 4 down, and 21 will go into 84 four times, 4 times 1 is 4, 4 times 2 that's 8, 00:14:32.136 --> 00:14:36.116 and when we subtract, we can see that we have a 0 remainder. 00:14:36.596 --> 00:14:43.366 The decimal in the inside should line up with the decimal in our answer. 00:14:44.416 --> 00:14:51.946 And so 2.814 divided by 2.1 will be 1.34. 00:14:55.086 --> 00:14:58.876 Pause the video and try these problems.