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Comparing Decimals: Tenths and Hundredths
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In this video, we’re going to learn how to use models and place value tables to compare numbers with up to two decimal places.
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These two children are taking part in a standing long jump competition.
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We can see that the boy has already jumped 1.15 meters or one and fifteen hundredths of a meter.
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And we’ve just watched the girl jump 1.2 meters or one and two-tenths of a meter.
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Now, both of these measurements are represented by decimal numbers.
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They have both a whole part and a fraction part.
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And these two parts are separated by a decimal point.
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So both children have jumped one and a bit meters.
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But in order for us to see who’s jumped the furthest, we’re going to need the ability to be able to compare these two decimals together.
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By the end of this video, we’ll have learned enough skills to be able to do this and to be able to say who’s jumped the furthest altogether.
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One way we could compare decimals together is to use models to help us.
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For example, if we want to find out which number is larger, 0.6 or 0.9, we could try sketching a model that shows six tenths and nine tenths.
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And then we could compare these parts visually.
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We could start by sketching a square like this to represent one whole.
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And then if we divided it into 10 equal parts, each part would be worth one-tenth or 0.1.
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So to model 0.6, we need to shade in six-tenths, and to model 0.9, we’d need to shade in nine tenths.
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Now we can see just by looking at the shaded parts which is the largest.
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Nine-tenths is greater than six-tenths.
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Now there are lots of different models that we can use to represent decimal numbers.
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Let’s have a go at comparing two decimals that have been modeled using an abacus.
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Complete the following using the symbol for is greater than, is less than, or is equal to.
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An abacus is a way of modeling a number.
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And in this question, we’ve got two to look at.
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We need to compare both numbers together and then choose the correct symbol to show whether the first number is greater than the second, whether it’s less than the second number, or whether the two numbers are exactly the same.
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Our first number shows two ones and three-tenths.
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It’s a decimal number.
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We know this because it’s made up of a whole part — two ones or a whole number — but also a fractional part.
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That’s a part that’s worth less than one.
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And three-tenths of this fractional part, three-tenths are less than one.
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And if we were to write this number using digits, we need a two in the ones place, then a decimal point to show that we’re moving into the fractional part, and then a three in the tenth place.
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Two ones and three-tenths is the same as 2.3.
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Now let’s look at our second number.
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This is made up of one one and seven-tenths.
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So we have less ones, but a few more tenths in this number.
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And again, it’s a decimal.
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This is because it’s made up of that whole part and a part that’s worth less than one.
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This time, we could represent this model using digits with a one in the ones place.
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Again, we’re going to need a decimal point to separate it from the fraction part and then a seven in the tenths place to show our seven-tenths.
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One and seven-tenths is the same as 1.7.
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So how are we going to compare these two decimals that have been modeled on each abacus?
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Should we compare the number of beads in the tenths place first?
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Or should we think about the ones?
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Well, we need to apply the same rule that we use when we compare whole numbers.
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And that’s to compare the digits that have the largest value first and work our way from left to right.
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So we shouldn’t start by comparing the tenths.
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A tenth is less than one.
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We need to start by looking at the number of ones that we have.
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Our first abacus shows two in the ones place, but, in our second abacus, there’s only one bead in the ones place.
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It doesn’t matter how many tenths each number has, because we’ve compared the ones.
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And we can see that the first number is larger than the second.
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2.3 or two ones and three-tenths is greater than 1.7 or one one and seven-tenths.
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The correct symbol to use in between these two models is the one that represents is greater than.
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Another way that we could compare decimals is by using a number line.
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For example, if we want to find out whether sixty-nine hundredths is greater than ninety-six hundredths, we could sketch a number line from zero to one and then label where each number belongs.
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0.69 or sixty-nine hundredths is just before 0.70 or seventy hundredths.
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But 0.96 or ninety-six hundredths is really close to one whole.
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So we’ve used our number line to show that sixty-nine hundredths is not greater than ninety-six hundredths.
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0.69 is less than 0.96.
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Let’s move on to a third way that we can compare decimals together, and you’ll be familiar with this from comparing whole numbers.
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This is to use a place value table to compare the digits in our numbers.
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Let’s go through an example question where we need to do this.
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Complete the following using the symbol for is equal to, is less than, or is greater than.
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In this question, we’re given two numbers, and they’ve been written for us in place value tables.
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Let’s read them.
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Our first number contains one one, four-tenths, and eight hundredths, 1.48.
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Our second number also has a single one, it also has four-tenths, but this time it has six hundredths, 1.46.
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These two numbers are decimals.
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They have a whole part.
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That’s the number of ones.
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But then they also have a part that’s worth less than one.
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And to separate the whole part from the fractional part, we use a decimal point.
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Now, in between our two decimals, there’s a box.
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And we need to complete the statement using one of the comparison symbols we’re given equal to, less than, or greater than.
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In other words, we need to compare both these decimals together.
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Now both of these decimals have been written in place value tables to help us.
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So we can compare them digit by digit.
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Now shall we start with the hundredths and work from right to left?
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Or shall we start with the ones and work from left to right?
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Well, if we take a moment to think about what each one of these columns is worth, we know that our ones column represents one whole.
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Now, as we move to the right, each new column is worth 10 times less than the one before.
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So a tenth is 10 times less than a whole.
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It’s what we get if we take a whole and split it up into 10 equal parts.
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And then a hundredth is 10 times less again.
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It’s what we get if we take one of our tenths and split that up into 10 equal parts.
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So knowing that, which part of our number do you think is most important?
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It’s the part that’s worth the most.
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We need to start by comparing our ones column and then work from left to right.
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Our first decimal number contains a one in the ones place.
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But can you see we’ve also got this in our second number too?
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So we can’t separate our numbers just by looking at the ones.
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We’re going to need to move on and look at the tenths.
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Both numbers have a four in the tenths place.
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We still can’t separate them.
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The beginning of both our decimals is the same, isn’t it, one and four-tenths.
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We’re going to need to compare the hundredth digits.
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Below these are the digits that have the least value in our numbers.
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They’re actually going to turn out to be the ones that make the difference.
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This tells us that the difference or a gap between our two numbers is very small.
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It’s just going to be a matter of hundredths.
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Our first number has an eight in the hundredths place, but our second number only has a six.
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And eight hundredths are greater than six hundredths.
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The difference between our two numbers is only two hundredths, which is a tiny difference.
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But it’s enough for us to be able to say which number is larger.
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We’ve compared these two decimal numbers digit by digit using place value tables.
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One, four tenths, and eight hundredths is greater than one, four tenths, and six hundredths.
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Or to say it a different way, 1.48 is greater than 1.46.
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The correct symbol to use in between these two decimals is the one that represents is greater than.
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So we’ve looked at three different ways we can compare decimals using a model, comparing them on a number line, and comparing them digit by digit using a place value table.
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But what if our decimals are represented in different ways?
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Let’s try a question where they are.
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Complete using the symbol for is less than, is equal to, or is greater than.
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Five-tenths what seventy hundredths.
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In this question, we’ve got two values that we need to compare together.
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And there are a combination of both digits and words.
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We’ve got five-tenths and then there’s seventy hundredths.
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Now a very easy mistake to make here would be just to look at the numerals and say to ourselves, “Well, I know that five is a lot less than 70, so five-tenths must be less than seventy hundredths.”
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Now, in a moment we’re going to compare these values, and it might be the case that the first one is less than the second, but it’s not because five is less than 70.
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And that’s because these values are talking about two different place value units.
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We’ve got five-tenths and seventy hundredths.
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Let’s sketch a couple of models to help us see what these values actually mean.
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We could represent five-tenths by taking a whole square splitting it into 10 equal parts and then shading five of them.
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There are no whole squares, but there are five-tenths of a square.
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So we’d write this value as 0.5 if we were writing it in digits.
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Our second value is a number of hundredths.
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Now, just like all the other columns, the hundredths place can only fit one digit in.
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But we’re told that this value is worth seventy hundredths.
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If we were to sketch this as a model again, we’d need our whole square and, this time, to split it into 100 equal parts.
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Then we’d need to shade in 70 of them.
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So once again, we’ve shaded in zero whole squares and we’d write seventy hundredths with a seven in the tenths place and a zero in the hundredths place.
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If we drew another model, can you see that seventy hundredths is actually the same as seven-tenths?
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That’s why that digit seven goes in the tenths place.
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Now we’ve sketched these two models; we can compare the values quite easily.
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We can see that five-tenths is less than seventy hundredths.
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As we’ve just seen, seventy hundredths is the same as seven-tenths.
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And we know that five-tenths must be less than seven-tenths.
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Another way we could find the same answer is by looking at our place value table and comparing the digits.
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Both numbers have zero ones, but our first number has five-tenths and our second number has seven-tenths.
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So it turns out that five-tenths is less than seventy hundredths.
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But it’s not because five is less than 70.
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To find the answer, we needed to work out what five-tenths and also seventy-hundredths were worth.
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And we use both models and also a place value table to help compare them.
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Five-tenths is less than seventy hundredths.
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And so the symbol that we need to use in between these two values to compare them is the one that represents less than.
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If you remember right at the very start of this video, we had a standing long jump competition with two decimal measurements that we wanted to compare.
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Decimals crop up quite a lot in sports: the length that somebody jumps, the time that it takes them to run a certain distance, or, as we’ll see in the next question, the number of yards that they can throw a football.
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Peyton Manning averaged 7.89 passing yards per attempt in 2006 and Carson Palmer averaged 7.76.
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Who had the higher average?
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This question is a really good example of how decimals are sometimes used in sport.
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Now, perhaps you’ve already heard of the two sportsmen that are mentioned in this question.
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But if not, Peyton Manning and Carson Palmer were both quarterbacks in American football.
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And one of the key things that a quarterback has to do is to make forward passes by throwing the football.
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And that’s where the two decimal measurements in this question come from.
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We’re told that Peyton Manning averaged 7.89 passing yards every time he attempted a throw and Carson Palmer averaged 7.76 yards.
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And our question asks us who had the higher average.
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In other words, we need to compare these two decimal measurements together.
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Let’s do so by drawing a place value table.
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We’re going to need to include columns for ones, tenths, and hundredths because both of these values are decimals with two decimal places.
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In other words, they show both tenths and hundredths.
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Now, back in 2006, Peyton Manning’s average was 7.89 yards.
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We know this is a decimal number because there’s a decimal point in it, but also because it contains a whole part, seven ones, which represents seven whole yards.
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And then there’s a fraction part, which is a part worth less than one, eight-tenths and nine hundredths.
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We could read this as 7.89 but also seven and eighty-nine hundredths.
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Now let’s complete Carson Palmer’s average, which we can see was 7.76.
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So that’s seven whole yards, a seven in the ones place, and then, in the fraction part, we’ve got seven-tenths and six hundredths, seven and seventy-six hundredths.
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So let’s find out which one of these numbers is larger.
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The column with the greatest value is going to be the ones column, so we need to compare these digits first.
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And we can see that both players threw seven whole yards.
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So far, our numbers are exactly the same, and this is often the case with sports measurements.
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When you’ve got two players that are very close together in ability, the first few digits might be the same.
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There’s often only a very small difference between them.
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So let’s move on to the tenths digits.
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7.89 contains eight-tenths, and 7.76 contains seven-tenths.
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And eight is greater than seven.
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And we can stop right there.
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We don’t need to move to the hundredths column.
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We can separate these two numbers by looking at their tenths.
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Because eight-tenths is greater than seven-tenths, we can say that 7.89 is greater than 7.76.
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Well, to say the numbers a different way, seven and eighty-nine hundredths is greater than seven and seventy-six hundredths.
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And so, the player that had the higher average was Peyton Manning.
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So what have we learned in this video?
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We’ve learned how to use models and place value tables to compare numbers with up to two decimal places.