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Arrange seven-halves, seven-quarters, seven-thirds, seven-eighths in descending order.
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So in this problem we’re given four fractions.
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And we’re asked to arrange them in descending order.
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That’s from largest to smallest now.
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And often, when we compare a set of fractions, it’s good to have the denominators the same.
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It’s very quick to compare them if the denominators are all the same.
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So one way to find the answer could be to convert these fractions so that they all have the same denominator.
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But in this particular question, we can find out the answer just by using our knowledge of fractions.
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Let’s have a look at the four fractions and see what we can spot about them.
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Firstly, nearly all of the fractions are improper.
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In other words, the numerator is larger than the denominator.
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This means that the fraction represents more than one whole.
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Seven-halves is more than one.
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Seven-quarters is more than one.
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Seven-thirds is more than one.
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In fact, the only fraction where the numerator is smaller than the denominator is seven-eighths.
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Seven-eighths is the only fraction we’re looking at that’s less than one.
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It’s the smallest fraction out of our list of four.
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So we haven’t had to convert anything, but we’ve already worked out that the smallest fraction is seven-eighths.
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Let’s cross it off our list and look at the remaining three fractions.
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We know that two-halves equal one whole.
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So how many wholes and how many halves do we have if we have seven-halves.
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Two-halves equal to whole.
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Four-halves equal two wholes.
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Six-halves equal three wholes.
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And so seven-halves equal three and a half.
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Now, let’s convert our next improper fraction into a mixed number.
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We know that four-quarters are in one whole.
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So how many wholes and how many quarters are in seven-quarters?
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Seven-quarters must be equal to one and three-quarters.
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Finally, how many wholes and how many thirds are in seven-thirds?
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We know three-thirds equal one whole.
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So six-thirds must be the same as two wholes.
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And so seven-thirds is the same as two and a third.
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We’re now in a position to be able to compare all of the fractions.
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We’ve converted them all into mixed numbers.
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So we can see what their values worth.
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We can also see why we didn’t need to convert them to have the same denominator.
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We can simply compare the number of ones because they’re all different.
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The mixed number with the greatest number of one’s is three and a half.
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This is the largest mixed number.
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And so it’s also the largest fraction.
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The largest fraction is seven and a half.
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The next largest number of ones is two, in the mixed number, two and a third.
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So we know the second largest fraction is seven-thirds.
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Because it’s the only one remaining, we know the third largest fraction must be seven-quarters.
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Look at how we could have predicted this before we started.
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The numerator in all of these fractions is the same, seven parts every time.
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But the denominator changes.
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Remember, the smaller the denominator with a fraction, the larger the parts.
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So if we’ve got seven parts, and they’re large parts, it’s going to be a larger number, a larger amount.
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And the opposite is true; the larger the denominator, the smaller the parts.
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So if we have seven parts and they’re small parts, because the denominators is a large number, then it’s going to be a smaller fraction.
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And that’s why these denominators go in order — two, three, four, eight.
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Because the numerator stays the same, all we have to do is to look at the denominators and think to ourselves, a smaller number means a larger piece, and then just put the denominators in order.
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And so the four fractions arranged in descending order are seven-halves, seven-thirds, seven-quarters, seven-eighths.