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Hi there! Welcome to Math Antics. In this video we are going to learn how to compare fractions.
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Hmmm… this fraction has 25% more fiber than this fraction…
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Oooo! But this fraction has trisodium phosphate!
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Well… it’s not quite like that.
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Comparing fractions just means telling which one is bigger.
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You know, just like we do with regular numbers when we use the greater-than, less-than, and equal-to signs.
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That sounds easy, right? But unfortunately, unlike regular numbers,
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it’s not always easy to tell which fraction is bigger just by looking at them.
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That’s because the value of a fraction depends on both the top AND bottom numbers and how they relate to each other.
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For example, if you have to compare these two fractions, 1 over 3 and 1 over 10,
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some of you might be tempted to say that 1 over 10 is bigger because you know that 10 is bigger than 3, right?
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But we need to remember that the fraction is really a number written like a division problem,
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and its value depends on that division.
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So in this case, the 1 over 3 is really the bigger fraction
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because its decimal value (what you get when you divide) is 0.333 but the value of 1 over 10 is only 0.1
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Okay, so comparing fractions isn’t quite as easy as comparing regular numbers,
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but that doesn’t mean it’s going to be that hard.
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We’re going to learn two methods for comparing fractions that make it very easy.
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The first method is called cross-multiplying,
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and it takes advantage of the fact that it’s easy to compare fractions with the same bottom numbers.
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If two fractions have the same bottom numbers, then we can just compare the top numbers.
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That’s because we are comparing the same size parts.
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We’re comparing fourths to fourths, eighths to eighths, tenths to tenths, and so on…
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And the top number just tells us how many of those parts we have,
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so it’s easy to see that 5 eighths is more than 3 eighths.
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But many times, you’ll have to compare fractions that have different bottom numbers. (or different size parts)
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Fortunately, there’s a trick we can do to make the comparison easy.
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In the Math Antics Videos about Common Denominators,
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we learn a simple method for changing “unlike fractions” (with different bottom numbers)
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into “like fractions” (with the same bottom number).
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Basically, it shows how you can multiply two unlike fractions
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by wholes fractions made from the different bottom numbers,
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so you end up with the same bottom number.
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This will give you two new ‘equivalent’ fractions that you can easily add, subtract, or compare.
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But, there’s a shortcut for comparing fractions.
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As long as we know that the bottom numbers of our fractions are the same,
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we don’t really need to know what that number is.
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We just need to know what the top numbers will be, since those are the ones that we’ll actually compare.
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So instead of multiplying each fraction by a whole fraction,
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we can just multiply the top number of each fraction by the bottom number of the other fraction.
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This is called ‘Cross Multiplying’ because if you draw a diagram of what you’re multiplying,
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it forms a criss-cross pattern.
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After you cross multiply, you will have two numbers that would be the new top numbers
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if you had made ‘like’ fractions, and those numbers will show you which fraction is greater.
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Let’s try this cross-multiplying method on an example or two.
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Let’s compare the fractions: 7 over 8 and 4 over 5.
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We start by multiplying the second fraction’s bottom number (5) by the first fraction’s top number (7)
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and that gives us 35 for the new top number on this side.
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You’ll always keep the answer on the side of the top number that you multiplied.
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…now for the other side.
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The bottom number (8) times the top number (4) gives us 32 for its new top number.
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Ah-ha! Now it’s easy to see that the fraction 7 over 8 is greater than the fraction 4 over 5
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because its new top number (35) is greater than the other new top number (32).
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Let’s do one more comparison by cross multiplying.
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Let’s compare 6 over 11 to 9 over 15.
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First we’ll multiply 15 by 6 to get the new top number of the first side, which is 90.
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Now you can use a calculator to do the multiplications if you need to.
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Next, we multiply 11 by 9 to get the second new top number, which is 99.
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So, that tells us that the second fraction (9 over 15) is greater than the first fraction because its new top number (99) is bigger.
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Pretty simple, huh?
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Okay, cross multiplying is pretty cool,
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but there’s another way to compare fractions that you need to know about.
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But this one is only really good if you can use a calculator.
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Remember, the reason that fractions are tricky to compare is because they’re really division problems.
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But if we want to, we can just do the division and get the answer,
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which is the decimal value of the fraction.
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So if you have two fractions to compare, you can just do the division
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(preferably using a calculator) and then compare the decimal values.
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For example, let’s say I offered to give you either 5/12 of a pizza or 7/15 of a pizza.
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Now, you happen to be really hungry, so you want to choose the biggest amount,
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but it’s not very easy to tell just by looking which is bigger: 5/12 or 7/15
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This is were decimal values can really help you out.
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If you convert the fractions to decimals by doing division,
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it will make it much easier to see which one is bigger.
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5 divided by 12 is about 0.42
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and 7 divided by 15 is about 0.47
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Yep, that makes comparing them much easier.
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Since 0.47 is greater than 0.42, it means that 7/15 is greater than 5/12.
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And that means that you’d rather have 7/15 of the pizza!
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Sometimes when you compare fractions this way,
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you’ll find two fractions that look different, but have the same decimal value: like 3/8 and 15/40.
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If you convert each fraction to a decimal, you’ll see that they both have the value 0.375
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Two fractions that have different top and bottom numbers, but the same value are called ‘equivalent fractions’.
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If two fractions are equivalent, then you can just use the equal sign to show the comparison between them, like this…
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Alright, so those are two great methods you can use to compare fractions.
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Cross multiplying is simple and works great, even if you don’t have a calculator.
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And comparing the decimal vales by dividing is easy if you do have a calculator.
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As always, practice makes perfect,
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so spend some time doing the exercises for this section, and I’ll see you next time.
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Learn more at www.mathantics.com