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Mixed Numerals and Improper Fractions
Mixed Numerals and Improper Fractions. In the Intro To Fractions lesson earlier we briefly discussed the issue of improper fractions versus mixed numerals. Both of these are options we have for writing a fraction that is larger than one. You may remember that an improper fraction.
Is a fraction who’s numerator is larger than it’s denominator. So, for example, 17 over three, or 40 over 11. A mixed numeral expresses this exact same information as an integer written next to a fraction that is less than one. For example five and two thirds is the mixed numeral equivalent of 17 over three. And three,seven and elevens is the mix numeral equivalent of 40 over 11.
The test may give you numbers and other form and may list the answer choice in either form. First of all it’s very important to be comfortable changing from one to the other. So, change these improper fractions to mix numeral form. Pause the video and just take a moment to do it yourself and then we’ll talk about it.
Okay, that first one, 28 over five. Well, the biggest multiple of five that is less than 28 is 25. So, I’m just gonna express the 28 as 25 plus three. Split up the fraction like that, the 25 over five becomes five, and then it’s five and three fifths. For 60 over seven, the biggest multiple of seven that is less than 60 is 56, so I’m gonna write that as 56 plus four.
56 over seven is eight. And so that’s eight and four sevenths. The biggest multiple of 13 that is less than 80 is 78. So, I’m gonna write this as 78 plus two and 78 is six times 13. So 78 over 13 is gonna be six, it’s gonna be six and two thirds all together.
Change these mixed numerals to improper fraction. Again, pause the video, work on these on your own, and then we’ll talk about these. Okay the way we do this, is thinking about it in terms of fraction addition. Because technically, between that integer and that fraction, in each mixed numeral, is and implicit addition sign. And so we find a common denominator.
- We’ll multiply that by two over two. And so that becomes 24 over two plus one over two. That’s 25 over two. We’ll multiply the eight by six over six, so it becomes 48 over six, plus one over six, equals 49 over six.
We’ll multiply the three by 19 over 19. Three times 19 is 57, so 57 over 19 plus three over 19 gives 60 over 19. Those are the improper fraction forms of those mixed numerals, so changing from mixed numerals to improper fractions or vice versa is one big idea. That’s an important skill.
Even more important than that is understanding the relative usefulness of each form. In other words, when would we want to have one form versus the other? It’s great that we can have either form. But what is strategic? When would we want to use one form versus the other form?
So mixed numerals are very useful if we need to locate the fraction on the number line. This could be helpful comparing the fraction in size to another number. That’s the principle use of mixed numerals. For adding and subtracting, it really doesn’t matter that much. The two forms are about equal in difficulty.
But here’s the big idea. In multiplying, dividing, and raising numbers to a power, mixed numerals are worse than useless and improper fractions are definitely the way to go. What do I mean by that? Worse than useless? Think of it this way, what is the value of one and five sevenths, squared?
Now, you see, if you think about that in terms of mixed numerals. First of all, very few people on earth could square a mixed numeral in their head correctly. Most people if they tried to do it, they would do something like square the one, square the five sevenths and add those together, something like that. In other words, almost anything that they would try would be wrong.
And so it just, it’s an invitation to make thousands of mistakes, because it’s almost impossible to do it correctly. Well by contrast, suppose we just change that mixed numeral to an improper fraction? 12 sevenths, what’s the value of 12 sevenths squared? Well that, we call can do in our heads, that’s 12 squared over seven squared, or 144 over 49.
This is much, much easier to do in one’s head. And so improper fractions are much better for cases involving multiplication, or division, or raising to a power. This has profound implications for problems with mixed numerals. If the question gives mixed numerals in the prompt, asks for a calculation of some kind, and gives all mixed numeral answers, do not assume that you do the calculation in mixed numeral form because mixed numerals.
Under many circumstances are worse than useless. Instead you change the prompt numbers to improper fractions, do the calculations with the improper fractions, then convert back to mixed numerals. Here is a practice problem. Pause the video and then we’ll talk about this. Okay, so this is multiplication.
We’re given the prompt numbers in mixed numerals, we’re given answers in mixed numerals, but do not assume that you are going to do the calculation in mixed numeral form. Most people who tried to do this calculation, would. Make all kinds of mistakes. It’s not very easy to multiply mixed numerals.
So, instead what we’re gonna do, we’re gonna change those to improper fractions. It seems like this might be a lot of extra work. It actually enormously simplifies the problem. So, one and one sixth. That’s seven sixth. One and 11 twenty firsts is 32 over 21.
Well now we’re gonna multiple those fractions. But, of course, before we multiple, we’re gonna cancel. Notice we can cancel factor seven in this seven of the 21. Then we can fact cancel a factor of two in this six and the thirty two. Well now that we’ve cancelled everything, we’re down to the lowest terms that we can get.
So now we’ll just multiply. And we get six over nine. Well now that we have our answer in improper fraction form, we’ll change this back to a mixed numeral. One and seven ninths. That’s the actual product.
One and seven ninths. So go back to the answer choices. And we select the correct answer choice, answer choice A. Here’s another practice problem, dividing mixed numerals. Pause the video, and then we’ll go through this together. Okay.
Same deal They give you mixed numerals, the answers are in mixed numerals. The naive test takers gonna think I have to divide the mixed the mixed numerals. Anyone who tries to divide mixed numerals is gonna do it incorrectly. It is almost impossible to divide mixed numerals correctly. So we’re not even gonna talk about that it’s a very. Hard conceptual thing.
Instead what we’re gonna talk about is let’s change those two mixed numerals to improper fractions. So rewrite them as improper fractions. I get to 45 over eight divided by nine over two. Of course, dividing by a fraction is the same as multiplying by it’s reciprocal 45 over eight times two over nine, cancel the factors of nine, cancel the factors of two, and I just get ordinary five fourths.
And of course now that we have that answer, I’ll just rewrite that as a mixed numeral, one and one quarter. So it turns out the quotient Is one and one quarter, answer choice A. Finally, another problem with this pause the video and then we’ll talk about this. Again, the same sort of thing, what we’re seeing here.
Is we have one and four fifths, a mixed numeral squared. Well no one can square a mixed numeral, but what we’re gonna do is change this to an improper fraction. Improper fraction, nine fifths squared, well that’s just nine squared over five squared. Or 81 over 25.
Well now that we have the answer, we’ll rewrite that as a mixed numeral, three and six 25ths, and look for that among the answer choices. And that’s answer choice E. In summary it’s important to be comfortable changing back and forth between mixed numerals and improper fractions or to be fluent in that conversion.
If you need to determine the position of a fraction of the number line, mixed numerals may be a little more helpful. For calculations involving multiplication, division or powers, mixed numerals are worse than useless. You have to use improper fractions. And if the problem gives you mixed numerals in the prompt and gives you mixed numerals in the answer choice, what you need to do is change the improper fractions.
Do your calculation and then change back.
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