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Mixed Numerals and Improper Fractions

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Mixed Numerals and Improper Fractions. In the Intro to Fractions lesson earlier, we briefly discussed the issue of improper fractions vs 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 whose numerator is larger than its denominator.

So for example, 17/3 or 40/11. A mixed numeral expresses this exact same information as an integer written next to a fraction that is less than one. So for example, 5 2/3 is the mixed numeral equivalent of 17/3. And 3 7/11 is the mixed numeral equivalent of 40/11. The test may give you numbers in either 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 mixed 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.

28/5, well, the biggest multiple of 5 that is less than 28 is 25. So I'm just gonna express the 28 as 25 plus 3. Split up the fraction like that. The 25/5 becomes 5, and then it's 5 3/5. For 60/7, the biggest multiple of 7 that is less than 60 is 56. So, I'm gonna write that as 56 plus 4.

56/7 is 8, and so that's 8 4/7. The biggest multiple of 13 that is less than 80 is 78. So, I'm gonna write this as 78 plus 2, and 78 is 6 times 13. So 78/13 is gonna be 6 2/3 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 it. 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 an implicit addition sign. And so, we find a common denominator. 12, we'll multiply that by 2/2.

And so, that becomes 24/2 plus 1/2. That's 25/2. We'll multiply the 8 by 6/6, so it becomes 48/6, plus 1/6 equals 49/6. We'll multiply the 3 by 19/19. 3 times 19 is 57, so 57/19 plus 3/19 equals 60/19.

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 in comparing the fraction in size to another number. That's the principal 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 if 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 can square a mixed numeral in their head correctly. And, most people, they tried to do it, they would do something like square the 1, then square the 5/7, and add those together, something like that.

In other words, almost anything that they would try would be wrong. And so, 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, 12/7, what's the value of 12/7 squared?. Well that, we all can do in our heads.

It's 12 squared over 7 squared, or 144/49. This is much much easier to do in ones head. And so, improper fractions are much better for cases involving multiplication or division or raising to a power. This has a profound implication 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 numerals form because mixed numerals, under many circumstances, are worse than useless.

Instead, you should change the prompt numbers to improper fractions, do the calculations with the improper fractions, then convert back to mixed numerals. Here's 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're gonna 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 1 1/6, that's 7/6. 1 11/21 is 32/21. Well, now, we're gonna multiply those fractions.

But, of course, before we multiply, we're gonna cancel. Notice we can cancel a factor of 7 in the 7 and the 21. Then, we can cancel a factor of 2 in the 6 and the 32. 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 6/9.

Well, now that we have our answer in proper fraction form, we'll change this back to a mixed numeral. 1 7/9, that's the actual product, 1 7/9. 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 taker is gonna think, uh-oh, I have to divide 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 45/8 divided by 9/2. Of course, dividing by a fraction is the same as multiplying by its reciprocal. 45/8 times 2/9. Cancel the factors of 9. Cancel the factors of 2. And I just get ordinary 5/4.

And, of course, now that we have that answer, I'll just rewrite that as a mixed numeral, 1 1/4. So it turns out the quotient is 1 1/4. Answer choice A. Finally, another problem of this ilk. Pause the video and then we'll talk about this.

Again, the same sort of thing. What we're seeing here is we have 1 4/5, 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, 9/5 squared, well, that's just nine squared over five squared, or 81/25.

Well, now that we have the answer, we'll re-write that as a mixed numeral, 3 6/25, 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 going to be fluent in that conversion.

If you need to determine the position of a fraction on 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 to improper fractions, do your calculation, and then change back.

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