Now we're going to get into a little bit more fractions. We're going to tackle another chocolate problem. This is going to be chocolate math 2. So here's our question, how many batches of brownies can you bake if a recipe requires 4/5 cup of sugar and you have 3 1/2 cups of sugar at home? Write your answer as a mixed number and I'll provide the boxes. Use any math reason that you know. Remember, the important part is just to try to reason, it doesn't matter if you're right, right now. We'll be coming back to this problem later. Just give your best guess now, and you can always change it and revise it later.
Before we attempt our problem, let's review mixed numbers and improper fractions. Mixed numbers are a way to count objects when we have a couple of objects and a part of that object. So here, I have 2 doughnuts and then I have whole doughnut into 3 equal pieces or thirds, and I'd have 2/3. So, our entire mixed number would be 2 and 2/3. Notice we have a mixture of whole numbers and fractions. That's why we call these mixed numbers. If I split all of my doughnuts up into thirds, I could count the pieces and I would have 8/3. That would be the improper fraction. It's improper because the number in the numerator is higher than the number in the denominator. So, let's see how we can go from the mixed number to the improper fraction. If I take my whole doughnuts and split them up into thirds, I would have six of them, 6/3. I can get that by multiplying 32. I still have 2/3 left over, so I need to add this 2/3 to my 6/3. 6+2 makes 8, so I get 8/3. Let's try going the other way. I can think about splitting 8/3 into 6/3 and whole doughnuts. I know 6/3 makes 2. and then I have my 2/3 left over. So, 2 and would have a remainder of 2 or 2/3 left over.
Now, let's look at mixed numbers with octagons. An octagon is just a shape with eight sides. What would be the mixed number and the improper fraction for these octagons? You want to think about splitting these octagons up into equal parts. You will also want to reduce your answer. So, for example, if I had 5 6/12, I'd really want to write that as 5 and I know six goes into 12 two times. So 6/12 reduces to 1/2.
If you got 3 3/4 that's great. I know I have three entire octagons and then I need to figure out this fraction. This is actually going to be 3/4. Let's see why. I could have split my octagons into eighths. So I know I have three whole octagons. And then I have 1/8, 2/8, 3/8, 4/8, 5/8, 6/8 of a hexagon. Remember, I know I can reduce 6/8. I know 2 goes into 6 and 8. So I can simplify it. And then I have 3/4. To find an improper fraction, we just need to count up all the 8ths. If I counted up all the 8ths, I would find out that I would have 30 over 8, or 30/8. Remember, I can reduce this fraction. So I get fifteen over four, or 15/4. But this doesn't quite make sense. My picture has eighths, but my answers have fourths. Well, that means I could have actually done this a little differently. I could have split up my octagons into fourths. This would be 1/4 of the octagon. Notice, when I have fourths, I can easily see that I have 3 have 3/4. 1/4, 2/4, 3/4 of this last hexagon, so 3 3/4. Counting the improper fractions is pretty easy, too. I can actually just use my mixed number. Because this mixed number is in fourths, I can figure out the improper fraction in fourths. I could multiply 4 times 3, which is 12. That makes sense because I have three octagons, and each of them has four pieces. Then, I have to add 3 more pieces, so, I add 12 + 3 and I get 15 over 4,
Let's try answering an easier form of the problem. This might help us think about what we should do. How many batches of brownies could you bake if a recipe requires 1/2 of a cup of sugar and you have 1 cup of sugar at home? This seems a little bit easier. This square is going to represent my 1 cup of sugar. I know 1/2 of a cup of sugar will make 1 batch of brownies, so I'm going to fill up half of my cup and make 1 batch of brownies. So, here's a 1/2 cup of sugar and I know I've made 1 batch of brownies. If I use another batch of brownies. Now, I've used my 1 whole cup of sugar and I don't have any sugar left at home. Well, I made 2 batches of brownies. Let's pair this with some math. I had 1 cup of sugar. And I wondered, how many times did 1/2 go into it? I knew I was able to make 2 batches of brownies, so 1 divided by a 1/2 must be 2. Remember, when we divide by a number, we can actually multiply by the reciprocal, we learned that from before. So, I could change this to look like 12/1. Well, remember, I could just put any number over 1. And then, multiply fractions like before. 12 is 2, and 11 is 1. 2/1 is just 2. Let's change this problem a little bit, and see what we can find out. What if, instead, a recipe called for 1/3 cup of sugar? I know if I use 1/3 cup of sugar, I'll make 1 batch of brownies. If I use another third of cup of sugar, I'll make another batch of brownies. And if I use my last third of cup of sugar, I'll make one more batch of brownies. So, let's put this with some math. I was wondering, how many times does I know that for every third cup of sugar, I get 1 batch of brownies, so I should get number, it's the same thing as multiplying by the reciprocal. I could change this to multiplication, 13/1. I know 1 is just 1/1 so now, I can multiply these fractions together. 13 is batches. I hope you're starting to see a pattern. Let's see if you can figure this one out.
So here's your quiz. How many batches of brownies can you bake if a recipe requires one-fourth of a cup of sugar and you have one cup of sugar at home? You can put your answer in this box here.
If you said 4 batches, way to go. I know for the first 1/4 cup of sugar, I'll make of sugar, I'll make another batch of brownies. Another 1/4 cup of sugar will give me another batch of brownies. And my last 1/4 cup of sugar will give me another batch of brownies. I started with 1 cup of sugar, and I wanted to see how many times did 1/4 divide into 1 cup. I can turn division into multiplication by multiplying by the reciprocal, so 1 times times 1, which is 1. 4 / 1 makes 4. The key idea in all these problems is that dividing by a fraction is the same as multiplying by the reciprocal of the fraction. We're going to use this idea in the next problem.
Let's make this problem a little more challenging. What if the brownie recipe called for 2/3 of a cup of sugar, and you had 1 cup of sugar at home? Before we get into the problem, let's make a prediction. Could we make more than one batch, exactly one batch, or less than one batch?
I had more sugar at home than my recipe calls for. My recipe only calls for 2/3 of a cup of sugar. If I had one cup of sugar at home, I could use 2/3 to make one full batch of brownies. And then, I'd have 1/3 of a cup left over so I know I can make more than one batch of brownies. If I only had 2/3 of a cup of sugar I could make exactly one batch of brownies. But I'll have extra leftover, I'll have a 1/3 of a cup of sugar that I can still use. If you said more than one batch of brownies, nice work.
Remember, I still have 1/3 of a cup of sugar that I can use at home to make brownies. I've drawn it over here. This fraction amounts. I can see that this 1/3 cup of sugar actually represents 1/2 of a batch of brownies. So I made one batch of brownies from 2/3 cup of sugar and I made sugar. And all together I used 1/3 and 2/3 which is 1 cup. I can see all together I made 1 1/2 batches of brownies. I used 2/3 cup of sugar to make one batch and then with the extra 1/3 cup of sugar I made 1/2 of a batch. I saw that, that piece was 1/2 of a batch of brownies. So 1 1/2 all.
Now, that we've covered more information about mixed numbers and improper fractions, let's revise our guess on our second chocolate math problem. How many batches of brownies can you bake if a recipe calls for 4/5 of a cup of sugar and you have 3 and 1/2 cups of sugar at home? You can put your answer here as a mixed number. Put your whole number here and the fraction part here. Remember, use any of your math reasoning. It's okay, you might not have all the tools to solve this problem yet. But just give your best guess and use any reason that you've learned so far.
Let's make another prediction. This time, how many batches of brownies can you bake if a recipe requires 3/4 of a cup of sugar and you have 2 cups of sugar at home. Again, before we solve this problem, let's guess about how many batches we'll make. Will we make less than one batch of brownies, exactly one batch, more than one batch but less than 2, exactly 2 batches or more than 2 batches? Go ahead, choose your answer.
Here's one cup of sugar, and I know 3/4 cup makes one batch of brownies. Here's another cup of sugar and I know this 3/4 cup of sugar makes another batch of brownies. So I know I can at least make 2 batches of brownies. I haven't used two full cups yet. I actually have some sugar that I can still use. So, I know I could've made more than two batches of brownies. If you chose more than two more batches, great job.
But what about the sugar I have left over? I've only used 3/4 of a cup and 3/4 of a cup. I still have 1/4 and 1/4 that I can still use to make more brownies. 1/4 and a cup of sugar that I can use to make brownies. This is the 2/4, so this 2/4 cup of sugar represents what fraction of the batch of brownies? You can enter your answer here.
This extra 2/4 cup of sugar actually represents 2/3 of a batch of brownies. I know one whole batch requires 3/4, so this
Okay. Let's wrap up and figure out exactly how many batches of brownies we can make if our recipe calls for 3/4 a cup of sugar and we have 2 cups of sugar at home. Go ahead and solve it.
If you counted the batches from the last problem we know we have 2 and 2/3 of a batch. To do that math, I would have done many times 3/4 goes into 2. Instead of dividing by a fraction, I'm going to multiply by the reciprocal. So I have 2 x now I can multiply 2 x 4 = 8. 1 3 is 3. So I have 8/3. I'm going to write this as a mixed number. I know 3 goes into 8 evenly 2 times, because 2 3 is 6. So I have 3 going into 8, 2 times. And then I'm left with two extra pieces. So, 2 and 2/3.
So let's go back and answer our Chocolate Math 2 question. How many batches of brownies can you bake if a recipe requires cups of sugar at home? You can put your answer here as a mixed number. Remember, you'll want to use what we've learned about mixed numbers and dividing fractions.
Nice work on practicing your multiplication of fractions. Now, let's try a couple of division problems. The first problem that we want to do is similar to the multiplication problem but we're going to divide. 3/4 / 5/7. You can put your answer here.
If you got 21/20, great job. Let's see how we got that. Well, we know that dividing fractions is the same as multiplying by the reciprocal. So, 3/4/5/7 is actually And we know when we multiply fractions together, we multiply their numerators together. So, 37=21. And we multiply the denominators together, =20.
Let's try a slight more difficult division problem, let's try 24/35 divided by 40/21. You can put your answer here
If you got 9/25, you got it right. We know to divide fractions, we want to multiply by the reciprocal. So, I can rewrite this problem as a multiplication problem as fraction, which is 21/40. So, instead of multiplying 2421 and trying to reduce that, I can factor before I multiply. So, written as 75, 21 can be written as 73, and 40 can be written as 410. From here, I notice that there's a seven in my numerator and in my denominator, and we know 7/7=1 and we know that 4/4=1. Now, we can multiply the numerators together, 63 is 18 and 510 is equal to I always want to make sure that my final answer is in simplified form, so I need to see if 18/50 is simplified. Well, I notice right away that 18 is an even number and divisible by two. So I can write 18 as 92 and I can write 50 as 252. Now that I have these in factored form, I can cross out 2/2. Now we get to our final answer which is 9/25.