In the last section we worked with multiplying and dividing fractions. We've done a lot with fractions. Let's continue and see what else we can do. Here's our third chocolate math problem. You want to frost a cake for a party. You frost 2/5 of the cake in vanilla icing and 1/4 of the cake in strawberry icing. What fraction of the cake is frosted? You might not know what to do with these two fractions, and that's okay. I want you to go ahead and take a guess. You can put your answer in these two boxes here, the numerator up top, and the denominator in the bottom. You also want to write down your answer. We're going to come back to it later, and you can see how your thinking might have changed.
We're going to use a strategy to tackle this problem just like we've done before. We're going to make this problem easier. So we're going to use different fractions instead of 2/5 and 1/4. Lets use 1/2 and down here. Before we answer this problem lets actually predict. So if we frost 1/2 of the cake in vanilla icing and one third of it in strawberry icing. What fraction of the cake is actually frosted? Would it be less than 2/5, exactly 2/5, or more than 2/5? Draw some diagrams to help you out. Let's actually consider a round cake. And here's what 2/5 of a round cake frosted would look like. Notice I have my whole chocolate cake and 2/5 of it has been frosted in this blue icing. So what do you think 1/2 and 1/3 of a cake covered in icing would look like? Make your choice.
Here are some cakes. Notice that for this first chocolate cake, I've split it up into thirds or three equal pieces. I have the second cake I have half of it covered in vanilla icing, 1/2. But I don't have two separate cakes here. I just have one cake covered in all that frosting. So here's what these pieces would look like together on one cake. So I have half of it covered in vanilla and a third of it covered in strawberry. If I compare the frosted part to the 2/5 part, I can easily see that this is more than 2/5. So if you said more than 2/5 nice work. It's not that obvious, but if you draw up diagrams it definitely helps support your reasoning. So let's try looking at this with some Math. I had half of the cake of vanilla icing and a third of the cake in strawberry icing. Many students think that adding fractions together is the same as adding numbers or integers together. They would say 1 + 1 makes 2 and 2 + 3 makes 5. Many students think 1/2 and 1/3 is 2/5. But, when I look at my diagram, I know that can't be right. I already have half of the cake in vanilla icing, and a little bit more in strawberry icing. I know this much is frosted, and that's way bigger than 2/5. So, we need another way to add these fractions together. This won't work.
Alright. So let's see how we can add 1/3 of strawberry icing and 1/2 of vanilla icing together. I need a way to compare into equal size pieces for both of them. Then I could add the parts together. Let's split up our cake into sixths. I'm going to split every third into 2 equal parts. Doing that, I can see I get 2/6 of the cake is covered in strawberry icing. So, how did I get from 1/3 to 2/6? We doubled the number of pieces we had altogether, I went from 3 pieces to 6. So I know I needed to multiply the denominator by 2. I also doubled the number of pieces I had. I had 1 out of 3. And now I have 2 out of 6. Notice I'm multiplying by the number 1. 2 / 2 is still 1. I'm just changing what this fraction looks like. I'm not changing its value. We can still see 1/3 is 2/6. I'm going to use the same approach to make cake into 6. To change from halves to 6, I need to multiply by 3 / 3. I know 2 3 gives me 6. And then, for the numerator, 1 3 = 3. Visually, I can see that when half of the cake frosted in vanilla icing is the same as 3/6th of the cake also frosted in vanilla icing. I multiplied by the fraction, I'm just changing what it looks like. Now that I have my cake split up into 6, I have 2/6t of it frosted in strawberry icing and 3/6 of it frosted in vanilla icing. Now I can just add my 6th's together. I add the same sized pieces, so I can just count them up. I can see that here's what it would look like on one cake. I have half of it covered in vanilla icing, and a third of it covered in strawberry icing. This would be 1/6 of the cake, 2/6 of the cake, 3/6 of the cake. when my fractions were just 1/3 and 1/2 I couldn't add them together, I had to get a common denominator first. I had to get them into the same size pieces. So how do I add fractions together once I have a common denominator? Let's try this with a quiz.
How did I add these two fractions together? To add fractions with common denominators, you blank the numerators together and blank the denominators. You could use the words add, subtract, multiply, divide, or do not change. You'll use any of these words to fill these boxes. What did we do here? How did we add these two fractions together? Go ahead, type in your choices.
Well, I noticed that down here I had 2/6 and 3/6. I had 2 plus 3, so I had five. So I added the numerators together. For the denominators I didn't change them. I was simply counting up how many sixths I had. I had 1/6, 2/6, 3/6 4/6, and 5/6, all together. So we do not change the denominator.
So we've done a lot of work with cake, and as much as I like cake, especially chocolate cake, lets see how else we can use our fractions. Even if I spend all that time baking, I probably want to go outside, so lets see our fractions in another context, walking. So here's our next problem with fractions. You walk 2/5 of a kilometer, and then 3/10 of a kilometer. How far did you walk? When thinking about this problem, I know I need to do something with these fractions. I'm trying to figure out the total distance I have to walk. So, I should probably add these fractions together. So I have 2/5 added to 3/10. Here again, I don't have common denominators. I can't just add these together. So let's see what we can do. But, you might be wondering, what's a kilometer? A kilometer is a measure of distance. We use the letters km to stand for kilometer. It is also the word, kilo. Kilo means 1000, so we know 1 kilometer is really 1000 meters, and you're probably wondering, what's that distance like? So, here's a picture of about a kilometer. In New York City, Central Park is about 849 meters across. 849 meters is just shy of about a kilometer, but a little bit less than that. If you wanted to see for yourself how long a kilometer is, you could go outside. It takes about 12 minutes to walk a kilometer, depending on how fast you walk. Alright, now that we know what a kilometer is, let's get back to problem solving. OK, I've rewritten our problem at the top. You walk 2/5 of a kilometer and then walk 3/10 of a kilometer. How far did you walk? Let's picture this using a kilometer. I'm going to draw a bar that represents a kilometer. So, this distance, or the full bar, would be one kilometer. 2/5 of a kilometer would only be this much. Let's see the same thing for 3/10. Let's split up our kilometer bar into tenths. So I have a full kilometer bar. I have tenths. And 3 of them are shaded to represent 3/10 of a kilometer. If I wanted to add 2/5 and 3/10 together, it'd look like this. I'd have t he 2/5 and then the 3/10. Don't follow the trap of the thinking this is 5 out of 15. I don't have 15 equally divided regions here. Even though I have 5 of them, they're not equally spaced. I can't add dealing with the same dimensions or links. I know 1/5 of a kilometer is actually longer than 1/10 of a kilometer. So, I can't simply add fifths and tenths together. Remember, I need to get a common denominator. So, what do you think? What would be the common denominator for 2/5 and 3/10? You can enter your answer in this box here. Just give me only the denominator. What you change tenths and fifths into?
If you said 10, nice work. This is a little bit tricky, so let's walk through how to actually do this. I'm going to show you two ways to find the common denominator. The first way is listing out the multiples. I have 5 for one denominator, and I have 10 for the other. I'm going to start by listing out the multiples of 5. I have 5, then 10, then multiple, I just add 5. For 10, I can list out its multiples as well, 10, 20, 30, 40, and 50, and it keeps going. To find the denominator, I want the lowest number that both 5 and 10 go into. When I think about the lowest number they both go into, I see that it's 10. So I know I'm going to need to split this bar into tenths and this one into tenths. Luckily, this one's already in tenths. Another way to find the common denominator is to find the factors of 5 and 10. I'm going to use a factor tree. I know the common factors for 5 and 10 are factors are 5, or 5 and 1, because 51 makes 5. The other factor of 10 has to be these numbers together, 152, I get 10. Either way, I can still find the common denominator.
All right, so we know we need to change fifths and tenths into tenths. So, to change fifths into tenths, I need to multiply by 2/2. And then, for 3/10, it's already in tenths, so, I don't need to change it. So, let's see what happens when I change 2/5 into tenths. I know if I multiply my fractions I'll get 4 tenths. Here's what that would look like, this would be 4/10 of a kilometer. Remember, I doubled the number of pieces that I had. I went from 5 pieces to 10 and I doubled the number of pieces that I had. I had 2/5 originally. And now I had 2 times 2, which is 4, or 4/10. Visually, we can see that of one kilometer bar. They really represent the same length. My 3/10 of a kilometer bar doesn't change, I'm just adding it to the 4/10.
So let's put this all together. What fraction of a kilometer did you actually walk? You can enter your final answer here. Put the numerator in this box, and the denominator in this box.
So, I really just want to add 4/10 and I'm going to fill it with both the 4/10 and the 3/10. This would be 4/10 of a kilometer, and this would be another 3/10 of a kilometer. So all together, I have
Let's try another kilometer problem. You walk 3/8 of a kilometer and 3/7 of a kilometer one day. How far did you walk? Before we solve this problem like the others, let's make a prediction. Did you walk less than a kilometer, exactly a kilometer or more than 1 kilometer. Use the two number lines to help you out. We have 1 full kilometer and this is 3/8 of that kilometer. I have 3 out of 8 parts of the kilometer. Here the kilometer's been split into seven equal pieces and I have best one.
Half a kilometer and half a kilometer makes one full kilometer. I know 3/8 is shorter than half a kilometer, and the same is true for 3/7. This would be the halfway point for one kilometer. So, if I add two things that are less than 1/2, I can't get a full kilometer. You must have walked less than one kilometer. If you chose less than one kilometer, nice work.
So, if we walk these two distances, we know we need to add them together to find our total distance. I can't add the fractions yet because I have unlike denominators. I want you to find the common denominator for these two fractions. Type in your answer here.
You might have taken some different approaches to find the least common denominator. Here's one way. We could have used the least common multiple. By listing the multiples of 7 and by listing the multiples of 8, we can see that the lowest common number that they go into is 56. but this a lot of work and i don't want to have to do this every single time. SO let's try using our factor tree. The only common factor of 8 and 7 is 1. and the numbers of the outside must be 8 and 7, since 8 times 1 is 8 And 7 1 is 7. If we multiply these factors together, we get
So I want you to find the equivalent fractions for 3/8 and 3/7, and then find their sum. Put your answers in the boxes.
We know our common denominator is 56. So I need to multiply 3/8 by 7/7 and I get altogether, I have 45/56. So altogether, we walked 45/56 of a kilometer. Maybe you have another method for finding the least common denominator. If you do, share it on the forum.
Now that we've walked through adding 2 fractions together, let's return to our original questions. You want to frost a cake for a party, you frost 1/4 of the cake in vanilla icing and 2/5 of the cake in strawberry icing. What fraction of the cake is frosted? So, I want you to try and answer this question using what we've learned. You can use diagrams to help support your answer, or if you already know the method and understand it really well, you can just apply that. Is it 3/5, choice.
I know 1/4 of the cake is frosted in vanilla icing, and 2/5 of it is frosted in strawberry icing. So I have 1/4 and 2/5. I need to add these 2 fractions together to figure out how much of the cake is frosted in total. My fractions do not have common denominators, I have 4ths and I have 5ths. So I need to get the same size pieces. To find the common denominator, I'm going to do the least common multiple method. So, I'm going to list the multiples of 4 and the multiples of 5. The smallest number that both 4 and 5 goes into is 20. I can also find that by using my factor tree method. When I think about the factors that 4 and 5 have in common, the only factor they share is 1. I know the other factor of 4 must be 4, since, 4 x 1 is 4. The other factor of 5, must be 5. Multiplying these together, I still get the twentieths, and fifths into twentieths. To change fourths into twentieths, I need to multiply 5 over 5. I know this because 20 is the fifth multiple of 4. I also know this because I have to multiply 4 times 5 to end up with 20. So 5 times 1 is 5, and 5 times 4 is 20. To change 2/5 into twentieths, I need to multiply by 4 over 4. Remember, I'm still multiplying by 1. 2 times 4 is 8, and 5 times 4 is 20. Now this is great. I can actually add the twentieths together. I have 5/20 and 8/20. I keep my denominator the same, and 8 plus 5 is 13. So I get
We've worked with frosted cake, but sometimes it might be better to know how much cake was not frosted. Let's look at a simple case of this, if 2/3 of the cake is frosted with vanilla icing, what fraction is not frosted? Well, let's start off by looking at a cake, here's a whole chocolate cake, and I'm going to ice 2/3 of it. So I've split my cake up in thirds and I iced 2 of those thirds, here's 1/3 here's 2/3. One way to think about this, is that the parts must add up to the whole. I know that I have 2/3 of the cake frosted in vanilla icing. And I have to add some amount to get a whole cake. So I know 2/3 + some amount = 1. Well this last piece is just one third, so I can just put frosted. And remember back from adding fractions we know we just need a count of the thirds, I have one third, two thirds and three thirds all together. So this piece that's not frosted is one third. Third. There's actually another way that we can think about this. Let's instead, start with a whole cake, and subtract off the parts that we don't want. We don't want the vanilla icing. So, let's subtract the pieces we don't want from the entire whole cake. I started with 1 whole cake, and I'm subtracting off 2/3. Well, I know Remember, I can just write 1 as 1 / 1. Because that's still 1 and then I can multiply this fraction by 3 over 3. I'm just going to turn 1 into 3rds or 3/3 so I have 3/3 - 2/3 which is 1/3. 1/3 of my cake is not frosted. And what's interesting about fractions is that is doesn't matter how much cake we're talking about, we could have started with a different cake, maybe we started with a rectangular cake. Here's a rectangular cake and I'm to ice 2/3.. Doing the same problem with a rectangular cake doesn't change my answer. I still have 2/3 of it frosted in vanilla icing and 1/3 that's left unfrosted. The amount of cake thats frosted or unfrosted depends on the size and shape of the original cake. The neat thing about fractions is that 2/3 could describe. This part of a round cake or this part of a rectangular cake. Fractions describe a part of a whole object.
If you save 2/5 of any money you earn in a month, what fraction can you spend? And I'm going to draw a circle to help you out. The circle is going to represent the total amount of money you've earned, and you can enter your answer in this box here. Put the numerator on top, the denominator on the bottom.
So, I'm going to start by taking my total amount of money and splitting it into fifths. Notice that 1 whole circle is 5/5. I'm going to save 2/5 of my money, so let's show that on another circle. And I'm going to save 2/5 of my total money. I don't know how much total money I have, but I know I'm saving this fraction of money. I'm trying to figure out what fraction I have left that I can still spend. Visually, I can see that I still have 3/5 of my money to spend. So, how did I get that? I did 5/5, that was the whole amount of money I had. And I need to subtract off 2/5. So, I have 5/5-2/5. So, I'm taking away 2/5 from my 5/5 so I just need to subtract the numerators. So, 5/5 taken away 2/5, leaves me with 3/5. I work with subtracting fractions just like I do with adding fractions. I only change the numerators. All I'm doing is taking my fifths and taking out two of them, leaving me with 3/5. If you got 3/5, nice work. Whether you know it or not, you've already started thinking algebraically. I have no idea how much money you can earn in a month but we let that amount represent this circle. It was an abstract form of reasoning. We knew that if you saved 2/5 of it, that you would have 3/5 of it left to spend on whatever you please. So, even though I don't know the total amount of money, I can stills say at least what fraction of that money can be spent. One of the key skills in Algebra is to find ways to work with unknown amounts or numbers, we'll see this throughout the entire course.
Let's try our hand at another unfrosted cake problem. If 1/6 of a cake is frosted with strawberry icing, and 3/8 is frosted with blueberry icing, what fraction of cake is not frosted? To set up this problem, let's give ourselves a picture of what's going on. Here are two rectangular cakes, and I have 1/6 of it frosted in strawberry icing. For this other cake I'm going to frost 3/8 of it in blueberry icing. Notice I've split up the cake into eight equal parts and I have three of those eight frosted, or 3/8. I'm not working with two cakes though, I'm working with one cake. So let's visualize these two cakes on one. I'm going to show that here. So notice I have 1/6 of the cake frosted in strawberry icing, and 3/8 of it frosted in blueberry icing, and we're trying to figure out what fraction is not frosted, the chocolate part. To setup this problem, we know we're going to start with one whole cake, than I need to subtract off 1/6 of the cake. And 3/8 of the cake. So, I have 1 minus 1/6 minus 3/8. I can't compare sixths to eighths right now, they're different sized pieces, so, I can't just subtract these numbers. I need to get a common denominator, just like we did when adding fractions. So, let's try this with a quiz and see if we can remember how to find a common denominator. What do you think the common denominator would be for these fractions? Remember, you can list the multiples or use the factor tree. You don't have to worry about the 1 when finding the common denominator. This is because we can change the 1 into these numbers are still equal to 1. So whats the common denominator for these fractions? You can put your answer in this box here. Remember to find the common denominator you can list the multiples or use the factor tree.
Alright, the common denominator is going to be 24. Here's how we can figure that out. I can start by listing my denominators, and then I can list the multiples of 6, and the multiples of 8. So I have 6, 12, 18, 24, 30, and 36. For 8, I have 16, 24, 32, and 40. I can see that the lowest common number is 24. Both 6 and like if you use the factor tree method. I know the common factors for 6 and 8 are two, and the other factor of 6 is 3, because 3 x 2 is 6. For 8, the other factor has to be 4. I know this because 2 x 4 is 8. If I multiply these factors together, I get 3 x 2 which is 6 times 4, which is 24. Hopefully you're getting the hang of finding the common denominator.
So, we know that we needed to change our pieces to 24ths. So I'm going to take my whole cake and make it into 24ths. I know one whole cake would be 24/24. But now, I need to change this 1/6 of the cake into that because I want to subtract off the pieces that are frosted. I want to be left with the unfrosted region. So, we know we need to change 1/6 to 24ths and we would change 3/8 into 24ths. What number do we need to multiply by to change 1/6 and to change 3/8? Go ahead, enter your numbers here in these boxes.
We need to multiply the 1/6 by 4/4. I know that if I do that, I'll get 24 in my denominator, and I'll get 4 in the numerator. My diagram of cake also supports this reasoning. If I thought about 1/6 of cake, I can see that it's the same as 4/24 of cake. For 3/8, we needed to multiply by 3/3. I know 83=24, and then 33 would give me 9. So, I have 4/24 in strawberry icing, and 9/24 ins blueberry icing. Now, I'm ready to subtract off my frosted pieces.
So, let's put this all together. What's the equivalent fraction we found for 1/6? The equivalent fraction we found for 3/8? And, what would be our final answer? Enter your numbers in these boxes. Remember, the numerators go up top and the denominators are on the bottom. Be sure you put the equivalent fraction for 1/6 here, and the equivalent fraction for 3/8 here.
We know the equivalent fraction for 1/6 is is 9/24. I'm taking away 4/24 and 9/24, so altogether, I'm taking away 13/24. I can see that visually, too. I have 1, 2, 3, 4, are the pieces I'm trying to subtract from my whole cake. So, 24/24 minus 13/24 leaves me with 11/24. That means, this region must represent 11/24 of my cake. That's the fraction that's not frosted. Lets see if this makes sense visually, too. I've split my cake into 24th and if I counted the chocolate pieces that are not frosted, I can see that I have 11 of those, so I have 11/24 of my cake not frosted.
Alright. So, here's our cake challenge. remaining half of the cake was eaten, what fraction of the original cake is left? You're going to need to combine what you've learned so far. Think about if you need to add, subtract, multiply, or divide. Also, think about how much cake you have to start. This is a tough problem so take your time and work slowly. Patient problem solving can often payoff in the end. Make sure you are reasoning through each step and use diagrams to support your reasoning. You can put your answer in this box here. Good luck.
Okay, so we started with one whole cake. We know when half of it was eaten, so I need to subtract off half of my cake. I also know the 3/10 of the remaining half of the cake was also eaten. So I have 3/10 of 1/2. I remember of means multiply. So I need to take 3/10 and multiply it by 1/2. And this 3/10 of 1/2 was eaten, so I need to subtract it off as well. So I have one whole cake minus 1/2 a cake minus 3/10 of and take out the words. So I have 1 for one whole cake minus 1/2 half and then minus 3/20. I got 3/20 by doing the multiplication here. I have 3 over 20 or to find a common denominator. I don't have to worry about the 1 when finding a common denominator, I can change 1 to 2/2 3/3 or easily. For 2 and 20 I know the least common multiple, or the least common denominator is 20. It's the smallest number that both 2 and 20 go into. I can also do this using my factor tree. I know the common factor for 2 and 20 is 2, the other factor of 2 is 1, and the other factor of 20 is 10, 1210 makes 20. So to change 1/2 to 20ths, I know I need to multiply by 10/10. So 1/2 is the same thing as 10/20. For 3/20, I don't need to multiply by anything. I already have need to change this 1. To change the 1 I'm going to multiply by 20/20, I know 20/20 is still 1 whole. So, 20/20-10/20 is know if I start with one whole cake and subtract off half of it I should still have half of it left. 10/20 is really the same as 1/2. Now I still need to subtract off my 3/20. So 10/20-3/20 is 7/20. If you said 7/20, excellent work.
Congratulations. I hope you've gained a stronger understanding of working with fractions. More importantly, I hope you have learned why and how we add, subtract, multiply and divide fractions. Try your hand at solving some problems with fractions and treat yourself to chocolate when you're done. You've worked hard.
Thanks Chris, for teaching us how to add and subtract fractions and finding the least common denominator. Now, let's practice what we learned again. Let's try a pretty easy problem. We're going to try You can put your answer for the numerator in that box and the denominator in that box. Good luck.
If you got 23 for the numerator and 24 for the denominator, you got it right. Let's see how to get that answer. I know when I add fractions, I have to have the denominators be the same. So the first thing that we need to do is find a common denominator between 12 and 8. Using the factor tree method I know that 4 is a common factor for both 12 and 8. I know that 4 3 is 12 and 4 2 gives me 8. So, I can multiply those together, which gives me a common denominator of 24. The next thing I need to do is create equivalent fractions that have 24 as the denominator so I need to create equivalent fraction for 7 / 12. Since we want to have 24 in the denominator I know I have to multiple the denominator by I also have to multiply the numerator by so an equivalent fraction for 7/12 is 14 / 24. Now lets convert 3/8. We want 24 as the denominator. 8 3 gives us 24, 8 3 is 24, 3 3 is 9. Now I have an equivalent fraction for 7/12 and I have an equivalent fraction for 3/8. I can now rewrite the problem using my equivalent fractions. I have 14/24 + 9 over 24. When I add fractions I keep the denominator the same and I add the numerators. 14 and 9 is because there are no factors in common between 23 and 24.
Good work on practicing adding fractions. Now, let's try to subtract two fractions. Let's try 5/27-2/45. You can put your numerator answer there and you're denominator there. Good luck.
If you got 19 for the numerator, and 135 for the denominator, good job. Let's see how we got that answer. When we subtract fractions, we also have to have a common denominator. So, the first thing I need to do is find that common denominator. When I use the factor tree method, I need a factor in common. For 45 and 27, that common factor is 9. I know that 95 is 45. And I know that 93 gives me 27. If I multiply those together, 395 I get 135. The next thing I need to do is to convert my 2 fractions that I'm subtracting into equivalent fractions with 135 as the denominator. So, I can take 5/27, And I want to have 135 as the denominator, so Now, let's create an equivalent fraction for 2/45. Since I want 135 as the denominator. I need to multiply 45 times 3 and now, I have 45 times 3 is 135, and 2 times 3 gives me 6. Now, I can rewrite my original subtraction problem with my equivilent fraction. So I have 25 over 135 minus 6 over 135. When I subtract fractions, I keep the denominators the same, and I subtract the numerators 25-6 gives me 19.
Now that we've practiced one addition and one subtraction problem with fractions, let's try a slightly more complicated problem. Let's try 5/12+7/20. And you can put your answer in this boxes.
For question 4, let's try one more subtraction problem. Let's try 7/12-5/28. You can put your answer in these boxes. Good luck.