Adding fractions word problem: paint 

Cindy and Michael need 1 gallon of orange paint for the giant cardboard pumpkin they are  making for Halloween. Cindy has 2/5 of a gallon of red paint. Michael has got 1/2 a gallon of  yellow paint. If they mix their paints together, will they have the 1 gallon they need? So let's  think about that. We're going to add the 2/5 of a gallon of red paint, and we're going to add that to 1/2 a gallon of yellow paint. And we want to see if this gets to being 1 whole gallon. So  whenever we add fractions, right over here we're not adding the same thing. Here we're adding 2/5. Here we're adding 1/2. So in order to be able to add these two things, we need to get to a  common denominator. And the common denominator, or the best common denominator to  use, is the number that is the smallest multiple of both 5 and 2. And since 5 and 2 are both  prime numbers, the smallest number's just going to be their product. 10 is the smallest number  that we can think of that is divisible by both 5 and 2. So let's rewrite each of these fractions with 10 as the denominator. So 2/5 is going to be something over 10, and 1/2 is going to be  something over 10. And to help us visualize this, let me draw a grid. Let me draw a grid with  tenths in it. So, that's that, and that's that right over here. So each of these are in tenths. These  are 10 equal segments this bar is divided into. So let's try to visualize what 2/5 looks like on this  bar. Well, right now it's divided into tenths. If we were to divide this bar into fifths, then we're  going to have-- actually, let me do it in that same color. So it's going to be, this is 1 division, 2, 3, 4. So notice if you go between the red marks, these are each a fifth of the bar. And we have two of them, so we're going to go 1 and 2. This right over here, this part of the bar, represents 2/5 of  it. Now let's do the same thing for 1/2. So let's divide this bar exactly in half. So, let me do that.  I'm going to divide it exactly in half. And 1/2 literally represents 1 of the 2 equal sections. So this  is one 1/2. Now, to go from fifths to tenths, you're essentially taking each of the equal sections  and you're multiplying by 2. You had 5 equals sections. You split each of those into 2, so you  have twice as many. You now have 10 equal sections. So those 2 sections that were shaded in,  well, you are going to multiply by 2 the same way. Those 2 are going to turn into 4/10. And you  see it right over here when we shaded it initially. If you Look at the tenths, you have 1/10, 2/10,  3/10, and 4/10. Let's do the same logic over here. If you have 2 halves and you want to make  them into 10 tenths, you have to take each of the halves and split them into 5 sections. You're  going to have 5 times as many sections. So to go from 2 to 10, we multiply by 5. So, similarly,  that one shaded-in section in yellow, that 1/2 is going to turn into 5/10. So we're going to  multiply by 5. Another way to think about it. Whatever we did to the denominator, we had to  do the numerator. Otherwise, somehow we're changing the value of the fraction. So, 1 times 5  is going to get you to 5. And you see that over here when we shaded it in, that 1/2, if you look at the tenths, is equal to 1, 2, 3, 4, 5 tenths. And now we are ready to add. Now we are ready to  add these two things. 4/10 plus 5/10, well, this is going to be equal to a certain number of  tenths. It's going to be equal to a certain number of tenths. It's going to be equal to 4 plus 5  tenths. And we can once again visualize that. Let me draw our grid again. So 4 plus 5/10, I'll do  it actually on top of the paint can right over here. So let me color in 4/10. So 1, 2, 3, 4. And then  let me color in the 5/10. And notice that was exactly the 4/10 here, which is exactly the 2/5. Let  me color in the 5/10-- 1, 2, 3, 4, and 5. And so how many total tenths do we have? We have a  total of 1, 2, 3, 4, 5, 6, 7, 8, 9. 9 of the tenths are now shaded in. We had 9/10 of a gallon of paint. So now to answer their question, will they have the gallon they need? No, they have less than a 

whole. A gallon would be 10 tenths. They only have 9 tenths. So no, they do not have enough of a gallon. Now, another way you could have thought about this, you could have said, hey, look,  2/5 is less than 1/2, and you could even visualize that right over here. So if I have something less than 1/2 plus 1/2, I'm not going to get a whole. So either way you could think about it, but this  way at least we can think it through with actually adding the fractions. 

Adding fractions with different signs 

Find the sum 3 and 1/8 plus 3/4 plus negative 2 and 1/6. Let's just do the first part first. It's pretty straightforward. We have two positive numbers. Let me draw a number line. So let me draw a  number line. And I'll try to focus in. So we're going to start at 3 and 1/8. So let's make this 0. So  you have 1, 2, 3, and then you have 4. 3 and 1/8 is going to be right about there. So let me just  draw its absolute value. So this 3 and 1/8 is going to be 3 and 1/8 to the right of 0. So it's going  to be exactly that distance from 0 to the right. So this right here, the length of this arrow, you  could view it as 3 and 1/8. Now whenever I like to deal with fractions, especially when they have  different denominators and all of that, I like to deal with them as improper fractions. It makes  the addition and the subtraction, and, actually, the multiplication and the division a lot easier.  So 3 and 1/8 is the same thing as 8 times 3 is 24, plus 1 is 25 over 8. So this is 25 over 8, which is  the same thing as 3 and 1/8. Another way to think about it, 3 is 24 over 8. And then you add 1/8  to that, so you get 25 over 8. So this is our starting point. Now to that, we are going to add 3/4.  We are going to add 3/4. So we're going to move another 3/4. We are going to move another  3/4. It's hard drawing these arrows. We're going to move another 3/4 to the right. So this right  here, the length of this that we're moving to the right is 3/4. So plus 3/4. Now where does this  put us? Well, both of these are positive integers. So we can just add them. We just have to find a like denominator. So we have 25 over 8. We have 25 over 8 plus 3/4. That's the same thing as we need to find a common denominator here. The common denominator, or the least common  multiple of 4 and 8 is 8. So it's going to be something over 8. To get from 4 to 8, we multiply by  2. So we have to multiply 3 by 2 as well. So you get 6. So 3/4 is the same thing as 6/8. If we have  25/8 and we're adding 6/8 to that, that gives us 25 plus 6 is 31/8. So this number right over here,  this number right over here, is 31/8. And it makes sense because 32/8 would be 4. So it should  be a little bit less than four. So this number right over here is 31/8. Or the length of this arrow,  the absolute value of that number, is 31/8, a little bit less than 4. If you wanted to write that as a mixed number, it would be what? It would be 3 and 7/8. So that's that right over here. This is  31/8. That's that part right over there. Now to that, we want to add a negative 2 and 1/6. So  we're going to add a negative number. So think about what negative 2 and 1/6 is going to be  like. So let me do this in a new color, do it in pink. Negative 2 and 1/6. So we're going to  subtract, or I guess we're going to say we're going to add a negative 1. We're going to add a  negative 2 and then a negative 1/6. So let me draw. So negative 2 and 1/6, we could literally  draw like this. Negative 2 and 1/6 we could draw with an arrow that looks something like that.  So this is negative 2 and 1/6. Now, there's a couple ways to think about it. You could just say,  hey, look, when you add this arrow, this thing that's moving to the left-- we could put it over  here, and you would get straight to negative 2 and 1/6. But we're adding this negative 2 and  1/6. It's the same thing as subtracting a positive 2 and 1/6. We're moving 2 and 1/6 to the left.  And we're going to end up with a number whose absolute value is going to look something like 

that. And it's actually going to be to the right. So it's not going to only be its absolute value.  Well, its absolute value is going to be the number since it's going to be a positive number. So  let's just think about what it is. This value right here, which is going to be the answer to our  problem, is just going to be the difference of 31/8 and 2 and 1/6. And it's the positive difference  because we're dealing with a positive number. So we just take 31/8. And from that, we will  subtract 2 and 1/6. So let's do this. So this orange value is going to be 31/8 minus 2 and 1/6. So 2 and 1/6 is the same thing as 6 times 2 is 12 plus 1 is 13. Minus 13/6. And this is equal to, once  again, we need to get a common denominator over here. And it looks like 24 will be the  common denominator, 24. And let me make it very clear. This is the 31/8. And this is the 2 and  1/6. This right here is the 2 and 1/6. So 31/8 over 24. You have to multiply by 3 to get to the 24  over here. So we multiply by 3 on the 31. That gives us 93. And then to go from 6 to 24, you  have to multiply by 4. We do that in another color. You have to multiply it by 4, so we have to  multiply by 4 up here as well. So 4 times 13, let's see. 4 times 10 is 40. 4 times 3 is 12. So that's  52. So this is going to be equal to 93 minus 52 over 24. And that is-- so 93 minus 52. 3 minus 2 is  1. 9 minus 5 is 4. So it is 41/24 and positive. And you can see that here just by looking at the  number line. This right here is 41 over 24. And it should be a little bit less than 2 because 2  would be 48 over 24. So this would be 48 over 24. And it makes sense that we're a little bit less  than that. 

Multiplying positive and negative fractions 

Let's do a few examples multiplying fractions. So let's multiply negative 7 times 3/49. So you  might say, I don't see a fraction here. This looks like an integer. But you just to remind yourself  that the negative 7 can be rewritten as negative 7/1 times 3/49. Now we can multiply the  numerators. So the numerator is going to be negative 7 times 3. And the denominator is going  to be 1 times 49. 1 times 49. And this is going to be equal to-- 7 times 3 is 21. And one of their  signs is negative, so a negative times a positive is going to be a negative. So this is going to be  negative 21. You could view this as negative 7 plus negative 7 plus negative 7. And that's going  to be over 49. And this is the correct value, but we can simplify it more because 21 and 49 both  share 7 as a factor. That's their greatest common factor. So let's divide both the numerator and  the denominator by 7. Divide the numerator and the denominator by 7. And so this gets us  negative 3 in the numerator. And in the denominator, we have 7. So we could view it as  negative 3 over 7. Or, you could even do it as negative 3/7. Let's do another one. Let's take 5/9  times-- I'll switch colors more in this one. That one's a little monotonous going all red there. 5/9  times 3/15. So this is going to be equal to-- we multiply the numerators. So it's going to be 5  times 3. 5 times 3 in the numerator. And the denominator is going to be 9 times 15. 9 times 15.  We could multiply them out, but just leaving it like this you see that there is already common  factors in the numerator and the denominator. Both the numerator and the denominator,  they're both divisible by 5 and they're both divisible by 3, which essentially tells us that they're  divisible by 15. So we can divide the numerator and denominator by 15. So divide the  numerator by 15, which is just like dividing by 5 and then dividing by 3. So we'll just divide by 15. Divide by 15. And this is going to be equal to-- well, 5 times 3 is 15. Divided by 15 you get 1 in  the numerator. And in the denominator, 9 times 15 divided by 15. Well, that's just going to be 9.  So it's equal to 1/9. Let's do another one. What would negative 5/9 times negative 3/15 be? Well,

we've already figured out what positive 5/9 times positive 3/15 would be. So now we just have  to care about the sign. If we were just multiplying the two positives, it would be 1/9. But now  we have to think about the fact that we're multiplying by a negative times a negative. Now, we  remember when you multiply a negative times a negative, it's a positive. The only way that you  get a negative is if one of those two numbers that you're taking the product of is negative, not  two. If both are positive, it's positive. If both are negative, it's positive. Let's do one more  example. Let's take 5-- I'm using the number 5 a lot. So let's do 3/2, just to show that this would  work with improper fractions. 3/2 times negative 7/10. I'm arbitrarily picking colors. And so our  numerator is going to be 3 times negative 7. 3 times negative 7. And our denominator is going  to be 2 times 10. 2 times 10. So this is going to be the numerator. Positive times a negative is a  negative. 3 times negative 7 is negative 21. Negative 21. And the denominator, 2 times 10. Well, that is just 20. So this is negative 21/20. And you really can't simplify this any further. 

Dividing negative fractions 

Let's do some examples dividing fractions. Let's say that I have negative 5/6 divided by positive  3/4. Well, we've already talked about when you divide by something, it's the exact same thing  as multiplying by its reciprocal. So this is going to be the exact same thing as negative 5/6 times the reciprocal of 3/4, which is 4/3. I'm just swapping the numerator and the denominator. So  this is going to be 4/3. And we've already seen lots of examples multiplying fractions. This is  going to be the numerators times each other. So we're going to multiply negative 5 times 4. I'll  give the negative sign to the 5 there, so negative 5 times 4. Let me do 4 in that yellow color.  And then the denominator is 6 times 3. Now, in the numerator here, you see we have a  negative number. You might already know that 5 times 4 is 20, and you just have to remember  that we're multiplying a negative times a positive. We're essentially going to have negative 5  four times. So negative 5 plus negative 5 plus negative 5 plus negative 5 is negative 20. So the  numerator here is negative 20. And the denominator here is 18. So we get 20/18, but we can  simplify this. Both the numerator and the denominator, they're both divisible by 2. So let's  divide them both by 2. Let me give myself a little more space. So if we divide both the  numerator and the denominator by 2, just to simplify this-- and I picked 2 because that's the  largest number that goes into both of these. It's the greatest common divisor of 20 and 18. 20  divided by 2 is 10, and 18 divided by 2 is 9. So negative 5/6 divided by 3/4 is-- oh, I have to be  very careful here. It's negative 10/9, just how we always learned. If you have a negative divided  by a positive, if the signs are different, then you're going to get a negative value. Let's do  another example. Let's say that I have negative 4 divided by negative 1/2. So using the exact  logic that we just said, we say, hey look, dividing by something is equivalent to multiplying by  its reciprocal. So this is going to be equal to negative 4. And instead of writing it as negative 4,  let me just write it as a fraction so that we are clear what its numerator is and what its  denominator is. So negative 4 is the exact same thing as negative 4/1. And we're going to  multiply that times the reciprocal of negative 1/2. The reciprocal of negative 1/2 is negative 2/1.  You could view it as negative 2/1, or you could view it as positive 2 over negative 1, or you could  view it as negative 2. Either way, these are all the same value. And now we're ready to multiply.  Notice, all I did here, I rewrote the negative 4 just as negative 4/1. Negative 4 divided by 1 is  negative 4. And here, for the negative 1/2, since I'm multiplying now, I'm multiplying by its 

reciprocal. I've swapped the denominator and the numerator. Or I swapped the denominator  and the numerator. What was the denominator is now the numerator. What was the numerator is now the denominator. And I'm ready to multiply. This is going to be equal to-- I gave both the  negative signs to the numerator so it's going to be negative 4 times negative 2 in the  numerator. And then in the denominator, it's going to be 1 times 1. Let me write that down. 1  times 1. And so this gives us, so we have a negative 4 times a negative 2. So it's a negative  times a negative, so we're going to get a positive value here. And 4 times 2 is 8. So this is a  positive 8 over 1. And 8 divided by 1 is just equal to 8.



Modifié le: lundi 7 mars 2022, 13:12