Adding fractions word problem: paint

Cindy and Michael need one 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 half a gallon of yellow paint if they mix their paints together will they have the one gallon they need so let's think about that we're going to add the 2/5 of red paint and we're going to add that to our 2/5 of a gallon of red paint and we're going to add that to half a gallon of yellow paint and we want to see if this gets to being one whole gallon so whenever we add fractions right over here we're not adding the same thing you're writing 2/5 here we're adding one halfs 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 & 2 and since 5 & 2 are both prime numbers the smallest number is just going to be their product 10 is the smallest number that we can think of that is divisible by both 5 & 2 so let's rewrite each of these fractions with 10 as a denominator so with 10 as the denominator so 2/5 is going to be something over 10 and 1/2 is going to be something something over 10 and to help us visualize this let me go out draw a grid let me draw a grid with tents in it so that's that and that's that right over here so each of these are intense these are 10 equal segments this bar is divided in two so let's try to visualize what 2/5 looks like on this bar well if we were to divide right now it's divided into tenths if we were to divide this bar into fifths then we're going to have so that's 1 actually let me do in that same color so it's going to be it's going to be so this is 1 division 2 3 4 so notice if you go between the red marks you are going these are each of fifth of the bar if you go between the red marks and we have two of them so we're going to go 1 & 2 this right over here this this part of the bar represents two fifths of it now let's do the same thing for one half so let's divide this bar exactly in half so exactly in half so let me do that so I'm going to divided exactly in half and one half literary map represents one of the two equal sections so this is one one-half now to go from fifths to tents you're essentially taking each of the equal sections and you are multiplying by two so to go from fifths to tents you're multiplying by two you have five equal sections you split each of those into two so you have twice as many you don't have 10 equal sections so those two sections that were shaded in what you were going to multiply by two the same way those two are going to turn into those two are going to turn into four tenths four tenths 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 two halves and you want to make them into ten tenths you have to take each of the halves and split them into five sections you're going to have five times as many sections so to go from two to ten to go from two to ten we multiply by five so similarly that one shaded in section that in yellow that's going to turn into five that 1/2 is going to turn into five tenths so we're going to multiply by five another way to think about it whatever we did to the numerator or whatever we did in the denominator we have to do the numerator whatever we did to the denominator we have to do the numerator otherwise we're changing somehow it work changing the value of the fraction so one times five is going to get you to five and you see that over here when we shaded it in that 1/2 if you look at the tenths is equal to one two three four five tenths and now we are ready to add now we are ready to add these two things four tenths plus five tenths 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 going to be equal to 4 plus five tenths 4 plus five four plus five tents and we can once again visualize that let me draw our grid again so four plus five tents I'll do it actually on top of the paint can right over here so let me color in four tents so one two three four and then let me color in the five tents and notice that was exactly the four tents here which is exactly the two-fifths let me color in the five tenths one two three four and five and so how many total tents do we have we have a total of one two three four five six seven eight nine nine of the tents are now shaded we had nine tenths of a gallon of paint so now to answer their question will they have the gallon they need no they have less than a hold a gallon would be ten tenths they only have nine 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 two fifths is less than a half and you could even visualize that right over here so 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