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Equivalent fraction word problem example 3

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You and your friends are making name cards to assign seats for dinner. You each are given an identical strip of blank paper and begin coloring parts of it with crayons. You start with blue, or you start with blue. Your friend Micah starts off with green, Joelle starts with the red, Holden starts with pink, And Kate starts with purple. After one color is used, whose strip has the same part of its area colored as yours? So let's look at my strip right over here. It has how many equal sections? 1, 2, 3, 4, 5, 6 equal sections. And how many of them have been colored blue? Well, let's see. 1, 2, 3, 4 of my equal sections have been colored blue. So I need to figure out which of these other folks have colored exactly 4/6 of their strip blue. Or another way of thinking about it, I want to say, whoever has an equivalent fraction colored, not blue, colored whatever their color might be. So let's first look at Micah, or "Mee-kah." So he's got 3 equal sections. He's got 3 equal sections, and he's colored, he's colored 2 of them, 2 of them green. Now is 4/6 and 2/3 the same thing? Well, he has 3 equal sections. I had 6. If I were to group each of these into groups of 2, so if I were to divide by 2, let me do that. If I were divide by 2, so I'm going to group this into groups of 2. Then I would have 1, 2-- so I have 1, 2, 3 equal sections. And then if I were to group the blue strips into groups of 2, then I would have 1, 2 of them that have actually been colored blue. So if I divide both the numerator and the denominator by 2, I get 2/3, which is the exact same fraction as Micah. Another way to think about it, is if you divide each of Micah's sections into 2. So if you divide this into 2, this into 2, and this into 2. Well, then how many total sections does he have? Well, he had 3, now each of those 3 have been divided into 2. So we can multiply by 2 to get 6. So this is multiplying by 2. And each of the 2 in green, there now each of those sections have now been divided into 2. So you can multiply by 2 as well, and you get 4. So 2/3 and 4/6 are the same thing. So Micah has colored in the same amount that I have. Now let's see whether any of these other fractions are the same. So Joelle has colored in 4, has colored in 4 out of 1, 2, 3, 4, 5, 6, 7. 4 out of 7. So it's not clear to me that we can somehow multiply or divide the numerator and the denominator here-- remember it has to be by the same number in order to not change the value of the fraction-- to somehow make that equal to 4/6. She's colored in the same number, but the size of each of her strips is different. So 4/7 is definitely different than 4/6. So we can rule Joelle out. Holden has colored in 3 out of 5 sections, 3 out of 5 sections. Once again, not obvious that I can multiply both the numerator and the denominator 3/5 times something to get you to 4/6, so these aren't equivalent. And then Kate has colored in, Kate has colored in 1, 2, 3, 4, 5, 6, 7, 8. 8 out of 11. And once again, I could try to get a numerator up here, 4, I could divide by 2. But then if I divide by 2 in the denominator, I'm going to get something strange. I'm going to get 4 over 11 divided by 2, which 11 divided by 2 is like 5.5. So this is clearly not the same thing as 4/6. We'll have rule this out as well. So Micah and I have colored in the same amount.