Topic D, E, & F: Foundations
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Akshay brought four boxes of chocolate truffles to a party. Each box contains 6 truffles. Every guest at the party ate exactly 3 truffles, and there were none left over. How many guests must there have been at the party, must have attended the party? So we're trying to figure out how many guests must have attended the party. So let's actually define a letter to represent that. So let's say that p, or let's say g, g for guests, let's say that g is equal to the number of guests at the party. Then we could actually set up a relation between the number of guests, the number of truffles each guest ate, and then the total number of chocolates. So what was the total number of chocolates that we have at this party? Well, he bought 4 boxes. And each box contains 6 truffles. So the total number of chocolates at the party must have been 4 times 6. 4 times 6 truffles must have been the total number of truffles at the party. So let me write this down. This is the number of truffles total. Now, what's another way of thinking about the total number of truffles at the party? Well, you have g guests. So you have g guests. And each guest ate 3 truffles. So g times 3 is also going to be the number of truffles at the party-- number of truffles total. So these two things need to be equal to each other. So we could figure out what 4 times 6 is. And then we say, well, 4 times 6 is going to be some number. And g times 3 has to equal that same number. What must g be? So let's think about that step by step. So let's just visualize 4 times 6. So here is one box of truffles. We get 6 truffles. So it's 1 times 6, 2 times 6, 3 times 6, and 4 times 6. Or another way of thinking about 4 times 6, that's the same thing as 6 plus 6 plus 6 plus 6, which is 6, 12, 18, 24. So what we have here on the left hand side is 24. So we get 24. And now we know that this is going to be equal to the number of guests at the party. This is going to be equal to the number of guests at the party times 3. So what times 3 is equal to 24? And another way of viewing this is if g times 3 is equal to 24, that means that 24 divided by 3 must be equal to g. So one way of thinking about it, if I were to divide these truffles, these 24 truffles, into groups of 3, 3 for each guest, well, the number of groups I'm going to have will tell me the number of guests I have. So let's do that. So let's divide it into groups of 3. So let's see. Here is one group of 3, right over here. One group of 3. And now I have another group of 3, so two groups of 3. And now I have three groups of 3. Here's four groups of 3. Here is five groups of 3, six groups of 3, seven groups of 3, and eight groups of 3. So if I were to take 24 things and divide it into groups of 3, I get 8 groups. So we see that g-- let me get my pen tool. We see that g, or I could say 24 divided by 3 is 8, which must be equal to the number of guests at the party. The other way of thinking about it is I'm like, hey, some mystery number here, g, that I'm trying to figure out, the number of guests at the party times 3 is equal to 24. So what times 3 is 24? Well, you could just think about all the multiples of three. 3 times 1 is 3. 3 times 2 is 6. 3 times 3 is 9, 12, 15, 18, 21, 24. That's three times 1, 2, 3, 4, 5, 6, 7, 8. 3 times 8 is 24, so g must be equal to 8, or 8 must be equal to g. Or we had exactly 8 guests at our party.