Missing angle problems
There is a half an apple pie left. You want to eat twice what your little brother eats, but you also need to save a slice for your mom. You can cut her a slice that is 30 degrees. What is the measure of your piece of the pie in degrees? So to tackle this, we just have to remember a few things. We have to remember that the degrees in a circle-- so if we were to go all the way around a circle that that would be 360 degrees. But we only have half a pie here. And actually, let me draw a little bit differently. So if we had a whole pie-- so if we started here, and we had a whole pie, we went all the way around, that would be 360 degrees. But we only have a half pie. So we only have 180 degrees and let me do that in a color you're more likely to see. That's hard to see as well. We only have 180 degrees of pie left, but let's just keep that in mind and think about how we're going to split it between ourselves, our brother, and our mom. So let's define some variables. Let's let x be equal to the degree measure of my brother's pie or your little brother's pie. So this is degree measure of brother's pie. And then what would the amount of pie you eat be? Well, it says you eat twice what your little brother eats. So 2x would be equal what you eat, and it's really the degree measure of what you eat. And then how much does your mom eat? Well, it says you cut her slice that is 30 degrees. So your mom's going to get a 30 degree slice, going to get a slice of something like that. So the amount that your brother eats, x, plus the amount that you eat, plus the amount that your mom gets, plus 30 degrees, is going to be equal to this half pie. And remember, all of these are degree measures. So it's going to be equal to 180 degrees. And just to visualize it over here, let me draw it down here where it's easier to see it. We have half a pie that we're dealing with. We're going to save 30 degrees for our mom. So that's 30 degrees right over there. Your brother is going to eat some amount. So that is x. And then you're going to eat twice that amount. So that is 2x. x is the measure of this angle, and then 2x is a measure of that angle. So you see that 30 degrees plus x plus 2x, or x plus 2x plus 30 degrees, is going to be equal to 180 degrees. Now, we can simplify this. If we have one of something and then have another two of it, how much do we have now? Well, I now have three x's. So 1x plus 2x is going to be 3x. So I have 3x plus 30 degrees is going to be equal to, of course, 180 degrees. Now, to solve for x, we can subtract 30 degrees from both sides. So minus 30 degrees, minus 30 degrees. And I could have just written-- I could have all assumed that I'm doing it in degrees and then just done it at the end. I keep writing it so I'll just keep going with that. And then we are left with 3x equaling 180 degrees minus 30 degrees is equal to 150 degrees. And now we can just divide both sides by 3, and we're left with x equaling 150 divided by 3 is 50 degrees. x is equal to 50 degrees. Now, we have to be careful. x is not what they're asking for. They're asking for the measure of your piece of pie in degrees. x is the degree measure of your brother's piece of pie. What you eat is 2 times that. So if x is 50 degrees, 2 times that is going to be 100 degrees. So what is the measure of your piece of the pie in degrees? It is going to be 100 degrees. So if we draw our pie again, if we draw our half pie, you have 30 degrees for your mom. So that's 30 degrees for your mom. You have 50 degrees for your brother. And then you have twice that for yourself, 100 degrees.