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Thank you We sneak And here's your shmoop du jour

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brought to you by coffee beans you wouldn't think they're

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big fans of insult comedy but they're always getting roasted

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A coffee shop sells two different types of coffee beans

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Taipei is eight fifty a pound and type he is

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nine twenty five pounds Malcolm wants to spend no more

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than a hundred bucks on coffee beans I'd also like

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at least fifty five percent of the coffee beans he

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purchases to be type a rounded to the nearest tenth

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of a pound How many pounds of type asian Malcolm

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by and here the tensile answer signed five six seven

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hey if someone's personality khun b type a why not

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a coffee bean We're sure they're some of them out

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there who are overly ambitious workaholics and well what can

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we say They love the grind So behind door number

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one we've got this type a product selling for eight

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fifty a pound and behind door number two there's a

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type b being going for nine twenty five pounds malcolm

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have one hundred bucks spending limit but he's also got

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a really weird thing about needing at least fifty five

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percent of his purchase To be the type a variety

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we want to know the maximum number of pounds of

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the type a b malcolm should buy How do we

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get there Well it always helps to turn word problem

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data into an equation here in math lands Unless you

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do that we use the variable a fur type a

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beans and b for type b b eight point five

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a plus nine point two five evils one hundred In

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other words than dollar amounts of all of the type

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a beans malcolm buys lost the dollar amount of all

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the type b b t vise will equal one hundred

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bucks However we have to fiddle with things here because

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malcolm's bizarro fifty five percent requirement dividing the number of

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pounds for type a beans by the total number of

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pounds should give us a number greater than or equal

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to point five five so a over the total pounds

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equals point five five The total number of pounds will

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be equal to a lost be so we can rewrite

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as a over the quantity A plus b equals point

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Five five Well would you look at that We've got

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ourselves a system of equation now we can start by

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taking that first equation and isolate a so we'll start

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by subtracting nine point two five b from both sides

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then divide both sides by eight point five to get

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a equals The quantity one hundred minus nine point two

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five b all over eight point five second inequality lather

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rinse repeat we'll multiply both sides by the quantity a

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plus b to get playing people's point five five a

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plus point five five d you still got an a

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on both sides of the equation though so we'll have

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to do something about that Let's subtract point five five

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a from both sides to get a point Four five

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day equals point five Five b and finally divide both

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sides by point for five to get a equals point

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Five five feet over Point for five now that all

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that's out of the way we can put on our

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substitution pants we've now determined that one hundred minus nine

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point two five feet over eight point five is the

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same as point five five feet over point four five

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so we'll set them equal to one another We'll multiply

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both sides by eight Point five to get rid of

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the fraction on the left which gives us one hundred

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minus nine point two five equals four point six seven

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five beef over point four Five Now to get rid

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of that fraction on the right we multiply both sides

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by point four five and wind up with forty five

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minus four point one six to five b equals four

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point six seven five feet add four point one six

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to five feet of both sides and who were getting

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so close we can feel it We're left with forty

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five people's eight point eight three seven five feet Then

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divide both sides by forty five and we get b

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equals five point o nine But keep in mind that

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we just found the value of b the malcolm should

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purchase around five point one pounds of the type b

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being the type a which rules out option a instead

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Let's take the five point o nine we just arrived

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at and see what we get when we plug that

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into our first inequality Well yet eight point five eight

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last nine point two five times five point o nine

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equals one hundred multiplying Through way get eight point five

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a plus forty seven point Oh eight equals one hundred

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and then we can subtract forty seven point oh eight

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from both sides Get eight point five a equals fifty

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two point ninety two Divide both sides by eight point

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five and looks like a comes out to about six

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point two two five which when we round that psych

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rock comes out tio six point to option c we

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can double check our numbers by figuring out how much

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these things are gonna run Malcolm well six point two

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times the eight fifty that it cost per pound comes

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out fifty two dollars and seventy cents and five point

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o nine times nine twenty five equals forty seven point

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eight Well if we had those two together we get

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ninety nine dollars seventy eight cents which will leave mouth

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um twelve cents left over to buy himself something nice

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