PSAT 1.16 Math Diagnostic. What is the maximum number of pounds of Type A that Malcolm should buy?

Math | Problem Solving and Data Analysis |

PSAT | PSAT Math Diagnostic |

PSAT Math | Math Diagnostic |

nine twenty five pounds Malcolm wants to spend no more

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

at least fifty five percent of the coffee beans he

purchases to be type a rounded to the nearest tenth

of a pound How many pounds of type asian Malcolm

by and here the tensile answer signed five six seven

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

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

there who are overly ambitious workaholics and well what can

we say They love the grind So behind door number

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

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

type b being going for nine twenty five pounds malcolm

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

a really weird thing about needing at least fifty five

percent of his purchase To be the type a variety

we want to know the maximum number of pounds of

the type a b malcolm should buy How do we

get there Well it always helps to turn word problem

data into an equation here in math lands Unless you

do that we use the variable a fur type a

beans and b for type b b eight point five

a plus nine point two five evils one hundred In

other words than dollar amounts of all of the type

a beans malcolm buys lost the dollar amount of all

the type b b t vise will equal one hundred

bucks However we have to fiddle with things here because

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

pounds for type a beans by the total number of

pounds should give us a number greater than or equal

to point five five so a over the total pounds

equals point five five The total number of pounds will

be equal to a lost be so we can rewrite

as a over the quantity A plus b equals point

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

ourselves a system of equation now we can start by

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

by subtracting nine point two five b from both sides

then divide both sides by eight point five to get

a equals The quantity one hundred minus nine point two

five b all over eight point five second inequality lather

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

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

plus point five five d you still got an a

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

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

a from both sides to get a point Four five

day equals point five Five b and finally divide both

sides by point for five to get a equals point

Five five feet over Point for five now that all

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

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

point two five feet over eight point five is the

same as point five five feet over point four five

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

both sides by eight Point five to get rid of

the fraction on the left which gives us one hundred

minus nine point two five equals four point six seven

five beef over point four Five Now to get rid

of that fraction on the right we multiply both sides

by point four five and wind up with forty five

minus four point one six to five b equals four

point six seven five feet add four point one six

to five feet of both sides and who were getting

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

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

divide both sides by forty five and we get b

equals five point o nine But keep in mind that

we just found the value of b the malcolm should

purchase around five point one pounds of the type b

being the type a which rules out option a instead

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

at and see what we get when we plug that

into our first inequality Well yet eight point five eight

last nine point two five times five point o nine

equals one hundred multiplying Through way get eight point five

a plus forty seven point Oh eight equals one hundred

and then we can subtract forty seven point oh eight

from both sides Get eight point five a equals fifty

two point ninety two Divide both sides by eight point

five and looks like a comes out to about six

point two two five which when we round that psych

rock comes out tio six point to option c we

can double check our numbers by figuring out how much

these things are gonna run Malcolm well six point two

times the eight fifty that it cost per pound comes

out fifty two dollars and seventy cents and five point

o nine times nine twenty five equals forty seven point

eight Well if we had those two together we get

ninety nine dollars seventy eight cents which will leave mouth

um twelve cents left over to buy himself something nice