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Another word problem involving a linear system.

paulmdrdo

Active member
May 13, 2013
386
during its annual picnic, a company supplies lemonade for all employees and their families. the picnic committee has purchased twice as many pint jugs as quart jugs and 8 fewer gallon jugs than quart jugs. how many jugs of each type are there if 22 gallons of lemonade were purchased? (there 2 pints to a quarts and 4 quarts to a gallon.)

let
$x$= number of quart jugs
$2x=$ number pint jugs
$x-8$ = number of gallon jugs.

can you help me how i will set up my equation?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: Another word problems.

You want to add up the number of gallons that each set of jugs can hold, and equate this to the total number of gallons purchased.

For example, if you have 12 quart jugs, how many gallons do they contain?
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
Re: Another word problems.

You will, of course, need to know how many pints there are in gallon and how many quarts in a gallon.
 

SuperSonic4

Well-known member
MHB Math Helper
Mar 1, 2012
249
Re: Another word problems.

during its annual picnic, a company supplies lemonade for all employees and their families. the picnic committee has purchased twice as many pint jugs as quart jugs and 8 fewer gallon jugs than quart jugs. how many jugs of each type are there if 22 gallons of lemonade were purchased? (there 2 pints to a quarts and 4 quarts to a gallon.)

let
$x$= number of quart jugs
$2x=$ number pint jugs
$x-8$ = number of gallon jugs.

can you help me how i will set up my equation?
Your equations are set up well. The next step would be to make sure your units are all the same - for this exercise I would recommend using pints.

Can you work out from the information given how many pints there are in a quart and how many pints in a gallon?
 

paulmdrdo

Active member
May 13, 2013
386
Re: Another word problems.


i will use the unit of gallons here

$x=$ number of quart jugs
$2x=$ number pint jugs
$x−8$= number of gallon jugs

using the given information i'll have
$\frac{1}{4}x=$ # of quart jugs(in gallons)

$\frac{1}{4}x=$ # of pint jugs(in gallons)

$\frac{1}{4}x-8=$ # of gallon jugs.

setting my equation,

$\frac{1}{4}x+\frac{1}{4}x+\frac{1}{4}x-8=22$

$\frac{3}{4}x=30$

$3x=120$ then, $x=40$

now i have,
$40$ quart jugs. converting it to gallons i'll have 10 gallons.
$80$ pint jugs equivalent also to 10 gallons.
$40$quartz-8= $(10-8)$gallons = 2 gallons

10gallons+10gallons+2gallons=22 gallons.

i think i got it right. but can you give me comment on my solution.
 

paulmdrdo

Active member
May 13, 2013
386
Re: Another word problems.

i also tried solving by choosing unknown represent $x=$# gallons jug and this is what i get,

$x=$# of gallon jug.
$x+8$=# of quart jug ---> convert to gallons
$2x+16=$#of pint jug----> convert to gallons

then,


$x=$# of gallon jug

$\frac{1}{4}x+2=$#of quart jug(in gal.)

$\frac{1}{4}x+2=$# pint jug (in gal.)

$x+\frac{1}{4}x+2+\frac{1}{4}x+2=22$

$\frac{2}{4}x+x+4=22$

$x=12$

now there is 12 gallon jug, 5 gallon jug(20 quart jug), 5 gallon jug(40 pint jug).

12+5+5= 22 gallons. the total gallons matches. but the individual gallons doesn't conform with my solution above. which one is correct? please help!
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: Another word problems.

during its annual picnic, a company supplies lemonade for all employees and their families. the picnic committee has purchased twice as many pint jugs as quart jugs and 8 fewer gallon jugs than quart jugs. how many jugs of each type are there if 22 gallons of lemonade were purchased? (there 2 pints to a quarts and 4 quarts to a gallon.)

let
$x$= number of quart jugs
$2x=$ number pint jugs
$x-8$ = number of gallon jugs.

can you help me how i will set up my equation?
Using your variables, I then obtained the following equation:

\(\displaystyle \frac{2x}{8}+\frac{x}{4}+x-8=22\)

\(\displaystyle \frac{3x}{2}=30\)

\(\displaystyle x=20\)

Hence, there are 40 pint jugs, 20 quart jugs, and 12 gallon jugs.
 

paulmdrdo

Active member
May 13, 2013
386
Re: Another word problems.

how do you get these terms $\displaystyle \frac{2x}{8}+\frac{x}{4}+x-8$

and also, can you pin point what's my mistake in my first and 2nd solution. thanks!
 
Last edited:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: Another word problems.

how do you get these terms $\displaystyle \frac{2x}{8}+\frac{x}{4}+x-8$
How many pints are in a gallon? How many quarts?