Jane had a bottle filled with juice. At first, Jane drank 1/5 the 1/4, followed by 1/3. Jane checked how much juice was left in the bottle: there was 2/3 of a cup left. How much juice was in the bottle originally?

Answer 1

Benjamin Altschul

Bottle originally had #5/3# or #1 2/3# cups off juice.

As Jane first drank #1/5#, then #1/4# and then #1/3# and GCD of denominators #5#, #4# and #3# is #60#

Let us assume there were #60# units of juice.

Jane first drank #60/5=12# units, so #60-12=48# units were left

then she drank #48/4=12# units, and #48-12=36# were left

and then she drank #36/3=12# units,

and #36-12=24# units left

As #24# units are #2/3# cup

each unit must be #2/3xx1/24# cup and

#60# units with which Jane started are equivalent to

#2/3xx1/24xx60=2/3xx1/(2xx2xx2xx3)xx2xx2xx3xx5#

#cancel2/cancel3xx1/(cancel2xxcancel2xxcancel2xx3)xxcancel2xxcancel2xxcancel3xx5#

= #5/3#

Hence bottle originally had #5/3# or #1 2/3# cups off juice.

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Answer 2

Aurora Arias

Based on the stated assumption:

#"1 bottle"= 3 1/13" cups"#

I chose the presentation to show the way of thinking when doing algebra.

#color(blue)("Assumption:")#

#color(blue)("The fractions are related to a full bottle each time")#

#color(blue)("Dr Cawas has opted for the different interpretation of")#

#color(blue)( 1-[1/3xx1/4xx1/5]" of a bottle left giving a different answer")#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("To determine how much of the bottle was drunk as a fraction")#

Total drank #-> 1/(color(red)(5))+1/(color(red)(4))+1/(color(red)(3))#

Consider the denominators. I chose to do it this way:

#color(red)(3xx4xx5)=60#

Convert all the denominators into #60^("ths")#

#[1/5 color(magenta)(xx1)] +[1/4color(magenta)(xx1)]+[1/3color(magenta)(xx1)] #

#[1/5 color(magenta)(xx12/12)] +[1/4color(magenta)(xx15/15)]+[1/3color(magenta)(xx20/20)] #

#" "12/60" " +" "15/60" "+" "20/60" "->" "(12+15+20)/60#

#" "color(blue)(= 47/60)#

Note that 47 is a prime number so this can not be simplified '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Determine the amount not drunk")#

#(1-47/60)" bottle "=" "2/3" cup"#

#color(blue)(13/60" bottle "=" "2/3" cup")#..........................Equation(1)

,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Determine volume of original full bottle")#

We need to change the #13/60# to 1. To do this we multiply by #60/13#

Multiply both sides of Equation(1) by #color(green)(60/13)#

#color(brown)(color(green)(60/13xx)13/60" bottle "=" "color(green)(60/13xx)2/3" cup")#

#60/60xx13/13" bottle "=" " 3 1/13" cup"#

#color(blue)("Full bottle had "3 1/13 " cups")#

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Answer 3

Jeremiah Andrews

Jane originally had 4 cups of juice in the bottle.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.