# 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?

Sep 30, 2016

Bottle originally had $\frac{5}{3}$ or $1 \frac{2}{3}$ cups off juice.

#### Explanation:

As Jane first drank $\frac{1}{5}$, then $\frac{1}{4}$ and then $\frac{1}{3}$ and GCD of denominators $5$, $4$ and $3$ is $60$

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

Jane first drank $\frac{60}{5} = 12$ units, so $60 - 12 = 48$ units were left

then she drank $\frac{48}{4} = 12$ units, and $48 - 12 = 36$ were left

and then she drank $\frac{36}{3} = 12$ units,

and $36 - 12 = 24$ units left

As $24$ units are $\frac{2}{3}$ cup

each unit must be $\frac{2}{3} \times \frac{1}{24}$ cup and

$60$ units with which Jane started are equivalent to

$\frac{2}{3} \times \frac{1}{24} \times 60 = \frac{2}{3} \times \frac{1}{2 \times 2 \times 2 \times 3} \times 2 \times 2 \times 3 \times 5$

$\frac{\cancel{2}}{\cancel{3}} \times \frac{1}{\cancel{2} \times \cancel{2} \times \cancel{2} \times 3} \times \cancel{2} \times \cancel{2} \times \cancel{3} \times 5$

= $\frac{5}{3}$

Hence bottle originally had $\frac{5}{3}$ or $1 \frac{2}{3}$ cups off juice.

Oct 3, 2016

Based on the stated assumption:

$\text{1 bottle"= 3 1/13" cups}$

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

#### Explanation:

$\textcolor{b l u e}{\text{Assumption:}}$

$\textcolor{b l u e}{\text{The fractions are related to a full bottle each time}}$

$\textcolor{b l u e}{\text{Dr Cawas has opted for the different interpretation of}}$

$\textcolor{b l u e}{1 - \left[\frac{1}{3} \times \frac{1}{4} \times \frac{1}{5}\right] \text{ of a bottle left giving a different answer}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{To determine how much of the bottle was drunk as a fraction}}$

Total drank $\to \frac{1}{\textcolor{red}{5}} + \frac{1}{\textcolor{red}{4}} + \frac{1}{\textcolor{red}{3}}$

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

$\textcolor{red}{3 \times 4 \times 5} = 60$

Convert all the denominators into ${60}^{\text{ths}}$

$\left[\frac{1}{5} \textcolor{m a \ge n t a}{\times 1}\right] + \left[\frac{1}{4} \textcolor{m a \ge n t a}{\times 1}\right] + \left[\frac{1}{3} \textcolor{m a \ge n t a}{\times 1}\right]$

$\left[\frac{1}{5} \textcolor{m a \ge n t a}{\times \frac{12}{12}}\right] + \left[\frac{1}{4} \textcolor{m a \ge n t a}{\times \frac{15}{15}}\right] + \left[\frac{1}{3} \textcolor{m a \ge n t a}{\times \frac{20}{20}}\right]$

$\text{ "12/60" " +" "15/60" "+" "20/60" "->" } \frac{12 + 15 + 20}{60}$

$\text{ } \textcolor{b l u e}{= \frac{47}{60}}$

Note that 47 is a prime number so this can not be simplified
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the amount not drunk}}$

$\left(1 - \frac{47}{60}\right) \text{ bottle "=" "2/3" cup}$

$\textcolor{b l u e}{\frac{13}{60} \text{ bottle "=" "2/3" cup}}$..........................Equation(1)

,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine volume of original full bottle}}$

We need to change the $\frac{13}{60}$ to 1. To do this we multiply by $\frac{60}{13}$

Multiply both sides of Equation(1) by $\textcolor{g r e e n}{\frac{60}{13}}$

$\textcolor{b r o w n}{\textcolor{g r e e n}{\frac{60}{13} \times} \frac{13}{60} \text{ bottle "=" "color(green)(60/13xx)2/3" cup}}$

$\frac{60}{60} \times \frac{13}{13} \text{ bottle "=" " 3 1/13" cup}$

$\textcolor{b l u e}{\text{Full bottle had "3 1/13 " cups}}$