Question

Two pipes \[A\] and \[B\] can separately fill a cistern in \[20\] minutes and \[30\] minutes respectively, while a third pipe \[C\] can empty the full cistern in \[15\] minutes. If all the pipes are opened together, in what time the empty cistern is filled?

Answer 1

Hint: This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. We need to know how to find what time the empty cistern is filled with the help of given information. Also, we need to know how to convert the given information in terms of mathematical expressions.

Complete step by step solution:
In this question, we have to find at what time the empty cistern will be filled.
Let’s make the given information into a mathematical expression
First, we have that
The pipe \[A\] can fill a cistern in \[20\] minutes
So, in one minute the fraction of cistern filled by the pipe \[A\] \[ = \dfrac{1}{{20}}\]
Next. We have that
The pipe \[B\] can fill a cistern in \[30\] minutes
So, in one minute the fraction of cistern filled by pipe \[B\] \[ = \dfrac{1}{{30}}\]
Next, we have that,
The pipe \[C\] can empty the full cistern in \[15\] minutes
So, in one minute the fraction of cistern emptied by pipe \[C\] \[ = \dfrac{1}{{15}}\]
So, we get
In one minute, the fraction of cistern filled, if all the \[3\] pipes are opened together is,
 \[ \Rightarrow \] In one minute the fraction of cistern filled by the pipe \[\left( {A + B} \right)\] \[ - \] in one minute the fraction of cistern emptied by pipe \[C\]
 \[ = \left( {\dfrac{1}{{20}} + \dfrac{1}{{30}}} \right) - \dfrac{1}{{15}}\] \[ \to \left( 1 \right)\]
Let’s solve the term present in the open bracket,
 \[\left( {\dfrac{1}{{20}} + \dfrac{1}{{30}}} \right) = \dfrac{{30 + 20}}{{600}} = \dfrac{{50}}{{600}} = \dfrac{5}{{60}} = \dfrac{1}{{12}}\] (Here we use cross multiplication)
So, the equation \[\left( 1 \right)\] becomes,
 \[
  \left( 1 \right) \to \left( {\dfrac{1}{{20}} + \dfrac{1}{{30}}} \right) - \dfrac{1}{{15}} = \dfrac{1}{{12}} - \dfrac{1}{{15}} = \dfrac{{15 - 12}}{{12 \times 15}} = \dfrac{3}{{180}} = \dfrac{1}{{60}} \;
 \]
So, we get
In one minute, the fraction of cistern filled, if all the three pipes are opened together is \[\dfrac{1}{{60}}\]
So, the final answer is,
If all the three pipes are opened together, then the empty cistern is filled in \[60\] minutes.
So, the correct answer is “ \[60\] minute”.

Note: Note that if two fraction terms are involved in addition or subtraction, we can simplify the term using cross multiplication. The formula for solving the fraction terms in cross multiplication is \[\left( {\dfrac{a}{b} + \dfrac{c}{d}} \right) = \left( {\dfrac{{ad + cb}}{{bd}}} \right)\] . Also, note that the denominator term would not be equal to zero. Note the following when we multiply the different sign terms,
A. \[\left( - \right) \times \left( - \right) = \left( + \right)\]
B. \[\left( - \right) \times \left( + \right) = \left( - \right)\]
C. \[\left( + \right) \times \left( + \right) = \left( + \right)\]