thinking skill

[cherrie3sg]

One tap fills a tank in 5 mins and the 2nd tap fills the same tank in 3 mins. how long does it take to fill the tank if both taps are on?

 

[soroban]

Hello, cherrie3sg!

One tap fills a tank in 5 mins and the 2nd tap fills the same tank in 3 mins.
How long does it take to fill the tank if both taps are on?

$\displaystyle \text{The first tap takes 5 minutes to fill the tank.}$
. . $\displaystyle \text{In one minute, it can fill }\tfrac{1}{5}\text{ of the tank.}$

$\displaystyle \text{The second tap takes 3 minutes to fill the tank.}$
. . $\displaystyle \text{In one minute, it can fill } \tfrac{1}{3}\text{ of the tank.}$

$\displaystyle \text{Together, in one minute, they can fill }\tfrac{1}{5} + \tfrac{1}{3} \:=\:\tfrac{8}{15}\text{ of the tank.}$

$\displaystyle \text{Therefore, together they can fill the tank in }\frac{1}{\frac{8}{15}} \:=\:\tfrac{15}{8} \:=\:1\tfrac{7}{8}\text{ minutes.}$

 

[BigGlenntheHeavy]

$\displaystyle cheerie3sg, \ another \ way \ to \ look \ at \ it.$

$\displaystyle One's \ rate \ multiply \ by \ the \ time \ it \ takes \ one \ to \ complete \ the \ job \ = \ unity(1).$

$\displaystyle Hence, \ Rate \ X \ Time \ = \ Fraction \ of \ Work \ Done.$

$\displaystyle In \ other \ words, \ if \ one's \ rate \ is \ 1/5, \ then \ it \ would \ take \ 5 \ minutes \ to \ complete \ the \ job$

$\displaystyle 1/5 \ X \ 5 \ = \ 5/5 \ =1 \ unity, \ job \ is \ completed.$

$\displaystyle Now, \ two \ taps \ working \ in \ unison \ with \ different \ rates \ will \ complete \ the \ job \ by \ the \ lesser \ of$

$\displaystyle their \ respective \ time, \ (common \ sense)$

$\displaystyle Ergo, \ rate \ of \ first \ tap \ X \ time(t) \ = \ fraction \ of \ its \ work \ done, \ = \ t/5$

$\displaystyle and \ rate \ of \ second \ tap \ X \ time(t) \ = \ fraction \ of \ its \ work \ done, \ = \ t/3$

$\displaystyle Therefore, \ the \ fraction \ of \ work \ done \ by \ the \ first \ tap \ combined \ with \ the \ fraction$

$\displaystyle of \ work \ done \ by \ the \ second \ tap \ must \ equal \ the \ whole \ piece \ of \ work, \ or \ 1;$

$\displaystyle hence, \ solution: \ t/5+t/3 \ = \ 1 \ \implies \ t \ = \ 15/8 \ minutes, \ the \ time \ it \ takes \ when \ they$

$\displaystyle are \ both \ working \ together.$

 

[chrisr]

One tap fills a tank in 5 mins and the 2nd tap fills the same tank in 3 mins. how long does it take to fill the tank if both taps are on?

If the taps were filling seperate tanks concurrently,
then after 3 minutes, one tank would be full and the other will be full to $\displaystyle \frac{3}{5}$ths capacity.

That's $\displaystyle \frac{8}{5}$ tanks filled.

$\displaystyle 3\left(\frac{5}{8}\right)\ mins....\frac{8}{5}\left(\frac{5}{8}\right)\ tanks\ =\ 1\ tank.$