74 views

A tank has $100$ liters of water$.$ At the end of every hour, the following two operations are performed in sequence$:$ $i)$ water equal to $m\%$ of the current contents of the tank is added to the tank $, ii)$ water equal to $n\%$ of the current contents of the tank is removed from the tank$.$ At the end of $5$ hours, the tank contains exactly $100$ liters of water $.$ The relation between $m$ and $n$ is $:$

1. $m=n$
2. $m>n$
3. $m<n$
4. None of the previous

retagged | 74 views

+1 vote
Initially the tank is having $100$ liters of water.

Every hour, the capacity changes as $C' = C \left(1+\frac{m}{100}\right)\left(1 - \frac{n}{100}\right)$

So, after $5$ hours we get $C ' = 100 = 100 \left(1+\frac{m}{100}\right)^5\left(1 - \frac{n}{100}\right)^5$

$\implies \left(1+\frac{m}{100}\right)^5\left(1 - \frac{n}{100}\right)^5 = 1$

$\implies \left(1+\frac{m}{100}\right)\left(1 - \frac{n}{100}\right) = 1$

$\implies 1+\frac{m}{100}-\frac{n}{100} -\frac{mn}{10000}= 1$

$\implies100(m - n) = mn$

Since, $m,n > 0, m - n > 0 \implies m > n.$
by Veteran (425k points)
0
Sir, if we take m=n=0 then also at the end of 5 hrs, the tank will have 100 litres.. I think  it is not given in the question that m,n should not be zero..
+1
When a sentence like "x amount of ... is taken" it implies that $x > 0$ unless otherwise stated.
+1 vote

NOTE :-

Two operations are performed in sequence means than after adding m%, from that total amount n% is subtracted.

--------------------------------------------------------------------------------------------------------------------------------------------

Let the Amount in the tank at any point be 100 liters.

(100 + m% of 100) - { n% of (100 + m% of 100) }

$\Rightarrow$$(100 + \frac{m}{100} * 100 ) - { \frac{n}{100} * (100 + \frac{m}{100} * 100 ) } \Rightarrow$$(100 + m$ ) - { $n + \frac{nm}{100}$}

$\Rightarrow$ $100 + m - n - \frac{nm}{100}$

$\therefore$ $m - n - \frac{nm}{100}$ is the net increase or decrease in the original value.

------------------------------------------------------------------------------------------------------------------------------------------------

If the net increase is 0 after each hour then the value will remain 100 liters after each iteration.

$\Rightarrow$ m - n - $\frac{nm}{100}$ = 0

$\Rightarrow$ m - n = $\frac{nm}{100}$

$\Rightarrow$ 100m - 100n = nm

$\Rightarrow$ 100m - 100n = positive number ($\because$ n and m are positive numbers so n *m will also be positive)

$\Rightarrow$ m>n.

------------------------------------------------------------------------------------------------------------------------------------------------

$\therefore$ Option $B.$ is the correct answer.

by Boss (21.7k points)
edited by
0
0

@Arjun Sir

0
3rd equation you had put + instead of - while expanding the last term.
0
Thankyou for correcting :)

+1 vote