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A container originally contains $10$ litres of pure spirit. From this container, $1$ litre of spirit replaced with $1$ litre of water. Subsequently, $1$ litre of the mixture is again replaced with $1$ litre of water and this process is repeated one more time. How much spirit is now left in the container?

1. $7.58$ litres
2. $7.84$ litres
3. $7$ litres
4. $7.29$ litres

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Quantity left after $n$ operations $=x\left(1 -\dfrac{y}{x}\right)^{n}$
where, $x=$ initial quantity
$y=$ amount of mixture withdrawn each time (this should be same every time)
$n=$ no. of times operation performed

$\quad = 10\left(1 -\dfrac{1}{10}\right)^{n}$
$\quad =10 \left(\dfrac{9}{10}\right)^{3}$
$\quad = 10\times 0.9\times 0.9\times 0.9$
$\quad = 10\times 0.729 = 7.29$ litres

Hence, option $D$ is correct.

From where did you get this formula?
$10 l$ initially

After $1^{st}$ replacement: $9l$ spirit + $1l$ water $(mixture)$

$1l$ mixture is again replaced. Means $\dfrac{9}{10}\times 1l=0.9l\ spirit\ removed$

After $2^{nd}$ replacement: $9-0.9=8.1$ spirit + $1.9l$ water

$1l$ mixture is again replaced. Means $\dfrac{8.1}{10}\times 1l=0.81l\ spirit\ removed$

After $3^{rd}$ replacement: $8.1-0.81=7.29$ spirit + $2.71l$ water

$\therefore\ 7.29l\ spirit\ is\ left$
Initial = 10L of spirit

stage 1:-

Take out 1 L of  S and add 1 L of W

9L of S + 1L of W

stage 2:-

1 L of this mixture contains 0.9 L of S and 0.1 L of W

Take out 1 L of above mixture and add 1 L of W

satge 3:-

Currently 10 L = 8.1 L of S + 1.9 L of W

1 L of this mixture contains 0.81 L of S and 0.19 L of W

Take out 1 L of above mixture and add 1 L of W

stage 4:

Currently 10 L = 7.29 L of S + 2.71 L of W

Hence the mixture contains 7.29 L of Spirit and 2.71 L of Water at final stage ie Ans is ==> D)

1 L of this mixture contains 0.9 L of S and 0.1 L of W

Take out 1 L of above mixture and add 1 L of W

Can you explain next step??

In case anyone needs elaboration of stage2

After taking out 1L of spirit and pouring 1L of water in stage 1, the ratio of spirit:water becomes 9:1.

So in 10L of mixture, 9L of Spirit and 1L of Water.

Similarly, in 1L of mixture, 9/10L=0.9L of Spirit and 0.1L of water will be there.

Now at this stage we are taking out 1L of mixture (containing 0.9L of spirit and 0.1L of water) and in the bucket there is 9L left of which 9*0.9L=8.1L is spirit and 9*0.1L=0.9L of water is there.

Adding 1L of water to this.

Spirit volume remains 8.1L and water becomes 0.9L+1L=1.9L.

Don't think about adding water. As how much of spirit is taken out that much water will be added.

First time you take out 1 litre of spirit or 1/10th of original capacity which is (10 litres*1/10) is taken out.. So 9 Litres remain. Rest is water.

Next time again 1/10th of the mixture which contains both spirit and water is taken out.
Lets consider only spirit. Total spirit was 9 Litres. 9 Litres * 1/10 = 8.1 litres spirit remains. Here 1/10 and not 1/9 because we are considering total mixture which is 10 litres only..

Again if we take 8.1 litres * 1/10th = 7.29 Litres of Spirit..
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