Question

A vessel which contains \[100{\rm{ }}litres\] of salt and sugar solution in the ratio of \[22:3\]. From the vessel \[40{\rm{ }}litres\] of mixture is taken out and \[4.8{\rm{ }}litres\] of pure salt solution and pure sugar solution, both are added to the mixture. What is the percentage of the quantity of sugar solution in the final mixture less than the quantity of salt solution?

Answer 1

Hint: Here we are given the amount of salt and sugar solution present in a huge volume we are also given that we are taking a small quantity of solution from it and a quantity of sugar and salt solution to it. Now we have to find the amount of salt and sugar present in the small quantity after adding an extra solution. From the sugar and salt values we will find the difference in the percentage of sugar and salt in the final solution.

Formula used:\[{\bf{Percentage}} = \left( {\dfrac{{{\bf{Value}}}}{{{\bf{Total}}{\rm{ }}{\bf{Value}}}}} \right) \times {\bf{100}}\].

Complete step-by-step answer:
Here, it’s given that a vessel contains \[100{\rm{ }}litres\] of salt and sugar solution in the ratio of \[22:3\]
So, salt is \[22\] out of \[25\] and sugar is \[3\] out of \[25\].
We have to find the initial amount of salt
To find the initial amount of salt in the given \[100{\rm{ }}litres\] we should multiply it with \[\dfrac{{22}}{{25}}\].
That is initial amount of salt in \[100{\rm{ }}litres = \dfrac{{22}}{{25}} \times 100 = 88{\rm{ }}litres\]
Again we should find the initial amount of sugar
To find the initial amount of sugar in the given \[100{\rm{ }}litres\] we should multiply it with \[\dfrac{3}{{25}}\].
So, initial amount of sugar in \[100{\rm{ }}litres = 100 - 88 = 12\,litres\]
Here we have taken out \[40{\rm{ }}litres\] of mixture from the total mixture and added \[4.8{\rm{ }}litres\] of pure salt and sugar solution to it.
Then we have to find the amount of sugar and salt in the newly formed mixture by adding \[4.8{\rm{ }}litres\] of salt and sugar.
Initially we have found that amount of salt is \[{\rm{88 }}litres\] now we have taken 40 liters from the original mixture therefore the amount of salt present in the \[40{\rm{ }}litres\] taken out is found by multiplying it with \[\dfrac{{22}}{{25}}\].
To find the amount of salt in the newly formed mixture we should subtract the initial amount of salt with salt present in the \[40litres\] and finally add it to \[4.8{\rm{ }}litres\] of salt.
Final amount of salt in solution \[ = 88 - \dfrac{{22}}{{25}} \times 40 + 4.8 = 57.6\,litres\]
Initially we have found that amount of sugar is \[{\rm{88 }}litres\] now we have taken 40 liters from the original mixture therefore the amount of sugar present in the \[40{\rm{ }}litres\] taken out is found by multiplying it with \[\dfrac{3}{{25}}\].
To find the amount of sugar in the newly formed mixture we should subtract the initial amount of sugar with sugar present in the \[40litres\] and finally add it to \[4.8{\rm{ }}litres\] of sugar.
Final amount of sugar in solution \[ = 12 - \dfrac{3}{{25}} \times 40 + 4.8 = 12\;litres\]
We have to find the percentage of sugar solution in final mixture
So, Percentage of sugar solution in final mixture less that quantity of salt solution \[ = \dfrac{{57.6 - 12}}{{57.6}} \times 100 = 79.16\% \]
Hence, Percentage of sugar solution in the final mixture is less than the quantity of salt solution is \[79.16\% \].

Note:A percentage is a fraction of an amount expressed as a particular number of hundredths of that amount. Like this question we have to concentrate on what we take out of and how much quantity we take those things. It will help us to solve this problem easily.
Also while finding the amount of sugar and salt present in the small quantity we should add 4.8 to the found value because we are given that 4.8 liters of solution is added to the small quantity.