Question

The ratio, by volume, of soap to alcohol to water in a certain solution is $2:50:100$. The solution will be altered so that the ratio of soap to alcohol is doubled while the ratio of soap to water is halved. If the altered solution will contain $100$ cubic centimeters of alcohol, how many cubic centimeters of water will it contain?

Answer 1

Hint: In this question we have been given with the ratio of $3$ liquids by volume for a certain solution and we have to find the ratio after there are certain changes to be made in the ratios, we have to find the cubic centimeters of water the solution contains given that in the altered solution there is $100$ cubic centimeters of alcohol. We will solve this question by first altering the ratio of soap to alcohol and then altering the ratio of soap to water, we will then make sure that the amount of soap is the same in both the ratios which will give us the new ratio and then we will find the volume of water.

Complete step by step answer:
We have the ratio of soap to alcohol to water as:
$\Rightarrow 2:50:100$
Consider a multiplier to be $x$ therefore, we have the ratio as:
$\Rightarrow 2x:50x:100x$
Now we are given that the ratio of soap to alcohol is doubled. We have the original ratio as:
$\Rightarrow 2x:50x$
We can write it in fraction form as:
$\Rightarrow \dfrac{2x}{50x}$
Now on doubling the ratio means we have to multiply it by $2$ therefore, we get:
$\Rightarrow \dfrac{2x}{50x}\times 2$
On simplifying the fraction, we get:
$\Rightarrow \dfrac{2x}{25x}$
Therefore, we can write the new ratio of soap to alcohol as:
$\Rightarrow 2x:25x$
Now we are given the ratio of soap to water halved. We have the original ratio as:
$\Rightarrow 2x:100x$
We can write it in fraction form as:
$\Rightarrow \dfrac{2x}{100x}$
Now on halving the ratio means we have to multiply it by $\dfrac{1}{2}$ therefore, we get:
$\Rightarrow \dfrac{2x}{100x}\times \dfrac{1}{2}$
On multiplying the fraction, we get:
$\Rightarrow \dfrac{2x}{200x}$
Therefore, we can write the new ratio of soap to water as:
$\Rightarrow 2x:200x$
Now the new ratio of soap to alcohol to water is:
$\Rightarrow 2x:25x:200x$
Now we know that in the altered solution there is $100$ cubic centimeters of alcohol and we have to find the amount of water.
We can write the ratio of alcohol to water as:
$\Rightarrow 25x:200x$
We can write it as fraction as:
$\Rightarrow \dfrac{25x}{200x}$
On using the cross formula, we get:
$\Rightarrow \dfrac{25x}{200x}=\dfrac{100}{a}$
On rearranging and simplifying, we get:
$\Rightarrow a=\dfrac{200x\times 100}{25x}$
On simplifying, we get:
$\Rightarrow a=800$, which is the volume of water in the altered solution.
Therefore, the volume of water in the altered solution is $800$ cubic centimeters which is the required solution.

Note: A ratio has to be with similar quantities for comparison, while comparison of two quantities the units of both the quantities should be the same. Ratios and proportions are used mostly when $2$ quantities are in terms of a fraction for example distance upon time or rupees per meter etc.