Question

$85$kg of a mixture contains milk and water in the ratio $27:7$. How much more water is to be added to get a new mixture containing milk and water in the ratio $3:1$?
A. $5kg$
B. $6.5kg$
C. $7.25kg$
D. $8kg$

Answer 1

Hint: In this problem we have a certain kilogram of a mixture of water and milk in the certain ratio. We need to find how much weight of water is to be added to get a new mixture containing milk and water in the given ratio. By using a cross multiplication method let us find the required weight of water.

Complete step-by-step solution:
Total mixture of milk and water $ = 85kg$
Let the amount of milk $ = x$
Therefore, the amount of water $ = 85 - x$
Given the ratio of milk and water $ = 27:7$
$ \Rightarrow \dfrac{{27}}{7} = \dfrac{x}{{85 - x}}$
Solve for $x$ using cross multiplication,
$ \Rightarrow 27\left( {85 - x} \right) = 7x$
Multiplying the terms,
$ \Rightarrow 2295 - 27x = 7x$
Keep $x$ terms in the right hand side,
$ \Rightarrow 2295 = 34x$
Rearranging the terms,
$ \Rightarrow x = \dfrac{{2295}}{{34}}$
Dividing the terms,
$ \Rightarrow x = \dfrac{{135}}{2}$
Since we have already taken $x$ is the amount of milk. So $\dfrac{{135}}{2}kg$ is the amount of milk, which means that we have,
Amount of water $ = 85 - \dfrac{{135}}{2}$
Taking LCM in the above equation, we have
Amount of water $ = \dfrac{{170}}{2} - \dfrac{{135}}{2} = \dfrac{{35}}{2}$
Therefore, amount of water $ = \dfrac{{35}}{2}kg$
Water to be added to make the ratio $3:1$.
Let the amount of water added be $n$
$ \Rightarrow \dfrac{3}{1} = \left( {\dfrac{{\dfrac{{135}}{2}}}{{\dfrac{{35}}{2} + n}}} \right)$
Now, solve for $n$. Again, start with cross multiplication,
$ \Rightarrow 3\left( {\dfrac{{35}}{2} + n} \right) = \dfrac{{135}}{2}$
Multiplying the terms,
$ \Rightarrow \dfrac{{105}}{2} + 3n = \dfrac{{135}}{2}$
Simplifying we get,
$ \Rightarrow 3n = \dfrac{{30}}{2}$
Hence we get,
$ \Rightarrow n = 5$
$\therefore $ $5kg$ of water is to be added to get a new mixture containing milk and water in the ratio $3:1$.

Option A is the correct answer.

Note: We can observe that, in mathematics a ratio indicates how many times one number contains another. The number in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive. A ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers. When two quantities are measured with the same unit, as is often the case, their ratio is a dimensionless number. A quotient of two quantities that are measured with different units is called a rate.