Answer
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Hint: We know that the comparison of two quantities in terms of ‘how many times’ is known as ratio.
For example: There are 24 girls and 16 boys in a class. Ratio of the numbers of girls to the numbers of boys $ = \dfrac{{24}}{{16}} = \dfrac{3}{2}$= 3:2
Therefore, we can get equivalent ratios by multiplying or dividing the numerator and denominator by the same number.
A ratio can be written as a fraction, thus the ratio 3:17 can be written as $\dfrac{3}{{17}}$.
Also, a percentage is a number or ratio that represents a fraction of 100. It is denoted by the symbol “%”.
While solving ‘x% of y’:
% means = divide by hundreds
of means = multiplication ‘$ \times $’
Increased or more than x% indicates the addition of x% of y to y; reduced or less than x% indicates the subtraction of x% of y from y.
Complete step-by-step answer:
Step 1: given that
The ratio of spirit and water in a mixture = 1:8
$ \Rightarrow $$\dfrac{{{\text{spirit}}}}{{{\text{water}}}} = \dfrac{1}{8}$
Step 2: multiply numerator and denominator by ‘x’
Equivalent ratio $ \Rightarrow $$\dfrac{{{\text{spirit}}}}{{{\text{water}}}} = \dfrac{x}{{8x}}$
$ \Rightarrow $volume of spirit in mixture $ = x$ volumes
Volume of water in mixture $ = 8x$ volumes
Hence, total volume of mixture (or solution) $ = 8x + x$
$ = 9x$volumes
Step 3: find the increase in volume of solution.
It is given that 25% of the volume of solution is increased.
$ \Rightarrow 25\% \;{\text{of}}\;9x$is increased
$
= \dfrac{{25}}{{100}} \times 9x \\
= \dfrac{{9x}}{4} \\
$
Step 4: calculate total volume of spirit in solution
It is given that the increase in volume of solution is by adding spirit only.
$\dfrac{{9x}}{4}$volume is added to the solution
$ \Rightarrow \dfrac{{9x}}{4}$volume of spirit is added to the solution.
$\therefore $total volume of spirit in solution $ = x + \dfrac{{9x}}{4}$
$
= \dfrac{{4x + 9x}}{4} \\
= \dfrac{{13x}}{4} \\
$
Step 5: find the ratio of spirit to the water
$
\dfrac{{{\text{spirit}}}}{{{\text{water}}}} = \dfrac{{\dfrac{{13x}}{4}}}{{8x}} \\
{\text{ = }}\dfrac{{13x}}{{32x}} \\
$
$ = \dfrac{{13}}{{32}}$
The resultant ratio of spirit and water is 13:32.
Note: The two quantities can be compared only if they are in the same unit. If they are not, they should be expressed in the same unit before the ratio is calculated.
The ratio 3:2 is different from 2:3. Thus, the order in which quantities are taken to express their ratio is important.
For instance, the ratio of water to the spirit is 32:13.
For example: There are 24 girls and 16 boys in a class. Ratio of the numbers of girls to the numbers of boys $ = \dfrac{{24}}{{16}} = \dfrac{3}{2}$= 3:2
Therefore, we can get equivalent ratios by multiplying or dividing the numerator and denominator by the same number.
A ratio can be written as a fraction, thus the ratio 3:17 can be written as $\dfrac{3}{{17}}$.
Also, a percentage is a number or ratio that represents a fraction of 100. It is denoted by the symbol “%”.
While solving ‘x% of y’:
% means = divide by hundreds
of means = multiplication ‘$ \times $’
Increased or more than x% indicates the addition of x% of y to y; reduced or less than x% indicates the subtraction of x% of y from y.
Complete step-by-step answer:
Step 1: given that
The ratio of spirit and water in a mixture = 1:8
$ \Rightarrow $$\dfrac{{{\text{spirit}}}}{{{\text{water}}}} = \dfrac{1}{8}$
Step 2: multiply numerator and denominator by ‘x’
Equivalent ratio $ \Rightarrow $$\dfrac{{{\text{spirit}}}}{{{\text{water}}}} = \dfrac{x}{{8x}}$
$ \Rightarrow $volume of spirit in mixture $ = x$ volumes
Volume of water in mixture $ = 8x$ volumes
Hence, total volume of mixture (or solution) $ = 8x + x$
$ = 9x$volumes
Step 3: find the increase in volume of solution.
It is given that 25% of the volume of solution is increased.
$ \Rightarrow 25\% \;{\text{of}}\;9x$is increased
$
= \dfrac{{25}}{{100}} \times 9x \\
= \dfrac{{9x}}{4} \\
$
Step 4: calculate total volume of spirit in solution
It is given that the increase in volume of solution is by adding spirit only.
$\dfrac{{9x}}{4}$volume is added to the solution
$ \Rightarrow \dfrac{{9x}}{4}$volume of spirit is added to the solution.
$\therefore $total volume of spirit in solution $ = x + \dfrac{{9x}}{4}$
$
= \dfrac{{4x + 9x}}{4} \\
= \dfrac{{13x}}{4} \\
$
Step 5: find the ratio of spirit to the water
$
\dfrac{{{\text{spirit}}}}{{{\text{water}}}} = \dfrac{{\dfrac{{13x}}{4}}}{{8x}} \\
{\text{ = }}\dfrac{{13x}}{{32x}} \\
$
$ = \dfrac{{13}}{{32}}$
The resultant ratio of spirit and water is 13:32.
Note: The two quantities can be compared only if they are in the same unit. If they are not, they should be expressed in the same unit before the ratio is calculated.
The ratio 3:2 is different from 2:3. Thus, the order in which quantities are taken to express their ratio is important.
For instance, the ratio of water to the spirit is 32:13.
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