A solution of 9% acid is to be diluted by adding 3% acid solution to it. The resulting mixture is to be more than 5% but less than 7% acid. If there is 460 L of the 9% solution, how many litres of 3% solution will have to be added?

Or

Solve the following system of linear inequalities graphically.

$3 x + 2 y \geq 24, 3 x + y \leq 15, x \geq 4$

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Solution

Let x L of 3% solution be added to 460 L of 9% solution of acid.

Then, total quantity of mixture = (460 + x)L

Total acid content in the (460 + x) L of mixtures

$= (460 \times \frac{9}{100} + x \times \frac{3}{100})$

$= \frac{4140 + 3 x}{100}$

It is given that, acid content in the resulting mixture must be more than 5% but less than 7%.

$∴$ 5% of (460 + x) < $\frac{4140 + 3 x}{100}$ < 7% of (460 + x)

$\Rightarrow (460 + x) \times \frac{5}{100} < \frac{4140 + 3 x}{100} < (460 + x) \times \frac{7}{100}$

$\Rightarrow$ 2300 + 5x < 4140 + 3x < 3220 + 7x

$\Rightarrow$ 2300+ 5x < 4140 + 3x

and 4140 + 3x < 3220 + 7x

$\Rightarrow$ 2x < 4140 - 2300

and 4140 - 3220 < 4x

$\Rightarrow$ 2x < 1840 and 920 < 4x

$\Rightarrow$ x < 920 and x >230

$\Rightarrow$ 230 < x < 920

Hence, the number of litres of 30% solution of acid must be more than 230 L and less than 920 L.

Or

Converting the given inequalities into equations, we have and

3x + 2y = 24, 3x + y =15 x = 4

and x = 4

Region represented by 3x + 2y $\geq$ 24

The line 3x + 2y = 24 meets the coordinates axes at A(8, 0) and B(0, 12) respectively. Join these points by a line. Clearly (0, 0) does not satisfy the inequality 3x + 2y = 24. So, half plane of line 3x + 2y = 24 does not contain origin.

Region represented by 3x + y $\leq$ < 15

The line 3x + y =15 meets the coordinates axes at C(5, 0) and D(0, 15) respectively. Join these points by a line. Clearly (0, 0) satisfies the inequalities 3x + y $\leq$ . So, half plane of line 3x + y = 15 contains origin.

Region represented by x $\geq$ 4.

The line x = 4 is parallel to Y-axis. Clearly, (0, 0) does not satisfies the inequality x $\geq$ 4. So, half plane of line x $\geq$ 4 does not contain origin.

It is clear from the graph that there is no common region corresponding to these inequalities. Hence, the given system of inequalities have no solution.

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