Question

A manufacturer has 460 litres of a 9% acid solution. How many litres of a 3% acid solution must be added to it so that the acid content in the resulting mixture be more than 5% but less than 7%?

Answer 1

Let x litres of a 3% arid solution be added to 460 litres of 9% acid solution. Then,

total quantity of mixture = (460 +x) litres.

Total acid content in (460 + x)litres of mixture

$=\{(460\times \frac{9}{100})+(x\times \frac{3}{100})\}litres=(\frac{207}{5}+\frac{3x}{100})$ litres.

Now, the add content in the resulting mixture must be more than 5% and less than 7%.

$\therefore 5\%of(460+x)<(\frac{207}{5}+\frac{3x}{100})<7\%of(460+x)$

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

$\Rightarrow 5(460+x)<4140+3x<7(460+x)$

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

$\Rightarrow 2300+5x<4140+3xand4140+3x<3220+7x$

$\Rightarrow 5x-3x<4140-2300and4140-3220<7x-3x$

$\Rightarrow 2x<1840and920<4x$

$\Rightarrow x<920and230<x$

$\Rightarrow 230<x<920$.

Hence, the required quantity of 3% acid solution to be added must be more than 230 litres and less than 920 litres.