# A solution of $8 \%$ basic acid is to be diluted by adding $2 \%$ boic acid. Solution to it. The resulting mixture is to be more than $4 \%$ but less than $6 \%$ boric acid. If we have 640 litres of $8^{\circ} \%$ Solution , how many litres of $2 \%$ solution is to be added ?

Toolbox:
• Same Quantity can be added (a subtracted ) to (from ) both sides of the inequality with out changing the sign of the in equality.
• Same positive quantities can be multiplied or divided to both side of the in equality with out changing the sign of the inequality.
• If same negative quantity is multiplied or divided to both sides of the inequality is reversed i.e $'>'$ sign changes to $'<'$ and $'<'$ changes $'>'$ .
Step 1:
Let x litres of $2 \%$ boric acid required to be added to 640 litres of $8 \%$ solution.
Then total mixture $=(x+640)$ litres
Percentage of boric acid in the mixture is $2 \% \;of \;x +8 \% of \;640 > 4 \% \;of \; (x+640)$
$=> \large\frac{2}{100} $$x +\large\frac{8}{100}$$ \times 640 > \large\frac{4}{100} $$(x+640) Multiplying birth sides by 100 => 2x +5120 > 4x+2560 Subtracting 2560 from both sides => 2x +5120 -2560 > 4x +2560 -2560 => 2x+2560> 4x Subtracting 2x from both sides 2560 > 2x Dividing by 2 on both sides 1280 > x -------(1) The percentage of boric acid must be less than 6 \% Hence 2 \% \;of \;x +8 \% of \;640 < 6 \% \;of \; (x+640) => \large\frac{2}{100}$$x +\large\frac{8}{100} $$\times 640 < \large\frac{6}{100}$$(x+640)$
Multiplying both sides by 100
$=> 2x +5120 < 6x+3840$
Subtracting 2x from both sides inequality and subtracting 2840 from both sides of inequality
$5120 -3840 < 6x -2x$
$=> 1280 < 4x$
$=> 320 < x$ ------(2)
From (1) and (2) the we conculde that $320 < x < 1280$
Thus the number of litres of $2 \%$ bonic acid solution to be added in more than 320 and less than 1280