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Home  >>  CBSE XI  >>  Math  >>  Linear Inequalities
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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 ?

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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
answered Aug 5, 2014 by meena.p
 

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