Question

A scientist has two acid solutions: Artesia, which is a $70\%$ acid solution, and Borromeo, which is a $20\%$ acid solution. (a) Find the amount of Artesia (in mL) that must be added to $500$ mL of Borromeo in order to produce a $60\%$ acid solution. (b) Find the amount of Artesia and Borromeo (in mL) that can be combined in order to form a $100$ mL solution that is $55\%$ acid. (c) Does there exist a combination of Artesia and Borromeo that is $80\%$ acid?

Answer 1

(a) 2000 mL

(b) 70 mL, 30 mL

(c) No

Explanation:

Given that Artesia is a 70% acid solution, and Borromeo is a 20% acid solution.

(a) Let x mL of Artesia is added to 500 mL of Borromeo to produce a 60% acid solution.

So, 70% of x + 20% of 500 = 60% of (x+500)

`\Rightarrow 0.7 * x + 0.2 * 500 = 0.6 * (x+500) \\\\\ \Rightarrow 0.7 * x + 0.2 * 500 = 0.6 * x+0.6 *500\\\\ \Rightarrow 0.7 * x -0.6 * x= 0.6 *500- 0.2 * 500 \\\\ \Rightarrow (0.7-0.6) * x = (0.6-0.2) *500\\\\\Rightarrow 0.1 * x = 0.4 *500\\\\`

`\Rightarrow x = (0.4 *500)/0.1=2000` mL.

(b) Let x mL of Artesia and y mL be combined in order to form a 100 mL solution having 55% acid solution.

So, x+y=100\Rightarrow y= 100-x

And, 70% of x + 20% of y = 55% of (100)\\\\

`\Rightarrow 0.7 * x + 0.2 * (100-x) = 0.55 * 100\\\\\Rightarrow 0.7 * x + 0.2 * 100- 0.2 * x = 0.55 * 100\\\\\Rightarrow 0.7 * x -0.2 * x= 0.55 *100- 0.2 * 100 \\\\\Rightarrow (0.7-0.2) * x = (0.55-0.2) *100\\\\\Rightarrow 0.5 * x = 0.35 *100\\\\`

`\Rightarrow x = (0.35 *100)/0.5=70` mL.

and y= 100-x= 100-70=30 mL.

(c) Let x mL of Artesia and y mL of Borromeo be combined in order to make 80% acid solution.

And, 70% of x + 20% of y = 80% of (x+y)

`\Rightarrow 0.7 * x + 0.2 * y= 0.8 * (x+y)\\\\\Rightarrow 0.7 * x + 0.2 * y = 0.8 * x +0.8 * y\\\\\Rightarrow (0.7-0.8) * x = (0.8-0.2) * y\\\\\Rightarrow -0.1 * x = 0.6 * y \\\\\Rightarrow \frac {x}{y} = -6 \\\\`

As x and y are amount of Artesia and Borromeo, so x>0 and y>0

`\Rightarrow x/y >0`, so the negative value of x/y is not possible.

Ad the negative value of x/y is not so x/y=-6 is not possible, hence, any combination of Artesia and Borromeo that is 80% acid solution is not possible.