At the Barber's

Let the required amount of 30% solution be x and let the required amount of 3% solution be y.

so, 30% of x + 3% of y = 12% of (x+y)

$\Rightarrow \quad \frac { 30x }{ 100 } +\frac { 3y }{ 100 } =\frac { 12(x+y) }{ 100 }$

$\Rightarrow 10x+y=4(x+y)$

$\Rightarrow 2x=y\quad \Rightarrow \frac { x }{ y } =\frac { 1 }{ 2 }$

so, the required proportion is 1:2

Let the required amount of 30% solution be x and let the required amount of 3% solution be y. so, 30% of x + 3% of y = 12% of (x+y) so, the required proportion is 1:2

It can be solved arithmetically but algebra does the job faster and more simply. Suppose to make up a 12-percent mixture we need x grams of a 3-percent solution and y grams of a 30-percent solution. Then in the first position we have 0.03x grams of pure hydrogen peroxide,and in the second 0,3y grams or,altogether 0.03x+0.3y.

As a result we have (x+y) grams of the solution in which there must be 0.12(x+y) grams of pure hydrogen peroxide.

We get the equation,

0.03x + 0.3y= 0.12(x+y) From this equation we find x=2y ,which means we have to take twice as much of a 3-percent solution as of a 30-percent solution.

So the required proportion is 1:2

Supposed the ratio of 30% solution is $x$ and the 3% solution is $y$

$\frac{30x+3y}{x+y}=12$

$30x+3y=12x+12y$

$18x=9y$

$\frac{x}{y}=\frac{9}{18}$

$\frac{x}{y}=\frac{1}{2}$

So, $1+2=\boxed{3}$

(0.3)(1)+(0.03)(x) = (0.12)(1+x) Solve for x to get x = 2. a+b = 1+2 = 3, Ans

Based on 1 unit volume of 12% $\ce{H2O2}$ solution:

Let: $a = 1 - x$ $and$ $b = x$

$30(1 - x) + 3x = 12$

$30 - 30x + 3x = 12$

$18 = 27x => x = 2/3$

$(1 - x ) : x = (1-2/3) : 2/3$

$a : b = 1 : 2$ $\Rightarrow$ $a + b = 1 + 2 = 3$