Chapter 11.2, Problem 88AYU

To determine

To find: The possible combinations of these powders that could be mixed in each pill.

Answer to Problem 88AYU

There are 12 possible combinations of these powders that could be mixed in each pill.

Explanation of Solution

Given:

Pharmacy A doctor’s prescription calls for the creation of pills that contain 12 units of vitamin $B_{12}$ and 12 units of vitamin E. Your pharmacy stocks three powders that can be used to make these pills: one contains $20\%$ vitamin $B_{12}$ and $30\%$ vitamin E; a second, $40\%$ vitamin $B_{12}$ and $20\%$ vitamin E; and a third, $30\%$ vitamin $B_{12}$ and $40\%$ vitamin E.

Calculation:

Let $x=$ amount of the 1st powder required ($20\%$ $B_{12}$, $30\%$ E)

Let $y=$ amount of the 2nd powder required ($40\%$ $B_{12}$, $20\%$ E)

Write an equation for each vitamin:

$.20x+.40y=12$; (Vit $B_{12}$)

$.30x+.20y=12$; (Vit E)

Multiply the 2nd equation by 2, subtract the 1st equation

Find $y$ using eq:

$.2x+.4y=12$

$.2\left(30\right)+.4y=12$

$6+.4y=12$

$.4y=12-6$

$.4y=6$

$y=\frac{6}{.4}$

$y=15$ units of the 2nd powder

Check solution in the 2nd equation

$.3x+.2y=12$

$.3\left(30\right)+.2\left(15\right)=12$

$9+3=12$