Chapter 11, Problem 103RE

To determine

How many of each size box should you buy?

Answer to Problem 103RE

  $\boxed{1,5,2}$

Explanation of Solution

Given information:

A cookie company makes three kinds of cookies, oatmeal raisin, chocolate chip, and shortbread, packaged in small, medium and large boxes. The small box contains $1$ dozen oatmeal raisin and $1$ dozen chocolate chips; the medium box has $2$ dozen oatmeal raisin, $1$ dozen chocolate chip, $1$ dizen shortbread; the large box contains $2$ dozen oatmeal raisin $2$ dozen chocolate chip, and $3$ dozen shortbread. If you require exactly $15$ dozen oatmeal raisin, $10$ dozen chocolate chips and $11$ dozen shortbread, how many of each size box should you buy?

Calculation:

Consider small box be $x$ , medium box $y$ and large box $z$ . The amount of oatmeal raisin , chocolate chips, short bread that each box contains is given.

The first equation representing the total number of raisin which is,

  $x+2y+2z=15$

Now total number of chocolate chips are,

  $x+y+2z=10$

Total number of shortbread will be,

  $y+3z=11$

Now we have system of three linear equations,

  $\{\begin{array}{l}x+2y+2z=15\\ x+y+2z=10\\ y+3z=11\end{array}$

Now subtract second equation from first to eliminate $x$ and $z$ .

  $\begin{array}{l}\underset{\_}{\begin{array}{l}x+2y+2z=15\\ x+y+2z=10\end{array}}\\ y=5\end{array}$

Now substitute $y=5$ in third equation,

  $\begin{array}{l}5+3z=11\\ 3z=11-5\\ 3z=6\\ z=2\end{array}$

Now find $x$ by substituting $y=5$ and $z=2$ in $x+y+2z=10$ ,

  $\begin{array}{l}x+5+2\left(2\right)=10\\ x=10-9\\ x=1\end{array}$

Hence, the number small,medium and large boxes to be brought are $\boxed{1,5,2}$ respectively.