Chapter 1, Problem 12CT

To determine

The number of each item sold in the given situation.

Answer to Problem 12CT

There are $75$ doughnuts, $30$ muffins and $25$ bagels.

Explanation of Solution

Given information: Foundational statements to formulate equations.

Calculation:

Let number of doughnuts, muffins and bagels be $x,yandz$ respectively.

As per the given situation the system of linear equations is,

  $\begin{array}{l}x=3z-(i)\\ x+1.5y+1.2z=150-(ii)\\ x+y+z=130-(iii)\end{array}$

Substituting $(i)$ in $(ii)$ ,

  $\begin{array}{l}3z+1.5y+1.2z=150\\ 4.2z+1.5y=150-(iv)\end{array}$

Substituting $(i)$ in $(iii)$ ,

  $\begin{array}{l}3z+y+z=130\\ 4z+y=130-(v)\end{array}$

Multiplying $(iv)$

  $10$ and $(v)$ by $10.5$ and subtracting,

  $\begin{array}{l}42z+15y=1500\\ 42z+10.5y=1365\\ ============\\ 4.5y=135\\ y=\frac{135}{4.5}\\ \\ y=30\end{array}$

Putting values of $y$ in $(v)$ ,

  $\begin{array}{l}4z+30=130\\ z=\frac{100}{4}\\ \\ =25\end{array}$

Putting values of z in $(i)$ ,

  $\begin{array}{l}x=3(25)\\ =75\end{array}$

Solution is $(75,30,25)$ ,

Thus, there are $75$ doughnuts, $30$ muffins and $25$ bagels.