The following is an exercise from MA$\Theta \; 1992$.
Mike and joey bought identical loaves of bread and packages of
bologna. Mike made sandwiches with 5 slices of bologna and had 4
slices of bread left when he ran out of meat. Joey made sandwiches
with 4 slices of bologna and had 4 slices of meat when he ran out of
bread. Each boy only started with one loaf, and each sandwich has two
slices of bread. How many slices of bread were in each loaf?
I am struggling with setting up the corresponding system of equations.
Let $Br$ and $Bo$ denote the total amount of bologna and bread respectively. The first person, Mike, made $\frac{Bo}{5}$ sandwiches because he uses five bologna slices per sandwich. The amount of bread used is therefore $\frac{Bo}{5} \cdot 2$. There are four slices of bread left at the end of his sandwich. So do we have $$Br – 4 = \frac{Bo}{5}\cdot 2$$
or $$4 = \frac{Bo}{5} \cdot 2$$ ?
The RHS of the equation says the total amount of bread used by mike. So the LHS should say how much bread is used up also. The exercise says there are $4$ slices left. Therefore, the amount of bread used up by mike is also written as $Br – 4$, right? If we say the LHS is $4$ instead of $Br – 4$ that is instead saying how much is left over which is not what the RHS is describing.
Now for Joey. Since Joey uses two slices of bread per sandwich, he makes $\frac{Bo}{2}$ total sandwiches. Also, since he uses four slices of bologna per sandwich then he uses a total of $\frac{Bo}{2} \cdot 4$ slices of bologna. The total amount of bologna he uses can also be written as $Bo -4$ as we are told he has four slices of bologna left over after assembly. Hence, we have the following equation
$$Bo – 4 = \frac{Br}{2} \cdot 4$$
The system of equations is therefore $$\begin{cases}Br – 4 = \frac{Bo}{5}\cdot 2 \\ Bo – 4 = \frac{Br}{2} \cdot 4\end{cases}$$
Solving this system I obtain: $Bo = 56$ and $Br = 28$
Best Answer
Let there be $x$ slices of bologna and $y$ slices of bread in a package.
Joey made $\frac y2$ sandwiches, used $2y$ slices of bologna and had $4$ left, so $x=2y+4$.
Mike made $\frac y2-2$ sandwiches and used $5\left(\frac y2-2\right)$ slices of bologna. Equating these $$5\left(\frac y2-2\right)=2y+4\\ \frac y2=14\\y=28\\x=60$$ So there are $28$ slices of bread and $60$ slices of bologna in a package. Mike made $12$ sandwiches and Joey made $14$.