I don't understand this system of inequalities word problem

Question

I don't get how to come up with the 2nd inequality in the answer explanation (not the $b+f$ inequality). Help is greatly appreciated.

Answer 1

To make one cake it is necessary to have $4$ cups of batter and $\frac{7}{2}$ cups of frosting. For two cakes: $8$ and $7$ cups, for three cakes: $12$ and $\frac{21}{2}$ cups, for $n$ cakes: $4n$ and $\frac{7}{2}n$ cups, respectively. So if there are $b=4n$ batters, there must be $f=\frac{7}{2}n$ frosts. Having more frosts than required is also possible, which implies leftover frosts.

Now if there are $b$ batters and $f$ frosts, then it can be made $n=\frac{b}{4}$ cakes provided there are enough frosts, that is $\frac{f}{\frac72}\ge n$. Now substituting first into second you get: $$\frac{f}{\frac72}\ge \frac{b}{4}.$$

Answer 2

As the image explains, for every $4$ cups of batter, there must be at least $\frac{7}{2}$ cups of frosting. Now, how can we express that? Well, we divide all the batter into groups of $4$ cups and for each one of those groups, the amount of batter has to be smaller (or at least equal, so we can make a cake) than the amount of frosting. But the amount of batter in those groups is just $\frac{b}{4}$ and the amount of frosting is just $\cfrac{f}{\frac{7}{2}}$. With that in mind one arrives at the inequality mentioned.