Canola Oil problem

Question

A retailer purchased 38 gallons of canola oil and wants to put the oil in smaller cans (all of the same size) for sale. He knows his customers will NOT be interested in buying less than 3/5 of a gallon or more than 4/5 of a gallon of oil at a time. He doesn’t want to put the oil in 3/5 – gallon cans or 4/5 – gallon cans because this would not allow him to fill a whole number of cans to full capacity, and would leave him with some oil he would not be able to sell. Advise the retailer on the capacity of cans all of which he would be able to fill to full capacity, so that no oil is left.

Answer 1

Basic approach. If the retailer has cans of capacity $38/k$, where $k$ is some positive integer, then he can fill exactly $k$ cans to capacity with his $38$ gallons of canola oil. So he needs to find $k$ such that

$$ \frac35 \leq \frac{38}{k} \leq \frac45 $$

There will be some range of $k$ that satisfies this double-ended inequality. Find it. (Hint: What happens to an inequality when you take the reciprocal of all values, if they happen to be all positive?)