Quote:
A total of s oranges are to be packaged in boxes that will hold r oranges each, with no oranges left over. When n of these boxes have been completely filled, what is the number of boxes that remain to be filled?
A) s-nr
B) s–(\(\frac{n}{r}\))
C) rs–n
D) (\(\frac{s}{n}\))–r
E) (\(\frac{s}{r}\))–n
A student of mine brought the above question to my attention. One approach I do not see outlined above, which can be useful on word problems that involve real-world measurements, is to
translate the algebraic expressions into language to ensure that the answer you select will be sensible. Consider the first option, for example:
(A) s - nr
If we substitute the word or phrase for its algebraic stand-in, we get the following:
(oranges) - [(boxes of oranges)(oranges/box)]
Ask yourself what sort of unit that bracketed multiplication would produce. The boxes would cancel out and you would be left with oranges. It should be clear that we cannot answer a question that asks us to count boxes by subtracting oranges from oranges.
(A) cannot be the answer. Now, consider the other options in a similar fashion:
(B) s - (n/r)
(oranges) - [(boxes of oranges)/(oranges/box)]
Even without doing the math, it is obvious that we cannot subtract
anything from oranges to get boxes, since the units would not match.
Get rid of (B).(C) rs - n
[(oranges/box)(oranges)] - (boxes of oranges)
What sort of a unit is oranges squared? Again, it should be clear that you cannot derive boxes from this made-up unit.
Nix (C).(D) (s/n) - r
[(oranges)/(boxes of oranges)] - (oranges/box)
We cannot get boxes by subtracting a "rate" of oranges per box.
Get rid of (D).(E) (s/r) - n
[(oranges)/(oranges/box)] - (boxes of oranges)
What do you do when you divide by a fraction? You invert the divisor and multiply, of course, and here, we will finally derive a sensible unit that will help to answer the question:
[(oranges) * (box/oranges)] - (boxes of oranges)
[(box)] - (boxes of oranges)
And it is perfectly reasonable to count boxes by subtracting a set of boxes from other boxes.
(E) must be the answer. Nothing more than a basic understanding of units and mathematics is necessary, and the question can be answered quickly and confidently.
See if you can apply the same logic to other word problems that use measurements and list algebraic answers. You may surprise yourself how handy the process of elimination outlined above may prove.
Good luck with your studies.
- Andrew