# Understanding "what fraction of" in a math problem

I am faced with a very problematic wording of a math problem. I can infer what the author meant from the solution, though, I was wandering for the correct linguistic interpretation of the sentence.

Damian needs 1/4 of the wood in the shed on the first day and 2/9 of the remaining wood on the second day to make a tree-house in his backyard. Given that 2/5 of the wood in the shed is unusable due to fungal damage, what fraction of usable wood is left once he has completed the tree-house?

Is that the fraction, (left usable wood)/(total usable wood) or (left usable wood)/(total wood)? does the fact that it is the fraction of 'something' implies that the whole must come from that something?

• Working through a math problem on EL&U is like flipping on ESPN and watching poker. Aug 6, 2019 at 11:13
• Does the author suggest the answer they expect? Aug 6, 2019 at 13:15
• @EdwinAshworth yes. When looking at the authors' solution, it is obvious that they did mean (left usable wood)/(total usable wood). Aug 6, 2019 at 13:30
• That's what the English would strongly suggest (arguably demand), but I've come across situations where default interpretations don't lead to the required answer. Aug 6, 2019 at 13:38

I believe that since the question only asks for the amount of usable wood left, it wouldn't make sense to calculate "What fraction of the total wood" since 2/5 of the total wood cannot be used.

Does the fact that it is the fraction of 'something' implies that the whole must come from that something?

Another example -

A four-piece pizza has macaroni on two of its pieces. Calculate the fraction of the pizza with macaroni.

So the fraction of pizza with macaroni must come from that whole piece of pizza.

• Given how badly the question is phrased, is the assumption that the calculation must make sense logical? Aug 6, 2019 at 11:15
• @EdwinAshworth - Badly phrased? It's completely clear to me what the maths question is asking. Aug 6, 2019 at 11:19
• Are the four pieces of pizza the same size? (Sorry, couldn't resist maths and pizza.) Aug 6, 2019 at 12:01
• @Andy T I assume your answer would be 11/36, as that is what the default interpretation suggests, but after being in maths moderation meetings (I taught maths for over a decade), I'd guess that 11/35 ('the fraction of the wood one finds in the shed in the final situation that is usable') is what the author expects. Aug 6, 2019 at 13:09
• @EdwinAshworth - Yes, 11/36. Where on earth have you got 11/35 from? I'm trying to work out what other calculation could be possible from the given wording, and I have no clue what you've interpreted. And I'm normally quite good at finding ambiguity. Aug 6, 2019 at 13:22

"what fraction of usable wood" means "how much of the usable wood, expressed as a fraction".

It can only mean (left usable wood)/(total usable wood).