Word problem that I need help explaining! Can someone break this down for me?

Question

There are $32$ students in Mrs. Vicky’s class. Some of the students took one pencil each from the box with pencils on the teachers desk. Then a third of the remaining students took $3$ pencils each, and there were no more pencils left in the box. How many pencils were there in the box at first?

I’m sure there is a super simple answer for this and I’m embarrassed that I can’t explain it myself! Please help!

Answer 1

I’ll give you a road map for a straightforward algebraic solution, though the problem can be solved without algebra using the hint given by Andreas Blass in the comments. Let $s$ be the number of students who took one pencil each, and let $p$ be the total number of pencils.

• In terms of $s$, how many students took $3$ pencils each?
• Again in terms of $s$, how many pencils did those students take altogether?

Now write down an expression in terms of $s$ for the total number of pencils taken by all of the students who took pencils. Simplify that expression. How is it related to $p$?