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A man has a stack of blank DVDS. on day one he uses one third of the DVD's. on day 2 he uses half the DVD's that were left from day one. At the end of day 2 there are 12 DVD's left. how many DVD's were in a box at the start of day one?
is the correct answer 72 dvds?
if x = no. of blank dvds at start
no. on day 1 = 1/3 x
no. on day 2 = 1/6x = 12
Outline:
Start: $x$ DVDs.
After day 1: $\frac23 x$ remain.
After day 2: $\frac12$ of $\frac23$ of $x$ remain. That is $\frac13 x$ remain.
Not quite. On day $2$, he used half of the DVD that are left, of which there are $\frac23 x$ after the first day.
$$\frac12 \cdot \frac23 x = 12$$
I'm afraid you've misinterpreted the second sentence. Rather, if $x$ is the number of blank DVDs at the start, then $\frac23x$ is the number remaining at the end of day $1.$ Can you take it from there?
Just as a general tip, once you've found an answer, see if it actually works in the context of the problem. In this case, let's try your answer of $72.$ Suppose he starts with $72$ DVDs. On the first day, he uses $\frac13$ of them. Since $\frac13$ of $72$ is $24,$ that means he uses $24$ of them, so he has $48$ left for the second day. Since he uses half of what's left on the second day, he uses $24$ the second day, as well, leaving $24,$ not $12.$ Thus $72$ is incorrect.
