Which Alice?

Solutions to the Puzzles
miles an hour is one mile in 15 minutes, or 1 1/2 miles in 22
1/2 minutes, so the man just caught the train.
HOW FAR AWAY IS THE SCHOOL? The difference between
being 5 minutes late and being 10 minutes early is a difference of 15
minutes. So the boy will save 15 minutes if he walks at the rate of 5
miles an hour rather than 4 miles an hour. Now, 5 miles an hour is
one mile in 12 minutes, and 4 miles an hour is one mile in 15
minutes, so walking at the faster rate, he saves 3 minutes in every
mile, which is 15 minutes in 5 miles. So the school is 5 miles away.
Let us check: If he walks at 5 miles an hour, he will take one hour. If
he walks at 4 miles an hour, he will take 1 1/4 hours (one hour for
the first 4 miles and a quarter of an hour for the last mile) — which is
one hour and 15 minutes. So there is indeed a 15-minute difference.
IS THIS PUZZLE SAD? Yes, it is a bit sad, in a way, because the
dealer miscalculated: He didn't break even; he lost $20 that day.
Let us see why: Consider first the painting he sold at a 10 percent
profit. He got $990 for the painting; how much did he pay for it?
Now the profit is not 10 percent of $990, but 10 percent of the
amount he paid. So $990 is 110 percent — or 11/10 — of what he
paid. This means he paid 10/11 of $990, which is $900. This
checks, because he paid $900, made 10 percent of $900, which is
$90, so he received $990. Therefore he made $90 on the first
painting.
Now let us consider the second painting: He lost 10 percent of
what he paid for it, so he sold it for 90 percent — which is 9/10 — of
what he paid. Therefore he paid 10/9 of $990, which is $1,100. Does
this check? Yes, because he paid $ 1,100, and 10 percent of $ 1,100 is
$110, so he sold it for $1,100 minus $110, which is $990.
So he lost $110 on the second painting, and gained only $90 on
the first one, so his net loss was $20.
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