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The Solutions.
Here are the three questions of interest (try to answer each rather quickly)

A bat and a ball cost \$1.10 in total. The bat costs \$1.00 more than the ball.
How much does the ball cost?

If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

In a lake, there is a patch of lily pads. Every day, the patch doubles in size.
If it takes 48 days for the patch to cover the entire lake, how long would it
take for the patch to cover half of the lake?

For question #1, one might intuitively say, “\$1.00 plus \$0.10 is \$1.10,
therefore a ball costs \$0.10“. While this is quite intuitive, it is also
incorrect. A person who is more thorough might respond that the ball actually costs \$0.05 (.05+(1+.05)=1.1). The correct answer is the ball costs \$0.05.

For question #2, an intuitive response would be that “5=5=5 so 100=100=100“.
However, if it takes a single machine 5 minutes to make a single widget, even a million machines
can make a million widgets in 5 minutes. The correct answer is therefore 5 minutes.

For question #3, the common intuitive response would be, “half the pond
would be covered in half the time, so 48/2=24 days“. However, this ignores the exponential growth
of the lily pads. Try working it backwards, if the pond is covered after 48 days,
and they double in size every day, then day 47 would be half covered. The correct answer is 47 days.
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