Percentages. Scary.
Started with Red Sox ticket prices (Section 3.2 in the book), to introduce absolute change (subtract – natural and easy) and relative change (divide – unnatural and difficult, but often more informative). Found
$375/$275 = 1.18
and interpreted that as an 18% increase – so getting to the “1+” idiom right away. I was tempted to point out that this ratio is dimensionless (the $ signs cancel) but resisted temptation.
Then some problems on the next homework. First
46,795 runners finished the 2011 New York marathon. The 1981 marathon had 13,203 finishers. What was the percentage increase during this time period?
This led to a good discussion of “which number goes on top?” and the answer “new/old” rather than “smaller/bigger” or the reverse. In this case you get about 3.5, or 350%, which is a 250% increase.
We finished be starting a really hard problem:
On January 14, 2013 The Boston Globe reported that boosting the income tax from the current rate of 5.25 percent to 5.66 percent would raise $1 billion annually.
Use the information in this sentence to calculate total taxable income and the tax revenue collected at the current rate of 5.25 percent.
The lesson to learn here is to start with the story behind the words, so that you know what the words mean, before trying to play with the numbers. In this case
current revenue = 5.25% of (taxable income)
projected revenue = 5.66% of (taxable income)
That nails down two of the numbers given in the problem. We struggled to see where the “$1 billion” belonged. I did a little tooth pulling and we arrived at
$1 billion = (projected revenue) – (current revenue)
There is in fact a little ambiguity in the statement. It’s possible (but not likely) that the billion is the projected revenue rather than the extra revenue. The best thing to do when you see ambiguity is to spell it out and then make what seems to be the most logical choice.
Then just a little algebra tells us
$1 billion = (0.0566 – 0.0525) * (taxable income)
leading to
taxable income = ($1 billion)/(0.0041) ~ $250 billion
End of class. Let’s see where folks get to for Thursday.
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