Percentages, picking up where we left off.
I’ll have them work on the exercises from last class while I return the homework. By then all will have assembled. I will try two artificial examples, one for reasoning backwards and one for percentage decrease, each done with the 1+ trick. Then they can return to the worksheet.
Points to make about the homework (I’ve written these out many times on individual papers but it’s worth saying them collectively too):
- Use common sense. It can’t cost $50 million to manufacture one penny.
- Round answers, but not too soon.
- 2 cents or $0.02 but not 0.02 cents
- 0.002 or 0.2 percent, but not 0.002 percent.
OK I did that. I think it worked. Turned the class loose to finish the worksheet. Most said they finished the second problem, so moved on the the third one. Here it is, to save you visiting the link:
On January 14, 2013 The Boston Globe reported that
Boosting the income tax from the current rate of 5.25 percent to 5.66percent 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 discussion at the end of class was rushed and very interesting. First there was a problem when a large fraction of the students didn’t know what “total taxable income” meant. So we went through that – a good reminder for them that you need to understand the words before you start playing with the numbers.
Then we encountered what I now realize is a genuine ambiguity. Does the $1 billion represent the total amount raised or the additional amount raised by the additional tax? I think the second reading is the one the author had in mind: … boosting … would raise …. But the first reading is possible. We started to solve the problem using the second assumption. Turn out that just a little bit of informal algebra helps.
0.0525 * (total taxable income) = (tax collected now)
and
0.0566 * (total taxable income) = (tax collected at new rate).
If you subtract the second equation from the first you get $1 billion on the right hand side so
(0.566 – 0.525) *(total taxable income) = $ 1 billion .
Then
total taxable income = ($1 billion)/0.0041 ~ $244 billion.
That answers one of the two questions asked. To find the tax raised at the current rate is easy now:
tax collected = 0.0525 * $ 244 billion = $12.8 billion.
The web site
www.mass.gov/bb/gaa/fy2013/index.html shows a 2013 budget of about $32 million. The site
www.mass.gov/informedma/revenue/says that about 23% percent of the budget comes from income tax. So the $13 billion dollar answer is in the right ballpark. I also know now that you can’t correctly read the question the other way. The amount raised by the income tax is way more than $1 billion.
I will point the students to this blog as a continuation of the lecture. Maybe some of them will read it. Maybe some will even comment.
blog home page