I’m a mathematics teacher, and I have some arithmetic problems for you to solve.

1. Mitt drives to the beach and back.

Q. Mitt Romney drives 50 miles from his home one of his homes to the beach and back one summer, with or without a dog on the roof. He averages 50 miles an hour on the way out, and he averages 10 miles an hour on the way back. What was Mitt’s average speed on the road?

A. It was 16-2/3 mph¹, but Mitt’s campaign insists he’s no slowpoke and that he averaged 30 mph. (They also say the trip explains his tan.)

2. “Going Dutch” with Mitt.

Q. You and Mitt Romney have lunch together at Chick-fil-A. The bill is \$15, and he only has a \$10 bill, so he pays 2/3 of the bill. Later, you have dinner together at the country club, and the bill is \$300. He puts in \$100, and this time you pay 2/3 of the bill. Are you even-Steven?

A. Of course not, but Mitt says you are, because he paid 1/3 once and 2/3 once.

3. Mitt Sees Red.

Q. In the four squares below, about what percentage of the area overall is red?

A. About 10%. Certainly way less than half. Mitt disagrees. He says the percentage of red is over 75%, and he surprises you by providing actual details instead of saying he has a secret calculation that solves the problem and comes out the way he wants it to. He says there are three 100% red squares and one 5% red square, for an average of 305%/4 or about 76%. Ann tells you to stop giving him a hard time.

Earlier today, the Romney campaign website posted a letter from PriceWaterhouseCoopers “PWC” LLP summarizing the Romneys’ tax returns for the years 1990-2009. The carefully-worded letter² states that

The average of the annual “effective federal personal income tax rates” as computed based on the returns as prepared during the period is 20.20%.

PWC computed each year’s “effective federal personal income tax rate” as instructed by the Romneys.

As you requested, we compute each annual “effective federal personal income tax rate” as total taxes owed divided by adjusted gross income as shown on the federal income tax returns as prepared.

First of all, the Romneys’ “adjusted gross income” (AGI) may be less than what most of us think of as simply “what the Romneys made.” For example, the expenses of carrying on a trade or business (such as dressage) can be deducted from total income. The Romneys’ tax rate on their total income is lower than that on their adjusted gross income. Maybe a little, maybe a lot.

More importantly, however, is how Mitt and Ann instructed PWC to compute a number they could call an “average.”

Despite appearances, no one has said that the Romneys paid 20.20% of their adjusted gross income in federal income taxes during the period 1990 – 2009. They probably paid less.

The 20.20% figure is not the Romney’s federal income tax rate for income they earned during the period 1990 – 2009. It’s the average of 20 separate annual rates. Mathematically, it’s a weighted average of their annual rates.

If you think the average of 20 annual rates is the same as the 20-year average annual rate, you’re wrong. By that flawed logic, Mitt averaged 30 mph, you and he are even-Steven for splitting meals, and the colored squares in the picture above are 75% red. Go away.

Further blurring the issue, the campaign today also posted a letter from Brad Malt, the “trustee of the Romney’s [sic] blind trust,” about Mitt’s taxes. Malt tries to say that PWC calculated something that it didn’t calculate.

Regarding the PWC letter covering the Romneys’  tax filings over 20 years, from 1990 – 2009:

Over the entire 20-year period, the average annual effective federal tax rate was 20.20%.

Malt omitted two short and crucial words: of the. PWC did not say that Romneys’ average annual effective federal tax rate was 20.20%. They said that if you average the 20 separate effective rates on their 20 federal tax returns, you get 20.20%. It’s a silly thing to calculate, but it’s what the Romneys told them (and presumably paid them) to do.

The Romneys’ overall tax rate might have been lower than 20.20%. It might have been higher, too, but given the outrage over the relatively low rates they pay compared to working Americans, if it were higher, he’d have said so. And Mitt won’t release his tax returns. We can only assume the worst.

If you value the truth, vote for Barack Obama on November 6, 2012.

¹ Let’s figure it out. Mitt drove a total of 100 miles. It took him 1 hour (50 miles at 50 mph) to drive the first half of the trip and 5 hours (50 miles at 10 mph) to drive the second half. He drove a total of 100 miles in 6 hours, so his average speed was 16-2/3 miles per hour.

² The letter is signed personally, in blue, by PriceWaterhouseCoopers LLP, because why wouldn’t they sign it? Corporations are people.