Driving the Second Mile

I think this problem was on my SAT exam.  Like the “fly-fly” problem, it is easy, if you just think about it.  A man drives one mile at 30 miles per hour (mph).  How fast would he have to drive another mile, in order to average 60 mph for the whole two- mile stretch?

A.    60 mph

B.     90 mph

C.     120 mph

D.    None of the Above

If you chose B, 90 mph, you would be wrong.  I know; 90 plus 30 equals 120, which divided by 2 is 60.  But it doesn’t work that way.  Consider that 60 mph is a mile a minute.  Therefore, for the man to average 60 mph for two miles, he would have to drive that distance in two minutes.  But he drives the first mile at 30 mph, thus taking up two minutes.  Consequently, in order to average 60 mph for the whole two miles, he would have to drive the second mile in no time at all, which is impossible.  Therefore, the correct answer is D, “None of the Above”.

Now, suppose we make the problem a little bit harder by asking how fast the man would have to drive the second mile to average 45 mph for the two-mile stretch?  Ironically, the reason this problem is harder is because it has a viable solution.

We already know that the man used up two minutes driving the first mile at 30 mph.   First, let’s figure out how much time it would take to average 45 mph for the whole two miles.   Note that 45 miles per hour divided by 60 minutes per hour is 45/60 or 3/4 of a mile per minute.  Now, 2 miles divided by 3/4 of a mile per minute is the same as 2 miles times 4/3 of a mile per minute, which equals 8/3 of a minute, or 2 and 2/3 minutes.  [ 2÷3/4 = 2×4/3 = 8/3 ]

Since the man used up two minutes driving the first mile, he only has 2/3 of a minute left to drive the second mile, in order to average 45 mph for both miles. Well, 1 mile divided by 2/3 of a minute is 1 mile times 3/2 of a minute, which is 1 and 1/2 miles per minute.  [ 1÷2/3 = 1×3/2 = 3/2 ]  And 1.5 miles per minute is 90 mph.  So, the answer is 90 mph, in order to average 45 mph for the two-mile stretch.  

How Far Does the Fly Fly?

Here’s another one of those problems that is tricky, only in that it seems to be difficult, yet it is easy as pie if you think about it in a certain way.  You don’t have to be a whiz in algebra to work this problem, but you do have to think clearly.  Like the bayou problem, I don’t recall where I first saw it.  Try to work it before you read my solution.

Two trains are heading toward each other on the same track, bound for a head-on collision.  When the trains are exactly two miles apart, a fly on the headlight of one train, which we will designate as train A, starts flying toward the other train, which we will designate as train B.  When the fly reaches train B, he turns around and races back to train A.  When he gets back to train A, he turns around and heads back to the train B again.  Back and forth he goes, flying ever shorter distances on each pass as the trains get closer and closer, until the trains collide.  The trains are going 30 mph, and the fly is going 60 mph.  The question is, how far does the fly fly before he meets his maker?

A typical engineer would probably make this problem much more difficult than it is.  For example, he might work out the distance-equals-rate-times-time thing and come up with a series of smaller and smaller distances and then take the sum of the series – not an easy thing to do.  However, I am not a typical engineer, so I will take an altogether different, and far simpler, approach.

Since there are 60 minutes in an hour, 60 mph is a mile a minute, and 30 mph is a mile in two minutes.  Since the trains are two miles apart at the beginning of our consideration, and since they are traveling toward each other at identical speeds, it is apparent that they will collide at the midpoint between them.  Therefore, each train travels one mile before colliding, and at 30 mph, it takes them two minutes to travel that distance.  Now, the fly flies for the same amount of time as the trains do before they collide; that is to say that the fly also flies for two minutes before the collision occurs.  Since the fly flies for two minutes at a speed of one mile per minute, the fly flies two miles.  Eureka!

An Unforgettable Character

My father-in-law, who goes by his initials L. W., is one of the more unforgettable characters I have ever had the pleasure to meet. A World War II fighter pilot, he returned from the war and went into the lumber business. He eventually founded his own lumber company and became very successful. He is retired now.

He sometimes has some unusual, or uncommon, opinions on things, often expressed with a dry sort of humour. For example, he once stated, “The only sustainable form of government is anarchy.” Another time, while lamenting the amount of taxes he had to pay, he said, “You know, poor folks really have it made.”

He sometimes had strong opinions about other people’s priorities, too. For example, one time he told me disgustedly, “Heck, I know some people who are driving Cadillacs, yet they don’t even have good whiskey in the house.”

He is not a very patient man. One evening a long time ago, when I was courting Elaine, we all drove over to visit a friend of his. “We” being L. W., his wife and Elaine’s mom Betty, Elaine and her identical twin sister Jane, and me. L. W.’s friend’s place was a ranch of sorts, and the approach to it was a dirt road. It was on that road at night that one of L. W.’s tires blew out. Elaine and Jane and Betty elected to walk the remaining distance to the ranch house, while L.W. and I were supposed to change the tire.

L.W. opened the trunk and got out the jack, and we discovered that it was nothing like anything either of us had ever seen. It was dark, so I was holding the jack manual under the trunk-lid lights to read it, when I heard a distinctive whew, whew, whew, whew, … . It turned out to be the sound that a jack makes when it is flying through the air, after someone entirely out of patience flings it away. We had to walk to the ranch house and get the car towed the next day.  Cell phones weren’t around in those days.

The last car L. W. owned was a Mercedes, but when I first met him in 1965 he was driving Cadillacs, and he held all other cars, especially Ford’s, in disdain. I was reading a “Consumer Report” one night over at his house, and I came across an article on the repair frequency for different kinds of automobiles. After awhile, I announced to L. W. that he might like to read the article, because the data indicated that Fords needed repairs a lot less frequently than Caddies or Chevys or other GM automobiles. He scoffed and said, “That’s because anybody who would drive a Ford wouldn’t have sense enough to know when it needed repairing.”

One time L. W. parked somewhere during some big event, and when he went to get his car afterwards, he couldn’t find it. He decided it was stolen and reported it to the police. A day or so later, he happened to be riding in a taxi by the same parking lot where, now, there were no events being held and the lot was empty. That is to say, nearly empty, as he saw his car all by itself way out in the middle of the lot. He immediately asked the cab driver to stop, and he got out right there and went to his car and drove off, headed for his country club to play golf. On the way there, though, he was stopped by the police for driving a stolen car. By the time he got things cleared up and got to the club, the word had somehow already gotten there ahead of him, and his friends kidded him to no end.

One time one of his lumberyard truck drivers drove under a large freeway sign that was not high enough for the truck. The truck knocked the sign down, and L.W. was required to pay for it. So, he took the sign and had it modified at his lumber yard to make a big saw for his lumber mill. That was the kind of unexpected way he had of sometimes handling a problem.

Will the Bayou Rise or Fall?

This is a classic problem. Perhaps you have heard it before. I don't remember where I got it. You might want to try to work it before you read my solution. Anyway, Pierre and Budreaux were sitting in a boat on the bayou when Pierre suggested thowing the anchor overboard. Budreaux said that if Pierre did that the anchor would take up space at the bottom of the bayou, causing the level of the bayou to rise a bit. But Budreaux said, on the other hand, without the anchor the boat would float higher, so maybe the level would go down. Pierre replied that maybe these two opposing things would cancel each other out. What is the right answer? Will the level of the bayou rise or fall when the anchor is thrown overboard?

Archimedes said that a floating object will displace a volume of water equal to the weight of the object.

The density of water is 62.4 lb/cf
Assume that a boat and its contents including an anchor weigh 1,000 lb
Then the amount of water displaced is 1,000 lb divided by 62.4 lb/cf = 16.03 cf
Assume the anchor weighs 50 lb
Then the anchor accounts for 50 lb divided by 62.4 lb/cf = 0.80 cf
of the volume of water displaced by the boat and its contents including the anchor.
Assume the anchor is made of cast iron, having a specific gravity of 7.03
Then the density of the anchor is 7.03 * 62.4 lb/cf = 438.67 lb/cf
And the volume of the anchor is 50 lb divided by 438.67 lb/cf = 0.114 cf

A sunken object displaces a volume of water equal to the volume of the sunken object.

When the anchor is thrown overboard it sinks, & the volume of water it displaces is 0.114 cf
Without the anchor, the boat and its contents weigh 1,000 lb - 50.0 lb
= 950.0 lb
Now the amount of water displaced by the boat and its contents is 950.0 lb divided by 62.4 lb/cf
= 15.22 cf
Thus, the total volume of water displaced by the sunken anchor, plus the boat and its contents
absent the anchor, is 0.114 cf + 15.22 cf = 15.34 cf
Since the volume of water displaced by the boat and its contents including the anchor is
16.03 cf, when the anchor is thrown overboard the total amount of water displaced
decreases by 16.03 cf - 15.34 cf = 0.69 cf
causing the level of the lake to fall.
This actually should have been apparent ever since we noted that when the
anchor is in the boat it accounts for 0.80 cf of water displacement, whereas when the anchor is
sunken it displaces only 0.114 cf. Note that 0.80 cf - 0.114 cf
= 0.69 cf, which is the same answer we got above,thus checking my answer.

Presidential Rarities

Presidents Who Were Elected by the House of Representatives

The president of the United States is not elected by popular vote, but rather by a group of 538 people referred to collectively as the Electoral College.  In the event that no candidate wins a majority of electoral votes, the House of Representatives decides.  Thomas Jefferson and John Quincy Adams were the only two presidents ever elected by the House. 

When the House elected Jefferson in 1801, most states still allowed their legislators to choose electors, and the Electoral College still voted for president and vice president on the same ballot.  Jefferson and Aaron Burr tied in electoral votes, and the House could not decide between the two men until the 36th ballot.  The voting went on for so long that there was serious concern that a president would not be elected before inauguration day.  This prolonged election was what led to the 12th Constitutional amendment, which changed the voting rules of the Electoral College.

When the House elected John Quincy Adams in 1825, Andrew Jackson had won far and away more popular votes, as well as more electoral votes, than any other candidate, but he did not win a majority of either.  Consequently, the election defaulted to the House, which, as stipulated by the Constitution, had to choose a president from the top three electoral vote winners.

Henry Clay was fourth place in electoral votes won and, therefore, was eliminated from competition in the Electoral College vote.  Clay, however, was a staunch political rival of Jackson's and threw his support in the House to Adams, who, as a direct result of Clay's support, was elected.  Thus, history recorded the irony of the least successful presidential candidate having been ultimately responsible for determining who would be president.

Presidents Who Won Fewer Popular Votes Than Their Nearest Opponent Won

The quirkiness of the Electoral College system has resulted in the election of four presidents who lost the popular vote, yet won the election by a majority of the Electoral vote, or won in the House of Representatives in absence of winning a majority of the electoral vote. These four people were John Quincy Adams, Rutherford B. Hayes, Benjamin Harrison, and George W. Bush.

John Quincy Adams

In the election of 1824, Andrew Jackson received 43.12 % of the popular vote, but John Quincy Adams won the election with only 30.54 % of the popular vote, the lowest of any president in history.  (Guess who was elected with the second lowest percent of the popular vote.  Answer:  Abraham Lincoln in 1860, with 39.87% Of the popular vote).  Incidentally, to the best of my knowledge, this was the first year that any electors were chosen by popular vote.  Prior to that time, state legislatures chose them.

Rutherford B. Hayes

In the election of 1876, Rutherford B. Hayes won the election with 47.87% of the popular vote, whereas Samuel J. Tilden lost with 51.01%.  This was one of the two most disputed presidential elections in American history, the other being the election of 2000.

Tilden won 184 electoral votes to Hayes's 165, with 20 electoral votes uncounted. These 20 electoral votes were in dispute in Florida, Louisiana, and South Carolina, and each party reported its candidate had won the state.  There were double sets of returns from these three states.  Congress passed a law forming a 15-member Electoral Commission to settle the result. Five members came from each house of Congress, and they were joined by five members of the Supreme Court.

As it turned out, the resulting committee was composed of seven Republicans, seven Democrats, and one independent, Supreme Court Justice David Davis.  However, just as the Electoral Commission Bill was passing Congress, the Legislature of Illinois elected Davis to the Senate.  He promptly resigned as a Justice to take his Senate seat.  All the remaining available justices were Republicans, so the four justices already selected chose Justice Joseph P. Bradley, who joined the other seven Republicans to result in an 8-7 vote in favor of Hayes, giving all 19 disputed electoral votes to Hayes, resulting in his 185-184 electoral vote victory.

The returns accepted by the Commission placed Hayes's victory margin in South Carolina at 889 votes, making this the second-closest election in U.S. history, after the 2000 election.  Also, Tilden became the first presidential candidate in American history to lose in the electoral college, despite winning a majority of the popular vote.

Benjamin Harrison

In 1888, Benjamin Harrison won the presidency with 47.79% of the popular vote, whereas Grover Cleveland lost with 48.68 %.  Cleveland, incidentally, was the only president in history to win two non-consecutive turns.  Harrison won the term in between.

George W. Bush

In 2000, George W. Bush, a Republican, won the election with 47.87% of the popular vote, whereas Al Gore, a Democrat, lost with 48.68 %.  This was the second of the two most disputed presidential elections in American history, the first being that of 1876 (Hayes versus Tilden).

Bush narrowly won with 271 electoral votes to Gore's 266 (with one elector abstaining in the official tally).  The election was noteworthy for a vitriolic controversy over the awarding of Florida's 25 electoral votes and its recount process.  It was the closest election since 1876.  The Supreme Court chose George W. Bush by a vote of 5-4.

Some people thought that the Supreme Court had no business getting involved, it being a State matter and not a Federal matter, since Article II, Section 1 states that “each State shall appoint, in such manner as the Legislature may direct, a Number of Electors …”  These people believed that the Supreme Court should have left it to Florida to make the call.  The problem was that Florida appeared to be at an impasse, thus creating a risk that a president would not be chose in time for Inauguration Day.

Father and Son Presidents

John Adams, our second president, was father to John Quincy Adams, who was our sixth president.  George H. W. Bush, our 41^st president, is father to George W. Bush, who was our 43^rd president.  These are the only two cases in history where a father and his son were both presidents.  Theodore and Franklin Roosevelt were just fifth cousins.

Carry-On Rules

My briefcase is not big at all; it is rather small as a matter of fact. It is actually a small, soft-sided, laptop case, so that tells you how big it is – or how small it is in this instance. Well, on a flight last week, I put my briefcase, which was the only carry-on bag I had, in the overhead bin above my seat. Some minutes later, an attendant ordered over the loudspeaker for everyone with big bags to put them in the overhead bins, and for everyone with small bags to put them under the seat in front of them. I ignored him, and after a minute or two, he came over to me and told me I would have to move my briefcase from the overhead bin and put it under the seat in front of me. I replied that since I had a valid ticket, I should be entitled to use as much overhead space as anyone else. I asked him why I, who was carrying a small bag, should have to be more uncomfortable than I otherwise would be, just so some other person could haul in a blimp of a bag and yet receive preference for the overhead bins.

He didn’t answer that. He just said, “You do know that you have to obey what the attendants say to do, or get thrown off the airplane, don’t you?” So, I gave up and said, okay, hand it down. But he let it go, and my briefcase stayed where it was. Elaine stated later that I would never win the “it’s not fair” argument (even though I apparently had won it on this one particular time), because the airlines decreed the policies, and they didn’t care about fair, and the attendants just did what they were told to do in order to keep their jobs. She said that I would be far more likely get my way by informing the attendants that I was an old man suffering from severe neuropathy and very poor blood circulation in my legs and that I needed to stretch them out, which a briefcase under the seat in front of me would prevent me from doing – all of which would be true.

Well, of course Elaine was right, generally speaking, but, my legs aside, I’ll tell you that this thing of the airlines giving people with big bags preference over people with small bags really bugs me. It got me to thinking, what if, for example, someone packed his small briefcase inside one of the largest allowable carry-on bags? He could put his briefcase in the bin above him without receiving any hassle from an attendant. Moreover, if a lot of other people started doing the same thing, the airlines might eventually adopt a fairer policy. One policy that would be fairer than the current one would be for the airlines just to reduce the maximum size of a carry-on bag that anyone is allowed to have and to diligently enforce that decision.

Yams and Sweet Potatoes

Yams and sweet potatoes taste more-or-less like each other, but they have little else in common. Sweet potatoes are botanically very distinct from yams. Sweet potatoes are members of the morning glory family, whereas yams are tubers of tropical vines that are closely related to lilies and grasses. Sweet potatoes are native to Central and South America, whereas yams are native to Africa and Asia. Sweet potatoes are dicotyledons; yams are monocotyledons.

If you live in North America, unless you specifically search for yams, you are probably eating sweet potatoes. It is not unlikely that you have never even tasted an actual yam. Yams are popular in Latin American and Caribbean markets; they are slowly becoming more common in U. S. markets.

Sweet potatoes are of two dominant types. One type has a thin, light yellow skin with pale yellow flesh that is not sweet, and it has a dry, crumbly texture. The second type is the one that is most often incorrectly called a yam. It has a thicker, dark orange-to-reddish skin with a vivid orange, sweet flesh and a moist texture.

Yams contain more natural sugar than sweet potatoes and thus are generally sweeter. However, yams are much starchier than sweet potatoes and not nearly as nutritious. Sweet potatoes are packed with vitamin A, which is considered critical in maintaining proper eye health. One sweet potato contains nearly eight times an adult's daily need of Vitamin A, and, because the vitamin is fat-soluble, the body can store it for later use.

Sweet potatoes contain several other vitamins and minerals in amounts not found in yams. Sweet potatoes contain significantly higher amounts of calcium, iron, and vitamin E, and twice as much protein per serving. Sweet potatoes are also strong sources of beta-carotene, manganese, and copper. Sweet potato varieties are classified as either “firm” or “soft”. When cooked, those in the firm category remain firm, whereas soft varieties, although actually firm when raw, become soft when cooked.

It is the soft varieties that are often (incorrectly) labeled as yams in the U. S. Firm varieties of sweet potatoes were produced in the U. S. before soft varieties. When soft varieties were first grown commercially, there was a need to differentiate between the two. African slaves called the soft sweet potatoes yams, because they resembled the yams in Africa. Thus, soft sweet potatoes were referred to as yams to distinguish them from the firm varieties of sweet potatoes. The U. S. Department of Agriculture requires sweet potatoes that are labeled as yams to also be labeled as sweet potatoes.