## Sunday, December 25, 2005

### Math puzzle

I found this math problem at Kitchen Table Math.

"769. An old brainteaser by Leonty Magnitsky: If a man drinks a barrel of water by himself in 14 days and the same barrel with his wife in 10 days, how many days would it take his wife to drink the barrel by herself?"

First of all, I find this easier to solve when the wording is changed to a barrel of vodka.

Here is my solution:

The sum of the fractional parts (man and wife) of drinking in one day is equal to the fractional part of both drinking in one day:

Let x = days wife takes to consume the whole barrel

1/14 + 1/x = 1/10

Multiply each side by the LCD to clear fractions:

10x + 140 = 14x
4x = 140
x = 35

Old Math said...

A man wants to drive from Denver to Detroit, turn right around and come back. He wishes to average 60 mph for the entire trip. When he gets to Detroit he finds that he has averaged only 30 mph for the first part of the trip. How fast must he drive to make the round trip at the desired 60 mph average?

(No offence to Detroit, reverse the role of the two cities for local consumption)

NYC Educator said...

Instructivist,

Sorry to go off topic, but how's your car search coming? I've checked out a few hybrids, and the only one that seems worth considering to me is the Prius. With a 3000 tax credit, it ends up costing as much as the Camry.

The Accord hybrid is a 6 cylinder, gets only 3 mpg better than the Accord 4 cylinder, and costs maybe 8000 bucks more.

Have you found a diesel you could recommend? I'm probably gonna buy a Prius if nothing else jumps out at me soon. It's a very cool car--I drove one today.

Author said...

This is a variant of the typical work rate problem which goes like this. John can mow the lawn in 3 hours and Jim can mow the lawn in 4 hours. How long does it take them to do it together?

I think it is useful to ask how fast is the man drinking beer? Answer: 1/14 barrel per day. The couple is drinking 1/10 barrel per day. Then take the difference. This gives how fast the wife is drinking. The wife's drinking speed can be inverted to give days per barrel.

The general principle is that we can consider work per unit time or time per unit work. How is the information presented to us? What is the best way to represent the information for the sake of solving the problem? What form does the problem want us to present the answer in?

In general, we should teach students that there are multiple ways of presenting the information given in a ratio.

Instructivist said...

Hola seĆ±or educator (un nombre que prefiero no usar).

I had my eyes on a VW diesel, specifically the Passat Wagon TDI. Now I find out that that car is no longer available in the U.S. That's a real shame. The vehicle gets terrific mileage.

Why this car was discontinued is something of a mystery. The dealers say it has to do with the quality of diesel fuel which makes it hard to meet emission standards. I called VW headquarters and was told the reason is a supposed mismatch between body and engine. All this is weird. In Europe diesels are a high percentage of all new cars. You would think the supposed mismatch would have been worked out by now.

It makes me angry that I cannot buy a diesel wagon in this country.

NYC Educator said...

I guess market forces are strong indeed. But perhaps there is some small hope--

I'm told you can get a Jetta diesel in NJ, but for some reason they aren't available in NY. It's easier than a trip to Europe, though.

Instructivist said...

The Jetta diesel exists but Jetta doesn't come in Wagon form. A wagon diesel is what I want.

"I guess market forces are strong indeed."

The dealers said the alotments they used to get sold quickly. It's not a question of demand. They claim the culprit is government policy. A car manufacturer supposedly can have only one diesel model in its lineup. Morevoer, they say EPA policy allows a lower diesel fuel quality to favor truckers. Consequently, higher grades are not available to help meet stricter emission standards for diesel cars. It's sounds like conspiracy-mongering. I don't know where the truth lies.

Old Math said...

"A car manufacturer supposedly can have only one diesel model in its lineup"

I don't know about you but I always believe everything a car dealer tells me....

NYC Educator said...

I've been to 4 dealers, looking at Priuses. Each one has them on display, and tells me how unusual it is they are in stock, and how no one else has them. None know anything about the US tax credit this year, which would be a tremendous selling point.

Sorry that VW is making it so tough for you. It does appear, though, that one way or the other all car manufacturers are going to have to start offering high-mileage vehicles in the very near future.

Instructivist said...

"I don't know about you but I always believe everything a car dealer tells me...."

Very funny. But in this case they are on to something. The problem appears to be the unavailability of low-sulfur diesel fuel.

I found this in a review:

Diesel as a solution to air pollution? No, I'm not kidding. Diesels emit much less carbon dioxide than gasoline engines. They do emit more oxides of nitrogen and particulates (soot), but technology similar to the catalytic converters used by gasoline engines is being developed to scrub those pollutants from diesel tailpipes. However, similarly to the manner in which a catalytic converter is destroyed by leaded gasoline, the devices to thoroughly clean diesel exhaust need low-sulfur diesel fuel, which will not be available here until 2007. Modern diesel cars are clean enough for the EPA, but not for the California Air Resources Board, so the Passat TDI (and all other diesel cars) will not be available in California, New York, Massachusetts, Maine, or Vermont until the advent of low-sulfur diesel fuel.

Barry Garelick said...

Old Math:

Your math problem. The total time to make a round trip from Denver to Detroit and average 60 mph would be 2d/60 where d is distance between Denver and Detroit. 2d/60 = d/30.

The time it takes to go from Denver to Detroit averaging 30 mph is d/30. Thus, the time has been used up to average 60 mph. No matter what speed he goes on the return trip, the time will be greater than d/30, so the average speed will be less than 60 mph.

Anonymous said...

Argh... this isn't even a brainteaser...

Let r1 be the rate the man drinks water in gallons per day.

Let r2 be the rate the wife drinks water in gallons per day.

Let B be the capacity of the barrel in gallons.

Then r1 x 14 = B
and (r1 + r2) x 10 = B

so

r1 x 4 = r2 x 10

or r2 = 0.4 r1

Then 0.4 r1 x X = r1 x 14, where X is the number of days it takes the wife to drink the barrel.