Huey, Dewey, and Louie find that they must get from their home in Duckburg to a friend’s house in Possumtown, a distance of 52 km. Louie has a motorcycle, but it can manage only one passenger at a time in addition to the driver. This cycle can only go 20 kph at best. The three of them hit on a plan: Louie will take Huey part of the way, and then leave him to walk the rest of the way (at 5 kph). Then Louie will return to pick up Dewey, who has been walking all the while at 4 kph, and they will ride the rest of the way to Possumtown. They carried out their plan, which brought them all to their destination at exactly the same time. Now, can you figure out how many hours it took for the whole adventure?

## Set 18

## Add Large Fractions

An equivalent of the expression with *xy* ≠ 0 is:

- 1
- 2
*xy* - 2
*x*^{2}*y*^{2}+ 2 - 2
*xy*+ 2/*xy* - 2
*x*/*y*+ 2*y*/*x*

## Bowling Ball

After the earthquake, Junior found that the bowling alley slanted upwards, so that he had to roll the ball upward toward the pins. The slant was gentle enough so that the pins would not fall over, but just barely. He found that when he rolled the ball it went with a velocity of 12 – 4t feet per second (where t is the number of elapsed seconds.) (a) How fast was the ball going after 1 second? (b) How fast was the ball going after 3 seconds? (c) How fast was it going after 4 seconds? (d) What does this mean? (e) How fast does Junior roll the ball on a flat lane?

## Box Volume Formula

A rectangular box with no top is formed by cutting squares from the corners of an

8 by 15 sheet of cardboard and then folding up the sides. If the square cutouts are each *x* by *x*, then (a) what formula (in terms of *x*) represents the volume of the resulting box? (b) Is this a quadratic function? (c) What is the domain of *x*-values that are relevant to this problem?

## Integer Less than 9 Million

What is the largest integer less than 9,000,000 whose smallest prime divisor is 3163?

## Motorists’ Speeds

Two motorists set out at the same time to go from Toledo to Ft. Wayne, a distance of 100 miles. They both followed the same route and traveled at different, though uniform, speeds of an integral number of miles per hour. The difference in their speeds was a prime number of miles per hour. After they had been driving for two hours, the distance of the slower car from Toledo was five times the distance of the faster car from Ft. Wayne. How fast did the two motorists drive?

## Number Sums Not 100

Here is a set of numbers: {93, 4, 2} . Each number is less than 100, and there is no way that you can add any of the numbers up to get exactly 100. Find the largest set like this. That is, find the set with the most numbers so that they’re all less than 100 and you can’t add up any combination of them to get exactly 100.

## Anthropologists, Cannibals

Three anthropologists and three cannibals are making their way through a jungle and come to a river. They find a canoe large enough to carry only two persons at once. Only one of the anthropologists and one of the cannibals knows how to paddle this type of vessel. How many crossings will be needed to get all the travelers across, without ever letting the anthropologists be outnumbered by cannibals on either shore?

## Fox, Goose, Corn, River

The following problem is taken from *The Schoolmaster’s Assistant,* being a Compendium of Arithmetic, both Practical and Theoretical, published in London in 1793, page 180:

A countryman having a fox, a goose, and a peck of corn, in his journey came to a river, where it so happened that he could carry but one item over at a time. Now, as no two items were to be left together that might destroy each other, he was at his wit’s end how to dispose of them: For, says he, though the corn cannot eat the goose, nor the goose eat the fox, yet the fox can eat the goose and the goose eat the corn. The question is, how must he must carry them across the river, so that none of them can devour another.

## Leah Envelopes Wrong

Leah wrote 4 letters and correctly addressed 4 envelopes. She put one letter into each envelope, but she was careless and got some of them mixed up. As it happened, one of these three things was how it came out: she got exactly 3 of them right, or she got exactly 2 of them right, or she got exactly one of them wrong. How many did she get right?