Varsity Math, Week 97

Varsity Math 97

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On their road trips across the country, the team members spend a lot of time driving alongside train tracks.

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Passing Fancy

LT sees two trains cross a bridge at the same time, heading in opposite directions. Pretty soon the discussion in the car leads to this problem.

How long does it take a 1.25-mile-long freight train going 30 miles an hour to pass a 0.25-mile-long passenger train going 60 miles an hour in the opposite direction, from the time that the engines first reach each other to the time that the last cars just clear each other?

My Sort of Train

Later, LT sees a big train yard with a dozen long parallel tracks that merge through a series of switches down to a single track. LT wonders aloud what the system is for. One teammate responds, “It’s for sorting cars. If a 12-car train comes in, they can put one car on each of the parallel tracks, and then re-link them in any order they like.” LT responds, “There must be more to it than that. They wouldn’t need so many tracks just to rearrange the order of cars.”

What is the smallest number of parallel tracks (that join to one track) required to rearrange a 12-car train into any desired order? You can assume that the parallel tracks are long enough that none of them will run out of space.

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Solutions to week 96

Fair Share. Let the children’s share (expressed as a fraction of the total estate) be c and the grandchildren’s be g. We are told that c/g = 1/c, whence g = c². Adding up the shares of all of the children and grandchildren, which must be the whole estate, we get 3c + 10c² = 1. But this quadratic factors as (5c-1)(2c+1) = 0. An inheritance share can’t be negative, so we conclude that c = 1/5, and each grandchild receives 1/25 of the estate.

Share and Share A-Different. What are the possibilities for the shares of the four children? Well, the smallest denominators would be 1/2, 1/3, 1/4, 1/5 — but these add up to more than one, which doesn’t work. And if we skip to 1/4, 1/5, 1/6, 1/7, we see that these add up to (105 + 84 + 70 + 60)/420 = 319/420, so the grandchild would get 101/420, which is larger than the share of three of the children, contrary to the problem’s specification. Making the denominators of the children’s shares larger would only make the grandchild’s share even larger, so we conclude that the children’s shares are 1/3, 1/4, 1/5, and 1/6, leaving a share of 1/20 for the grandchild.

Recent Weeks

Week 96: Fair Share & Share and Share A-Different, solutions to Proper Place & Earliest Bird

Week 95: Proper Place & Earliest Bird, solutions to Penny Race & Quirky Quads

Week 94: Penny Race & Quirky Quads, solutions to Great 48 & Southern Circuit

Week 93: Great 48 & Southern Circuit, solutions to Sort of Hanoi-ing & Amphibian Marathon

Week 92: Sort of Hanoi-ing & Amphibian Marathon, solutions to Hard States & Great Star

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

Come back next week for answers and more puzzles.