Amy and Phyllis are standing at opposite points on a round pond on the path that circles the pond at its edge. At the same time, they start racing around the pond in opposite directions. They meet for the first time after Phyllis has traveled 100 yards, and they meet again 60 yards before Amy has completed her first lap. What is the circumference of the path around the pond?
Set 12
Speedboat
Orville’s speedboat will run 4 hours at full throttle or 7 hours at half throttle. We went out to East Sister Island at full throttle. It took one hour. We returned at half throttle. It took 3 hours. Then we refilled the fuel tank. It took 3.8 gallons of diesel fuel to refill the tank. How much fuel does the tank hold?
Factor Fourth Degree Equation
The number of distinct ordered pairs (x, y), where x and y have positive integral values satisfying the equation x4y4 – 10x2y2 + 9 = 0, is:
- 0
- 3
- 4
- 12
- infinite
Quadratic Leading to x = y = z
Show that all real solutions to the equation:
x2 + y2 + z2 = xy + yz + zx
satisfy x = y = z .
Jessamine Bush
Here is a problem allegedly proposed by ancient Hindu mathematicians: The square root of half the number of bees in a swarm has flown out upon a jessamine bush; 8/9 of the whole swarm has remained behind; and one female bee flies about a male that is buzzing within a lotus flower into which he was lured in the night by its sweet scent, but is now imprisoned in it. Tell me the number of the bees in the swarm.
Quadratic, Difference of Roots is 1
Consider x2 + px + q = 0, where p and q are positive numbers. If the roots of the equation differ by 1, then what is p? (Your answer will be in terms of q.)
- √4q + 1
- q – 1
- -√4q + 1
- q + 1
- √4q – 1
Dorothy Eggs
Dorothy and Toto were carefully driving their small veggie-oil-powered truck containing three baskets of eggs to market, when a car going at full speed collided with the truck. Dorothy was unhurt, but Toto got a broken leg. The eggs were destroyed. The car’s driver admitted his responsibility and offered to pay for the eggs. Dorothy said he must pay Toto’s vet bill too. The driver, aghast, asked, “Toto too?” Dorothy compromised and said she’d pay half the vet bill if the driver could answer a simple problem for her:
The number of eggs was somewhere between 50 and 100. If they were counted out 2 at a time or 3 at a time, none would remain at the end. But if they were counted out 5 at a time, there would be 3 left over.
The driver was required to say how many eggs there had been in the baskets. What answer would get him off the hook for the full vet bill?
Pickanumba, Result is 2
Think of any number, multiply it by 2, then add 4, multiply by 3, divide by 6, subtract the number you started with, and the result will be 2. Explain with algebra why this works.
Horse, Mule, Golden Trinkets
A horse and a mule were traveling together heavily laden with bags of golden trinkets. After a time, the mule complained to the horse that his load was too heavy. “What’s the matter?” said the horse. “You shouldn’t complain, for had I taken one of your bags I would have had to carry twice as many bags as you. On the other hand, if you agreed to take one of mine, we’d have the same number of bags.” How many bags did each have?
Apollo Ticket Prices
For a special show, the Grand Apollo Theater charged ticket prices as follows: adults, $10; students, $3; and children under 12, 50 cents. The theater collected a total of $100 from an audience of 100 people. How many adults, students, and children were in the audience?