In a rectangular prism, if the length and width are each increased by p%, by what percent must the height be decreased so that the volume remains the same?
- p
Set 11
Audio Equipment
Sue got a lot of money for graduation from a mysterious, rich aunt that she’d never known she had. Her first thought was AUDIO EQUIPMENT, but then she figured she really ought to put it away for a rainy day. She divided it into two parts. One part had $250 more than the other part. She invested the smaller part at 6% interest, and the larger part at 8%. A year later, the total interest she had accumulated was $72.50. How much money did she get from her mysterious, rich aunt for graduation in the first place?
Ratio with Two Variables
If the ratio of 2x – y to x + y is 2/3, what is the ratio of x to y?
- 1/5
- 4/5
- 5
- 6/5
- 5/4
Two Cubes, Volume = Edges
The combined volume of two cubes of different sizes, each having edges of integer values, is equal to the combined length of all their edges. What are the dimensions of the two cubes?
Particle Moving
A particle moves in a straight line so that its speed is constant for a mile and then changes abruptly for the next mile and is constant for that mile, then changes again, and so on. It moves so that its speed for a given mile varies inversely as the total integral number of miles previously traveled. If the second mile is traversed in 2 hours, how long does it take to travel the nth mile?
Formula: Recursive to Closed
If f(1) = 2 and f(n) = f(n – 1) + n, then what’s a formula for f(n)?
All the Money You Need
If you are tired of feeling broke all the time, here is some algebra that will cheer you up.
Let m be the amount of money that you actually have now,
and let n be the amount of money that you think you need.
A, their average, is given by
A = (m + n) / 2, as usual.
Now, follow this argument and see how much better you feel.
m + n = 2A
Multiply both sides by (m – n).
(m + n)(m – n) = 2A(m – n),
m2 – n2 = 2Am – 2An.
m2 – 2Am = n2 – 2An,
Now, add A2 to both sides.
m2 – 2Am + A2 = n2 – 2An + A2.
Thus (m – A)2 = (n – A)2.
So (m – A) = (n – A), and then m = n.
So you have all the money you need.
Isn’t mathematics wonderful? But how can this be? Can you find a flaw with this reasoning?
Ball-Point Pens on Sale
A certain make of ball-point pen was priced at 50¢ at Chuck’s Bargain Bin, but nobody was buying it. Chuck reduced the price by putting the pens on sale, and then all the remaining pens sold for a total of $31.93. What was the sale price of a pen?
Nasty Numbers
A nasty number is a positive integer having at least four different factors such that the difference between one pair of factors equals the sum of another pair of factors. For instance,
6 is nasty because 6 – 1 = 2 + 3
24 is nasty because 12 – 2 = 4 + 6
30 is nasty because 15 – 2 = 10 + 3.
Find the other six nasty numbers that are less than 180.
Broken Radical Key
The √ key on your calculator is broken. (So is the y½ subroutine.) How can you find √173 to 3 decimal places using your calculator and nobody else’s?