Mark:
GUS
50 lbs
(Amazing - but true!)
Let the loss weight be x lb.
(99%)(100) - (98%)(100 - x) = x
(0.99)(100) - (0.98)(100 - x) = x
99 - (98 - 0.98x) = x
99 - 98 + 0.98x = x
1 + 0.98x = x
1 + 0.98x - 0.98x = x - 0.98x
1 = 0.02x
1 / 0.02 = 0.02x / 0.02
50 = x
100 - x = 100 - 50 = 50
The weight of the dehydrated potatoes was 50 lb.
A king sits on gold...who sits on silver?
a lone ranger.
(Silver was the name of the Lone Rangers horse)
PAINTING
80 minutes
It would take Adam and Beth 4/3 hours (1 hour and 20 minutes) to paint the room without Cory's help.
Let's let A, B, and C be the number of hours required for each of Adam, Beth, and Cory to paint the room by themselves, and let's let X be the time required for Adam and Beth working together. Then 1/A, 1/B, and 1/C are the fractions of the room that each can paint alone in one hour, and 1/X is the fraction of the room that Adam and Beth would paint in an hour of working together. Then we have:
1/X = 1/A + 1/B
Now the time needed for all three together is one-half the time for Cory alone, which means that the total fraction of the room they all paint in an hour is twice the fraction Cory would paint alone:
2*(1/C) = 1/A + 1/B + 1/C
2/C = 1/X + 1/C
1/C = 1/X
X = C
We're also told that all three working together take 6 hours less than Adam working alone -- that means that all three together paint the fraction 1/(A-6) of the room in one hour, so:
1/(A-6) = 1/A + 1/B + 1/C
1/(A-6) = 1/X + 1/C
1/(A-6) = 2/X
A = (X/2) + 6 = (X+12)/2
Similarly, the time for all three together is only one hour less than for Beth alone, so:
1/(B-1) = 1/A + 1/B + 1/C
1/(B-1) = 1/X + 1/C
1/(B-1) = 2/X
B = (X/2) + 1 = (X+2)/2
Putting our information together, we have
1/X = 1/A + 1/B = 2/(X+12) + 2/(X+2) = (4X+28)/[(X+12)(X+2)]
0 = (4X+28)/[(X+12)(X+2)] - 1/X = (3X2+14X-24)/[(X+12)(X+2)X]
For this fraction to be zero, its numerator must be zero, so
3X2 + 14X - 24 = 0
(3X-4)(X+6) = 0
Clearly X can't be negative (Adam and Beth aren't that fast at painting!), so the only solution that makes sense is X = 4/3.
BASKETBALL
81-78
The final score was 81 to 78. Getting that answer is a matter of using some algebra to keep track of information and then eliminating possibilities.
First, let's start with the three-point shots. Team B made 41.666% of theirs, which works out to 5/12. Now they couldn't have taken as many as 24 3-point shots, since the two teams together only took 20 -- so, they must have taken 12 shots and made 5 of them. But that means Team A took the other 8 3-point shots, and (since they made 50% of theirs) made 4 of them. So far, then, we have 15 points for Team B and 12 points for Team A.
Now let's let X be the number of Team A 2-point baskets. Then the number of Team B 2-point baskets must be 45-X. The number of 1-point free-throws by Team A is also X, so we can calculate Team A's final point total as:
(points from 3-pointers) + (points from 2-pointers) + (points from free-throws)
= 12 + 2X + X = 3X + 12
If we let Y be the number of free-throws made by Team B, then Team B's point total is:
(points from 3-pointers) + (points from 2-pointers) + (points from free-throws)
= 15 + 2(45-X) + Y = 105 - 2X + Y
Since Team A won by 3 points, we then have the equation:
Team A score = Team B score + 3
3X + 12 = 105 - 2X + Y + 3
5X = Y + 96
Y = 5X - 96
The only clue we haven't accounted for yet is that the number of free-throws made by Team B (Y = 5X - 96) must be a prime number and must be less than the number of 2-point shots made by Team B (45 - X). We need to check various values of X to see which one fits this last requirement. Clearly, Y can't be negative, so our last equation above tells us that the smallest that X can be is 20. We'll just make a table to look at the possibilities:
Code:
X Team B free-throws
5X-96 Team B 2-pointers
45-X
20 4 25
21 9 24
22 14 23
23 19 22
24 24 21
By the last row of this table, the value of 5X-96 is already greater than the value of 45-X, something we know can't happen. So, X must be either 20, 21, 22, or 23. But X=23 is the only one of these possibilities that makes 5X-96 a prime number (look at the middle column of the table). So, X=23 is correct, which means Team A's final score is 3X+12 = 81. Team B's score must then be three less, or 78.
SEQUENCE
J.J.M.A.M.F.J.
It cannot be denied that yesterday was Mark's day. However, yesterday has passed, my yesterday is still to come.
A greeting card company makes a LOT of different valentine card designs - somewhere between 200 and 500 to be more precise. As you might expect, there are some common themes in these many different designs: many have heart shapes on them, many have poems, and exactly half of the card designs are on a red background.
The total number of non-red cards that have either a heart shape or a poem (or both) is the square root of the number of red cards that feature both a heart shape and a poem. All of the red designs have either both a heart shape and a poem or neither. Of all the card designs that have neither a heart shape nor a poem, only one-ninth are red.
How many card designs are there
A man arrives at work between 8:00 and 9:00 a.m. at exactly the moment the minute hand and hour hand of the clock point in the same direction. He takes a quick break for lunch from exactly noon to the next time the clock hands point in opposite directions. He leaves work between 4:00 and 5:00 p.m. at exactly the moment the clock hands point opposite one another.
How many hours did he work that day
(Your answer (some chance) should be precise to a fraction of a second!)
