+885 1. Paula can paint a room in six hours. Paula starts painting the room at
9:00 AM. At 11:00 AM, Carlos accompanies Paula, and both of them,
working together, finish painting the room at 1:30 PM. Assuming both
painters always paint at a constant rate, how many hours can Carlos
paint the same room working by himself?
2. An alphagram is a string of letters, such as ABC or AABEGLR,
whose letters are in alphabetical order. Find the number of three-letter
alphagrams.
4. The numbers 1, 2, 3, 4, . . . are written starting with 1, spiraling outward
in a counterclockwise direction, as shown below. What is the
sum of the four numbers directly adjacent to the number “2018” in
this spiral?
.................. 26
13 12 11 10 25
14 3 2 9 24
15 4 1 8 23
16 5 6 7 22
17 18 19 20 21
5. A frog starts at “0” on a number line. Each second, it flips a fair coin,
and moves forward 1 unit if the coin shows heads, and forward 3
units if the coin shows tails. What is the probability that the frog will
eventually land on “6?” Express your answer as a common fraction.
1)
6 - 2 = 4 hours left for Paula to finish by herself
1/4 + 1/C =2/5, where C = Carlos' time
C =20/3 = 6 2/3 - Carlos' time in painting 2/3 of the room!!!
So, to paint the whole room by himself, it would take him: 6 2/3 x 3/2 = 10 hours !!!!.
EDITED!!. I simply reversed the time they finished the room together!!. Oh, my!!
+129852 1. Paula can paint a room in six hours. Paula starts painting the room at
9:00 AM. At 11:00 AM, Carlos accompanies Paula, and both of them,
working together, finish painting the room at 1:30 PM. Assuming both
painters always paint at a constant rate, how many hours can Carlos
paint the same room working by himself?
Paula paints 1/6 of the room every hour
By the time Carlos joins in, Paula has painted 2 (1/6) = 1/3 of the room
And after Carlos joins....she paints an additional 2.5 hrs
So...by 1:30 PM....she has painted 1/3 + (2.5)(1/6) = 1/3 + (5/2)(1/6) =
1/3 + 5/12 = 4/12 + 5/12 = 9/12 = 3/4 of the room
So.....Carlos must paint the other 1/4 of the room in 2.5 hours
So...the time it would take him to paint the room by himself must be 4 times this =
10 hrs
+129852 Sorry, ant....I would just be guessing on the rest and that usually leads to wrong answers....maybe some other people know the solutions to the rest of these !!!!
+26393 4. The numbers 1, 2, 3, 4, . . . are written starting with 1, spiraling outward
in a counterclockwise direction, as shown below.
What is the sum of the four numbers directly adjacent to the number “2018” in this spiral?
.................. 26
13 12 11 10 25
14 3 2 9 24
15 4 1 8 23
16 5 6 7 22
17 18 19 20 21
\(\begin{array}{cccc} \text{Start at $(0, 0) = 1$ }\\\\ && \boxed{2019\\\tiny{(22,16)}} \\ && \uparrow \\ & \boxed{1843\\\tiny{(21,15)}} \leftarrow & \boxed{2018\\\tiny{(22,15)}} & \rightarrow \boxed{2201\\\tiny{(23,15)}} \\ & & \downarrow \\ & & \boxed{2017\\\tiny{(22,14)}} \\ \end{array} \)
\(1843+2017+2019+2201 = 8080 \)
+26393 Computing...
1.
\(\text{Numbers}:\\ \begin{array}{rrrrrrr} 31 & 30 & 29 & 28 & 27 & 26 & \\ 32 & 13 & 12 & 11 & 10 & 25 & \\ \ldots & 14 & 3 & 2 & 9 & 24 & \\ \ldots & 15 & 4 & 1 & 8 & 23 & \\ \ldots & 16 & 5 & 6 & 7 & 22 & \\ \ldots & 17 & 18 & 19 & 20 & 21 & \\ \end{array} \)
2.
\(\text{Coordinates of the lattice points}:\\ \begin{array}{rrrrrrr} \text{Start at $(0, 0) = 1$ }\\\\ (-3,3) & (-2,3) & (-1,3) & (0,3) & (1,3) & (2,3) & \\ (-3,2) & (-2,2) & (-1,2) & (0,2) & (1,2) & (2,2) & \\ \ldots & (-2,1) & (-1,1) & (0,1) & (1,1) & (2,1) & \\ \ldots & (-2,0) & (-1,0) & (0,0) & (1,0) & (2,0) & \\ \ldots & (-2,-1) & (-1,-1) & (0,-1) & (1,-1) & (2,-1) & \\ \ldots & (-2,-2) & (-1,-2) & (0,-2) & (1,-2) & (2,-2) & \\ \end{array}\)
3.
\(\text{Algorithem to get the lattice point coordinates:} \\ \begin{array}{|r|r|} \hline \text{lattice point number} & \text{coordinate }(x,y) \\ \hline 1.& (0,0) \\ 2.& (0,1) \\ 3.& (-1,1) \\ 4.& (-1,0) \\ 5.& (-1,-1) \\ 6.& (0,-1) \\ 7.& (1,-1) \\ 8.& (1,0) \\ 9.& (1,1) \\ 10.& (1,2) \\ 11.& (0,2) \\ 12.& (-1,2) \\ 13.& (-2,2) \\ 14.& (-2,1) \\ 15.& (-2,0) \\ 16.& (-2,-1) \\ 17.& (-2,-2) \\ 18.& (-1,-2) \\ 19.& (0,-2) \\ 20.& (1,-2) \\ 21.& (2,-2) \\ 22.& (2,-1) \\ 23.& (2,0) \\ 24.& (2,1) \\ 25.& (2,2) \\ 26.& (2,3) \\ 27.& (1,3) \\ 28.& (0,3) \\ 29.& (-1,3) \\ 30.& (-2,3) \\ 31.& (-3,3) \\ 32.& (-3,2) \\ \ldots \\ 1843.& (21,15) \\ \ldots \\ 2017.& (22,14) \\ \ldots \\ 2018.& (22,15) \\ \ldots \\ 2019.& (22,16) \\ \ldots \\ 2201.& (23,15) \\ \ldots \\ \hline \end{array}\)
4.
short algorithm to get (x,y) the index in a grid-net of the spiral in c++:
int n_max = 2818;
int y_coordinate = 0;
int x_coordinate = 0;
for(int i = 0; i < n_max; ++i) {
// output Number i+1
// output x_coordinate
// output y_coordinate
if(abs(y_coordinate) <= abs(x_coordinate) && (y_coordinate != x_coordinate || y_coordinate >= 0))
y_coordinate += ((x_coordinate >= 0) ? 1 : -1);
else
x_coordinate += ((y_coordinate >= 0) ? -1 : 1);
}
Find the lattice point coordinate of 2018.
We find (22,15).
The four numbers directly adjacent to the number “2018” in
this spiral are (21,15), (22,14), (22,16), (23,15).
The Numbers are 1843, 2017, 2019, 2201.
