Split - problem solving and reasoning

[ctmckoay]

Hi,

I've got a couple of queries if someone can help. Appreciate if there can be workings/methods to support deriving the answer.

1) A worm crawls up a 47cm wall 3cm each day, but slips back 2 cm each night. After how many days will he reach the top ?

Answer : 45 days

(2) At halftime, Albert Park School had scored all of the points so far in their annual match against Hawthorn School. In the second half, each side scored three points. At the end of the match, Albert Park had scored 90% of the points. What fraction of the point in the match was scored in the second half ?

Answer : 1/5

(3) The sequence 2,3,5,6,7,10.....consists of all natural numbers which are neither perfect squares nor perfect cubes. Find the 54th term of this sequence.

Answer : 63

 

[Denis]

Re: problem solving and reasoning

1) A worm crawls up a 47cm wall 3cm each day, but slips back 2 cm each night. After how many days will he reach the top ?
Answer : 45 days

At net of 1 per day, worm is at 44cm after 44 days.
On 45th day, goes up 3cm: so at 47cm; so reached top in 45 days:
does not matter if worm goes down after...kinda trick question...

 

[ctmckoay]

Re: problem solving and reasoning

1) A worm crawls up a 47cm wall 3cm each day, but slips back 2 cm each night. After how many days will he reach the top ?
Answer : 45 days

At net of 1 per day, worm is at 44cm after 44 days.
On 45th day, goes up 3cm: so at 47cm; so reached top in 45 days:
does not matter if worm goes down after...kinda trick question...

Thanks Denis. Appreciate your feedback.

 

[Denis]

Re: problem solving and reasoning

(2) At halftime, Albert Park School had scored all of the points so far in their annual match against Hawthorn School. In the second half, each side scored three points. At the end of the match, Albert Park had scored 90% of the points. What fraction of the point in the match was scored in the second half ?
Answer : 1/5

x = points by AP in 1st half
1st half: x , 0
2nd half: 3, 3

x + 3 = .9(x + 6)

Wanna finish it?

 

[soroban]

Re: problem solving and reasoning

Hello, ctmckoay!

(2) At halftime, Albert Park School had scored all of the points so far in their annual match against Hawthorn School.
In the second half, each side scored three points.
At the end of the match, Albert Park had scored 90% of the points.
What fraction of the points in the match was scored in the second half ?

Answer: 1/5

In the first half AP scored \(\displaystyle x\) points; HS scored 0.

In the second half, both teams scored 3 points.


The scoring looks like this:

. . \(\displaystyle \begin{array}{c||c|c||c|} & \text{1st half} & \text{2nd half} & \text{Total} \\ \hline \text{AP} & x & 3 & x+3 \\ \hline \text{HS} & 0 & 3 & 3 \\ \hline\hline \text{Total} & x & 6 & x+6 \\ \hline \end{array}\)

\(\displaystyle AP\) scored \(\displaystyle x+3\) of the \(\displaystyle x+6\) points.
We are told that this is 90%.

. . \(\displaystyle \frac{x+3}{x+6} \:=\:\frac{9}{10} \quad\Rightarrow\quad 10(x\,+\,3) \;=\;9(x\,+\,6) \quad\Rightarrow\quad 10x\,+\,30 \;=\;9x\,+\,54 \quad\Rightarrow\quad x \;=\;24\)


Now the scoring is:

. . \(\displaystyle \begin{array}{c||c|c||c|} & \text{1st half} & \text{2nd half} & \text{Total} \\ \hline \text{AP} & 24 & 3 & 27 \\ \hline \text{HS} & 0 & 3 & 3 \\ \hline\hline \text{Total} & 24 & 6 & 30 \\ \hline \end{array}\)


In the second half, 6 points were scored out of the total of 30 points.

. . \(\displaystyle \text{The fraction is: }\;\frac{6}{30} \;=\;\frac{1}{5}\)

Edit: Too slow again . . . *sigh*
.

 

[Denis]

(3) The sequence 2,3,5,6,7,10.....consists of all natural numbers which are neither perfect squares nor perfect cubes. Find the 54th term of this sequence.
Answer : 63

Go here:
http://www.research.att.com/~njas/sequences/index.html
Enter A094784 in box: your sequence will be shown...
along with some exotic gezintos too complicated for me!

If the sequence skipped only squares (not cubes), then this does it:
nth term = n + RND[SQRT(n)]
Examples:
13th term = 13 +RND[SQRT(13)] = 13 + RND(3.605....) = 13 + 4 = 17
29th term = 29 +RND[SQRT(29)] = 29 + RND(5.385....) = 29 + 5 = 34

 

[ctmckoay]

Thanks all. You are a champ.