October 2001 Test expl: Section 3, #19, 22, 23, 25. Section 6, #7,10

Jimmy2588 · 08-27-2005, 07:38 PM

Heres another post, thanks in advance again to all who help me

. I am slowly starting to recognize my weaknesses.

Section 3:

19. Of 650 cities surveyed, each city has an art museum, or a nature museum, or both. Of the 320 cities that had art museyms, 1/4 also had nature museums. What is the total number of cities surveyed that had nature museums?

(I tried making a venn diagram and got the left circle to be 240, then the space between to be 80, then I couldn't get thr right circle.)

22. In an art class, there were just enough staplers, rulers, and glue bottles so that every 2 students had to share a stapler, every 3 students had to share a ruler, and every 4 students had to share a glue bottle. If the sum of the number of staplers, rulers, and glue bottles used by the class was 65, how many students were in the class?

(I thought of this as a ratio problem.. I did it weirdly, but here is how I did it.. i got x/2 + y/3 + z/4 = 65, got the common denominator, which is 12, then it occurred to be that it SHOULD be 6x+4x+3x = 65. x = 5. then I plugged in an got 60 for everything. The answer is 60, but I ended up doing 60+60+60 = 180, which is obviously wrong. Would someone please explain how to this the "right" way?)

23.

STEP1: Choose a number n greater than 2
STEP2: If n is prime, add 5, if not, add 2
STEP3: Divide the result of step 2 by 2
STEP4: Find the greatest integer less than or equal to the result of step 3

In the chart above, if the number n chosen in step 1 is 39, what number will be the result of step 4?

(Here are my steps: Step 1-39+2=41. Step 2- 43. Step 3 - 43/2 (21.5). Step 4- 21?, Answer is supposed to be 20)

25. how many positive integers less than 100 are multiples of 5 and are equal to 3 times an even integer?

Section 6:

7. Six points are placed on a circle. What is the greatest number of different lines that can be drawn so that each line passes through two of these points?
a. 12
b. 15
c. 25
d. 30
e. 36

10. If three different circles are drawn on a piece of paper, at most how many points can be common to all three?
a. none
b. one
c. two
d. three
e. six

(I hate these stupid ones that ask you to draw something!)

Thanks for your time!

CDN_dancer · 08-27-2005, 08:46 PM

19. 1/4 of 320 is 80
Therefore, 80 cities have both museums, 240 have only art, 330 have only nature. So the total number of cities with nature museums is 410.
(Hopefully I did that right....)

23. This one, you just repeated one of the steps.
Step 1: 39
Step 2: 41 -->39+2
Step 3: 20.5 -->41/2
Answer: 20

*Working on other problems.

CDN_dancer · 08-27-2005, 09:12 PM

25. There are 19 integers less than 100 that are multiples of 5 (the 20th multiple of 5 is 100). 9 of those intergers are even (you can eliminate the other 10 because 3*an even number can not equal an odd). That leaves you with 10, 20, 30... From there, only 30, 60, and 90 are divisible by 3.

*Please let me know if I did any of those wrong.

CDN_dancer · 08-27-2005, 11:19 PM

7. The answer should be B). If you think about a circle with 6 points, the first point can be connected with 5 other points, the second can be connected with 4 different points, and so on. 5+4+3+2+1=15

10. I think the answer should be C), but I'm not entirely sure.

Jimmy2588 · 08-28-2005, 02:07 AM

thank you for your replies. here are the answers..

Section 3:
19. 410 (done)
22. 60
23. 20 (done)
25. 33 (i still dont understand this one

)

Section 6:
7. B (done)
10. C

CDN_dancer · 08-28-2005, 02:09 AM

Your welcome.
Dang, got #25 wrong. Wasn't sure of that one anyways.

tanman · 08-28-2005, 02:20 AM

Are you sure about the answer for #25? There aren't even 33 positive integers less than 100 are multiples of 5, let alone the other conditions. CDN_Dancer's is the same thing I got but I worked it slightly differently. If the number is a multiple of 5 and divisble by 3 times an even number, that means it has to be a multiple of 5, 3, and 2. This means it has to be a multiple of 30 and there are only three multiples of 30 less than 100 - 30, 60, and 90

tanman · 08-28-2005, 02:23 AM

Ahh. I realized the Oct 2001 test is online and took a look at #25. The question asks for numbers less than 1000, not less than 100. There are 10 multiples between 0 and 300, 10 between 301-600, 10 between 601-900 and 3 (930, 960, and 990) between 901 and 1000, giving you a grand total of 33

tanman · 08-28-2005, 02:27 AM

22 Your method looks right. There are x students in the class, so there are x/2 staplers, x/3 rulers and x/4 glue bottles. If there are 65 items total, x/2 + x/3 + x/4 = 65. Multiply by the LCM of 2,3 and 4 to get 6x + 4x + 3x = 780. So 13x=780 and x = 60

Jimmy2588 · 08-28-2005, 11:21 AM

sorry for the misprint, 1000 integers

.. would you mind explaining that one again? Also, for the stapler and ruler one, how did you get 6x + 4x + 3x = 780? Where did the 780 come from?

Flipsta_G · 08-28-2005, 12:38 PM

You must remember that when you multiply by a common denominator on one side of the equation, you also multiply it by the other side of the equation. This is where you went wrong, Jimmy. Since x/2 + x/3 + x/4 = 65, when you multiply by 12, you also have to multiply 65 by 12. This yields 780. 780/13 = 60. You also messed up by using 3 different variables (x,y,z) rather than just one, x.

Flipsta_G · 08-28-2005, 12:43 PM

10. If three different circles are drawn on a piece of paper, at most how many points can be common to all three?
a. none
b. one
c. two
d. three
e. six

How many different points can 2 different circles have in common? The answer is 2. So the answer cannot be higher than 2. But can we find a way to get a third circle on those 2 points of intersection? Yes! We kust have to make it small enough, directly in between the 2 other circles, so that its higherst and lowest points hit the 2 intersection points. The answer is 2 or C.

Flipsta_G · 08-28-2005, 12:47 PM

For 23, you messed up by assuming that 39 is prime. It is not, because 13 *3 = 39. Therefore, in step 2, you should add 2, not 5.

Jimmy2588 · 08-28-2005, 12:58 PM

Flipsta! where would I be in life without you, haha. Thank you for pointing out my mistakes!

and for the weird step question, I added 2, not 5.. I think I repeated a step, which is what CDN said.

Thanks everyone for the explanations!

janan · 02-22-2006, 02:06 AM

i think Q#25 can also be solved in this way

1st part of question,

How many positive integers less than 100 are multiple of 5,
if you take those multiple upto to 90 (5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85 ,90)

No. of multiples is 18.

now come to the 2nd part of the question,

the number of multiple is 3 times and even number so

3x6= 18 ( total no. of multiple of 5 less than 100)

I hope this work.

Waiting for response