a few more questions

langris5 · Senior Member

a few more questions

langris5 · Senior Member

here is another problem (i think it is a similar type)

2. 2/3 + 4/5 + 1/6= ? (mixed number)
answer: 1 19/30
(this is #11 from 2003 UIL district 2 test )

langris5 · Senior Member

here is one involving probability again...

3. A die is tossed, what is the probability that it shows a prime number? (this problem cames in many different forms....some i could do, some i couldn't, is there a general formula for this?)

answer: 1/2 or .5
(# 53 on 95-96 UIL invitational A)

langris5 · Senior Member

4. Find the modulus of 3+4i

answer:5
(from 95-96 UIL invitational A)

langris5 · Senior Member

5. 1/2-1/4-1/8=
(i could do this kind of problem fairly quickly, but i don't know the trick to it, and sometimes the problem gets a little complicated, like...the numerators are numbers other than 1 or totally different numbers,)

answer: 1/8
(from UIL 9562)

langris5 · Senior Member

last one....

143(base 5) / 4 (base 5)=?
answer:22
(from 2003 UIL regional)

thanks so much for the help, these are my last minute questions b4 going to district this weekend!

webmaster · Administrator

Quincy,

Welcome back and GOOD LUCK!!! I mean that.

Well, you threw me a question I had never seen which is why it took awhile to reply. I have been working on it and came up with something, but I think there is probably an easier way. The way I came up with though is easier than finding an LCM and adding fractions the long way, so here goes. Know I will keep working on it and see what I can come up with.

Anytime you have a problem like:

a/b + b/c + c/d, where each letter is one plus its previous, you can use this formula:

2 + (a^2*d - (d+2))/[b*c*d]

Let me try to explain a little. The whole number is USUALLY 2. If a=1 then the whole number would be 1, so know this only works for a>1. So 1/2 + 2/3 + 3/4 would not work for this method. (You can still use it but you get -> 2 (-1)/12 instead of 1 11/12)

The numerator is found by: a^2*d - (d+2). This is a little easier but not much since most of the time you will have to reduce the fraction. So don't write down the numerator until you know the whole fraction is in lowest terms. (*An easy way to know if you will have to reduce is look to see if the LCM of the denominators is b*c*d. If it is, then you will not have to reduce. If it isn't you will have to reduce.*)

The denominator is the product of all the denominators in the answer. (You can see that most of the time the fraction will be reduced.)

Ex [1] 3/4 + 4/5 + 5/6 = __________(mixed number)

We know the whole number is 2 so write 2.
The numerator is 3^2*6 - 8 or 54 - 8 = 46.
The denominator is 4*5*6 = 120.
46/120 = 23/60. This is the fraction part of the answer.
The answer is 2 23/60.

Ex [2] 6/7 + 7/8 + 8/9 =__________(mixed number)

Write down 2.
The numerator is 6^2 * 9 - 11 = 324 - 11 = 313.
The denominator is 7*8*9 = 504.
The answer is 2 313/504.

In Ex[2] you can know that the fraction will not reduce because the LCM of 7, 8, and 9 is simply 7*8*9. So you don't have to worry about having to reduce the fraction.

I hope this helps. I will keep working on these types of problems. But with everything I have done so far, this is by far the easiest.

Webmaster

webmaster · Administrator

2. I don't think there is a short cut to this one. I know that it looks like there might be but the numbers do not follow any sort of pattern and the numbers are easy to add the conventional way (finding an LCM and adding across). If there were a trick to it, it would look more like problem #1, where there is obviously something there, and it would likely be difficult to add the conventional way. So I would...

Change 2/3 + 4/5 + 1/6 = (20+24+5)/30 = 49/30.

Since the question wants a mixed number that is simply 1 19/30. This is how I would do the problem. Keep an eye out for more problems like this. If you can find more examples, then we might be able to determine if there is indeed a trick to this by comparing patterns that evolve.

webmaster · Administrator

#3 - I always liked probability especially when it dealed with dice. There are some easy memorizations you need to know.

If you are working with 1 die, there are 6 possibilities: 1,2,3,4,5,6. So the probability of rolling any one of those is 1/6. So to find the probability of rolling a prime we need to add the probabilities for 2, 3, and 5. Or 1/6+1/6+1/6 = 3/6 = 1/2.

If we are dealing with sum of dice you need to know that there are 36 possible combinations: 6^2. So the denominator will always be 36. If you need to know why there are 36 you can see below.

I will make a list of all the possible combinations. On the left will be the sum and the right in the form of a:b is a possible combination of 2 dice. If this doesn't make sense tell me and I will explain in a different way. The last number in () is the number of different combinations that are possible to reach the sum. This is the number we are concerned about. This will be the numerator of the answer.

If the sum of the numbers is:

2 -> 1:1 (1)
3 -> 2:1, 1:2 (2)
4 -> 1:3, 3:1, 2:2 (3)
5 -> 1:4, 4:1, 2:3, 3:2 (4)
6 -> 1:5, 5:1, 2:4, 4:2, 3:3 (5)
7 -> 1:6, 6:1, 2:5, 5:2, 3:4, 4:3 (6)
8 -> 2:6, 6:2, 3:5, 5:3, 4:4 (5)
9 -> 3:6, 6:3, 4:5, 5:4 (4)
10 -> 4:6, 6:4, 5:5 (3)
11 -> 5:6, 6:5 (2)
12 -> 6:6 (1)

With this diagram I am sure you can see the pattern and memorize it very easily. What this means is that the probablity of rolling a sum of 7 is 6/36 or 1/6. The probablity of rolling a 12 is 1/36. So for the probability of rolling a prime number we need to add all the probabilities for 2, 3, 5, 7, and 11. Or 1+2+4+6+2 = 15. So the answer is 15/36 which reduces to 5/12.

If you are interested we can also look at the product of the dice. For instance, you can find the probability that the product of the dice is a multiple of 7, or 2, etc.

webmaster · Administrator

#4 - Modulus

When dealing with imaginary numbers we often put them in the form a+bi, where a is the real number and bi is the imaginary number. If you were to plot this on a graph it would be the same as plotting a normal (x,y) coordinate except you use (a,b). The modulus means the distant from the origin or (0,0) to the point (a,b). How do you do this? I will let you find the answer for yourself. I will give you a hint: You are working with a right triangle.

Good luck!

webmaster · Administrator

#5 - For this problem to avoid repeating myself I will give you a link to my website:

http://www.math-magic.com/sequences/add_dec_finite.htm

If the numerator is different than 1, it would change the whole problem. See if you can find an example to show me to help me come up with different examples. But if the numerators are all 1, then follow the instructions from the above link.

---------------------------------------------

Sorry about what I said above - I didn't realize that I only put a section in there for adding and not subtracting. I thought I had also did subtracting. So I am appending my previous post.

When subtracting fractions in the form of 1/a - 1/a^2 - ... - 1/a^n, we can use the following:

The numerator of the answer is a^(n-1) - a^(n-2) - ... - a - 1.
The denominator is the last denominator in the question.

Basically what this says is take the next to last denominator and subtract every value to its left from that then subtract 1.

Ex [1] 1/2 - 1/4 - 1/8 - 1/16 = __________

To find the numerator we use: 8-4-2-1 = 1.
The denominator is 16.
The answer is 1/16.

Ex [2] 1/4 - 1/16 - 1/64 - 1/256 = __________

To find the numerator we use: 64-16-4-1 or 64-20-1=43.
The denominator is 256.
The answer is 43/256.

Ex [3] 1/11 - 1/121 = __________

The numerator is 11-1 = 10.
The denominator is 121.
The answer is 10/121.

In problems like Ex [1] where the denominators are 2^n, the numerator of the answer is ALWAYS 1 no matter how far the question goes. That makes the problem very quick.

webmaster · Administrator

Last Question:

I am assuming since the answer is 22 that the answer also needs to be in base 5. For this problem the only thing you can do is change the numbers to base 10 then divide then change that number back to base 5. Don't worry though it is always easy and takes just fractions of a second to convert usually.

For this problem the numerator is 143 base 5. In base 10 this is 5^2 * 1 + 5*4 + 3 or 48. 4 base 5 is still 4 in base 10. So divide 48 / 4 we get 12. Change 12 to base 5 we get 22. It is usually very easy.

Most of the time only one number will need to be changed (and that is usually the numerator). The writers of the number sense test usually pick a number to divide by that is the same in base 10 as it is in base b, making this problem all the more easier. Like in this example the base is 4. This is the same as base 10.

I hoped I helped a little. Good luck at district and let me know how you do. Oh yeah, I think I was unclear about what I wanted. I liked how you asked the questions before where they were all in one block and not separate posts. I only meant after I answered your questions if you had new ones to open a new thread (which means a new topic). That would make things easier for me. I think if people read all your questions and they were in one post, it would be easier than a different post for each question.

Webmaster

langris5 · Senior Member

thanks so much for your reply, so for the modulus problem, i am pretty sure it is a 3,4,5, Pythagorean triple. So the answer is 5.

webmaster · Administrator

Yup,

In general a modulus is found by using sqrt(a^2 + b^2).

If you have any more questions don't hesitate to ask.

Webmaster