Math Help Please?

Question

I am having trouble with these three difficult types of math problems. If anyone knows how to work the examples below, that would be extremely helpful. (if possible could you explain to me how you solved them) thanks so much!!

1. The average of two numbers, one of which is "q", is "pq". If a third number is added, the average of all three numbers is equal to "(p + 2pq) / 3". What is the third number?

2. In a room, 5 chairs are placed to accomodate 3 people, one person to a chair. How many seating arrangements are possible?

3. In a geometric series, each term is a constant multiple of the preceding one. If the first three terms in a geometric series are -2, x, and -32, what is the sixth term in the series?

Thanks again to anyone who even attempts at helping me out!!

Answer 1

1) First problem does not make sense. It is not stated does p mean. Also, you do not say if "pq" is some number p multiplied by q or whether it is a positional expression like "23" [twenty three] meaning 2 times ten and 3 times one.

2) If you pick a seat for the first one, you have 5 options. Once you have done this, you are left with 4 options for the second and then only 3 for the third. The answer is 5*4*3.

3) Say the constant is q. Then x = -2*q and -32 = x*q = -2*q^2. Hence by solving the last equation for q you get that q^2=16 and hence q is either 4 or -4 [you can't tell! both are valid solutions!]. Now the sixth term is -2*q^5 which is either -2*(4^5) or -2*(-4)^5 = 2*(4^5), depending on whether x is negative, or positive. So there are two solutions.

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And trust me, the answer to the second problem is not what the people above me say it is. It is not combination number but variation number since the three people have to be treated as being three different individuals and hence their if you just reorder them using the same seats, this reasults in a different seating arangement.

I am still not sure what does problem 1 even mean. I have a guess, but I believe you owe us some more explanation as to what does "pq" mean.

Answer 2

Solution to no. 1

assuming the 2 numbers are x and q, then

[(x + q) / 2] = pq , lets simplify this

to remove 2 from the left side of the equation;

2[(x + q) / 2] = 2pq

x + q = 2pq

assuming the third number is y, then

[(x + q + y) / 3] = (p + 2pq) / 3 , since 2pq = x + q ( as previously mentioned), then

[(x + q + y) / 3] = [p + (x + q)] / 3 , lets simplify

3[x + q + y / 3] = 3[p + (x + q)] / 3

(x + q + y) = p + (x + q)

(x + q) + y = p + (x + q)

y = p + (x + q) - (x + q)

y = p

anyway hope it helps :)

Answer 3

2. The answer is 5c3 = 10.
5c3 is the number of ways you can seat 3 people with 3 cairs. This is a branch of math called combatronics.

3. -32/x = x/-2
Hence, x^2 = 64 and x = 8
So the sixth term is -2 * -4^5

1. the number added is p
q+x / 2 = pq
q+x+y / 3 = p+2pq / 3
2pq+y = p+2pq

Therefore,
p = y

Answer 4

1.Average of two numbers=pq
Hence,their sum is 2pq
One of the number is q
Therfore the second number is 2pq-q
The average of all three numbers is (p+2pq)/3
therefore,their sum is 3*(p+2pq)/3
=p+2pq
we have already got the first two numbers
which are q and 2pq-q
Therefore,the third number is
p+2pq-(q+2pq-q)
=p+2pq-2pq
=p
2.The answer will be 5C3=5!/3!2!
=5*4/2
=10
3.The first term is -2
common ratio is x/-2 which is again equal to -32/x
or x^2=64
x=8
Hence the common ratio is 8/-2=-4
the sixth term of the series
= -2*(-4)^5
=-2*-1024
=2048

Answer 5

1. Average=sum of all numbers/no. of numbers

average of 2 numbers =pq
total of two numbers=2*pq=2pq
one of those numbers is q,then the other no. =2pq-q

average of three numbers=(p+2pq)/3
total of those nos.=(p+2pq)/3*3=p+2pq

total of 3numbers-total of two nos.=third number
p+2pq-2pq=p

P is the third number.
I could do only the first sum.

Answer 6

1) p
2) 18
3) -2048

hope this helps but i am not so sure about 2