CPhill

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First job total pay = 16 * 400 =  $6400

Second job total pay = 1300 + 1300*1.10 + 1300*1.10^2  + 1300*1.10^3 = $6033.30

So    $ [ 6400 - 6033.30 ]  =  $366.70

We need to use some logic, here

Let n = the number of neighbors

Let q be the number of oranges that each neighbor received

Let r be the number of apples that each received.......and we have the following system:

150  = q*n + 17  →    q*n  = 133      (1)

100  = r*n + 5 →       r*n =  95        (2)

Subtract (2) from (1)

q*n - r*n =  38     factor out n

n (q - r)   =  38

Since 38 factors as 2 * 19   or 19 * 2 ........

We can assume that n = 2 or n = 19

But...if n = 2    then  "q"  and "r"  in both (1) and (2)  would be fractions

So.....let us assume that n = 19

Then  q = 133/19 =  7

And r  = 95/19  = 5

So....there were 19 neighbors

Another way to do this is to factor both 133 and 95

133 = q * n = 7 * 19

95 =   r * n  =  5 * 19    and it would be clear that n = 19  in both (1)  and (2)

Determine the gradient k of the straight line passing through the point (4,0) and is perpendicular to the line 4x+17y+3=0. Reply with a shortened fraction k = a/b

First....find the slope of the given line.....

4x + 17y + 3  = 0       get y by itself........subtract 17y from both sides

4x + 3  = -17y             divide both sides by -17

(-4/17)x - 3/17    =  y     the slope of this line is   (-4/17)

A line perpendicular to the given one will have a negative reciprocal slope =  17/4

So...if k is th gradient, we have that k = 17/4

[ The fact that this line passes thorugh (4,0)  does not affect the slope ]

1) 2 standard 6-sided dice are rolled. What is the probability that the sum rolled is a perfect square?

The possible perfect squares are 4 and 9

A "4"  can be obtained in 3 ways

A "9" can be obtained in  4 ways

And the total number of outcomes = 36   so.......

The probability of a perfect square being rolled  =  [ 3 + 4 ] / 36 =    7/36

Given A(-5,-4) and B(7, 18), find the point two-fifths of the way from A to B.

This can be found thusly :

The x coordinate is given by ;

-5 + [2/5] [ 7 - (-5)]  =   -5 + [2/5] [12]  =  -1/5  = -.2

And the y coordinate is given by :

-4 + [2/5] [ 18 - (-4) ]  =  -4 + [2/5] [22]  = 4.8  

Proof :

The total distance between the points =  2√157

And the distance between  (-5, 4)   and ( -.2, 4.8)  = √  [4.8^2 + .8^2]  ≈ 10.024

So.....10.024 / [2 √157] ≈  .4    ≈   2/5

Two gears work together. There are 32 teeth on one gear and 36 on the other gear. The first gear makes 128 revolutions per second. How often, during a 24 hour period, will the same two teeth from each gear be together?

I saw geno do something like this one the other day......

32 = 2^5

36 = 3^2 * 2^2

The least common multiple between 36 and 32  =  3^2 * 2^5    =  288

So....... the two teeth are together when the first gear makes 288/32  = 9 revolutions   and the second gear makes 288/36 = 8 revolutions

And it takes the first gear 9/128 seconds to make 9 revolutions

And there are  60*60*24 = 86400 seconds in  a 24 hour period.......so......

In 24 hours.......the same two teeth from both gears will be together

86400  / [ 9/128] =  1,228,800   times

 (x^2+9x+3)(x^3-4x+7/x)  =  

x^5 + 9 x^4 - x^3 - 36 x^2 - 5 x + 21/x + 63

I assume you can take it from here

78 = 2 x 39  = 2 x 3 x 13

97   is a prime itself

The first term of a geometric series is 2, the nth term is 486 and the sum of the n terms is 728. find r and n.

We have two  things we can work with, here........first......the sum of the series 

728 = 2 [1 - r^n] / [ 1 - r]

364 = [ 1 - r^n ] / [1 -r]

364 -364 r  = 1 - r^n

364r - 364   = r^n - 1

364r  - 363    =  r^n    (1)

And, second.......the nth term

486  = 2(r)^(n-1)

243  = r^(n-1)

243  = r^n / r

243r  = r^n     (2)

So....equating (1)  with (2), we have

364r - 363   = 243r   simplify

121r - 363  = 0

121r = 363       divide both sides by 121

r = 3

And using (2)

243(3)  = 3^n

729  = 3^n  

And n = 6

Sorry.....I get the same result as Melody........check

Sum =  2[ 1 - 3^6] / [ 1 -3]  =  728

And the 6th - and last -  term is given by

2*3^(6 -1)  =

2*3^5  =

2* 243  = 

486

I'm not sure of the answers generated by WA........for instance.........their calcuation for "n"  results in a decimal approximation......it should be an integer!!!!......I'd go with the solutions Melody and I provided