geno3141

31²  =  961

2²  =  4

And the question is?

a  =  6

a² + b²  =  c²     --->     6² + b²  =  c²     --->     36 + b²  =  c²   

a + b + c  =  20     --->     6 + b + c  =  20     --->     b + c  =  14     --->    b  =  14 - c

36 + (14 - c)²  =  c²     --->     36 + 196 - 28c + c²  =  c²     --->     232 - 28c  =  0

                                    --->     28c  =  232     --->     c  =  58/7 

A)  age >= 13

B)  filled circle at 13 with an arrow to the right

      or:  use ' [ ' at 13 with an arrow to the right.

Dean bought both a skateboard and a helmet:  S + H

The helmet was only 5/7^ths the cost of the skateboard; so H = (5/7)S

Therefore, the cost of the skateboard and the helmet was S + (5/7)S.

The helmet cost $44.00 more than the skateboard.

Also, the helmet cost 2/7^ths more than the skateboard [ S - (5/7)S = (2/7)S ].

So:  (2/7)S  =  $44.00    

Multiplying both sides by 7:  2S  =  $308.00     --->     S  = $154.00

The helmet cost:  (5/7) x $154.00  =  $110.00

Dean's amount  =  $154.00 + $110.00  =  $264.00

Since he received 8 shares, each share is worth $33.00.

Clare received 3 shares =  $33.00 x 3  =  .....

Eden received 9 shares = $33.00 x 9  =  ....

Total given to all threee was  =  ....

2/3 x 3/4 x 4/5 x 5/6 x ... x n/(n + 1)  =  1 / 500

Notice that the denominator of the first fraction cancels the numerator of the second fraction and this pattern continues throughtout.

So, the only numbers left on the left-hand-side are 2 and n + 1.

This means that 2 / (n + 1)  =  1 / 500     --->     1000  =  n + 1     --->    n  = 999

So:  n / (n + 1)  =  999 / 1000

Let their heights be a, b, c, d, and e with a > b > c > d > e

Since the tallest three friends have an average height exactly equal to the height of one of them, the average height of these 

three must be b.

Also, a must be as much greater than b as c is less than b; call this amount x.

Thus:  a = b + x

           b = b

           c = b - x

Since the average of the tallest four is the height of one of them; this amount must be c.

So d = b - 4x.                 [Check:  ( (b + x) + (b) + (b -x) + (b - 4x) ) / 4  =  (4b - 4x) / 4  =  b - x.]

The average height of all five is the height of one of them; this amount must be b - 4x.

So e = b - 16x     [Check: ( (b + x) + (b) + (b - x) + (b - 4x) + (b - 16x) ) / 5  =  (5b - 20x) / 5  =  b - 4x.]

The second-tallest friend is 16 cm taller than the second-shortest one:

     b  =  16 + (b - 4x)     --->     0 = 16 - 4x     --->     4x  = 16     --->     x  =  4cm

How much taller is the tallest friend than the shortest one?

     tallest = b + x  =  b + 4     shortest = b - 16x  = b - 64 cm

Thus, the tallest friend is 68 cm taller than the shortest friend.

CPhill --

I got your answer ??? but a different number.

Number Eugene starts with:  E

Number Frank starts with:     F

Eugene gave away 1/3 of his stamps to Frank ...

Now Eugene has:      (2/3)E

and Frank has:          F + (1/3)E

Frank then gave 1/5 of his stamps to Eugene ...

Now Eugene has:      (2/3)E + (1/5)( F + (1/3)E )

and Frank has:           (4/5)( F + (1/3)E )

They now have the same number of stamps:

        (2/3)E + (1/5)( F + (1/3)E )  =  (4/5)( F + (1/3)E )

          (2/3)E + (1/5)F + (1/15)E  =  (4/5)F + (4/15)E

multiply by 15:     10E + 3F + E  =  12F + 4E

                                   11E + 3F  =  12F + 4E

                                             7E  =  9F

Since  E + F  =  772      --->     F  =  772 - E

                                             7E  =  9(772 - E)

                                             7E  =  6948 - 9E

                                            16E  =  6948

                                                E  =  434.25

Final result:  ?????

Problem:  (1/3)(x + 3)²

The anti-derivative is:  (1/3)·2·(x + 3)  =  (2x + 6)/3  (by using the chain rule)

If you multiply out the original problem:  (1/3)(x + 3)²  =  (1/3)(x² + 6x + 9)

which has:  (1/3)·(2x + 6)  =  (2x + 6)/3  as its anti-derivative; the same as before.

Now:  evaluate  (2x + 6)/3  when x = 2

    and evaluate it when x = 1

    and subtract the second answer from the first.

If you write out several of the terms:   3, 5, 2, -3, -5, -2,    3, 5, 2, -3, -5, -2,    3, 5, 2, ...

you can notice that the sum of the first six terms is 0, 

the sum of the next six terms is 0, etc.

Dividing 1616 by 6 gives a quotient of 269 and a remainder of 2.

This means that there are 269 groups each of which sum to 0

and two remaining terms 3 and 5 which sum to 8.

The sum of all thee terms is 8.