CPhill

M - 12 = (3/7)M

M - (3/7)M =  12

(4/7)M = 12

M = 12 (7/4) =  $21   is what he started with

He has   21 - 12 = $9  left

Let the number that Tom had = T

Then Peter had 4T

When tom got 48  pencils, Tom only had twice as many

So

2 *(T + 48)  = 4T

2T + 96   = 4T

96 =  4T - 2T

96 = 2T

96/2  = T =  48

a )   Tom had   48 + 48  = 96 at the end

b )  Peter had  4 (48) =  192

Together, they had     96 + 192 =  288

Let  Hunter's seashells be H

Then Adam has 2H

When  Hunter finds  15 more we have that he now has twice as many as Adam....so....

H + 15  =  2 ( 2H)    simplify

H + 15  = 4H

15 = 4H  -H

15 = 3H

15 / 3 =  H =  5

Adam had  2H =   2 (5)  =  10 shells

Simplify as

sqrt  [  x^2 -6x + 9  - x^2 + 30x - 225 ]  =

sqrt [ 24x - 216 ]

This will be smallest when  the expression inside the radical =  0......so.....

24x  - 216  = 0

24x  = 216

x  = 216  / 24  =   9

The smallest number in the  domain is  x  = 9

WolframAlpha  shows the min to  be   -1/2   when a = -1  and b = 3

The base 3 integers  with 4 digits go from

3*3 + 0*3^2  + 0 * 3^1  + 0   = 3000₃ = 27

to

2*3^3  + 2*3^2 + 2*3 + 2 = 2222₃  =  80

The base 5 integers with two digits  go from

1*5 + 0  =  10₅ =  5

to

4*5  + 4 = 44₅  =  24  

There is no overlap ....so.....no average is possible

a/3 + 1  = a + 3                

1  - 3 =   a - a/3

1 - 3  = (2/3)a

-2 = (2/3)a

-1 = (1/3) a

a  = -3

The interior angle and its corresponding exterior angle are supplementary ( sum to 180°)

So.....

9n + n = 180        

10n = 180

n  = 180  /10   =  18°

And   the sum of the measures of the exterior angles =  360°

So....the number of sides  =  360  / 18   =   20

The equation for the  number of  diagonals in a  regular polygon is

n ( n - 3) / 2      where n is the number of  sides

So we have

n (n -3)  / 2  =  2n

n ( n - 3) = 4n

n^2 - 3n  = 4n

n^2 - 7n =  0

n ( n - 7)  = 0

The second factor gives us what we need

n - 7 = 0

n = 7

See the graph here : https://www.desmos.com/calculator/f0s0j3wor8

The base of the trangle   =   (2 - -2) = 4

The height = (3.6 - 2)   = 1.6

The area =  (1/2) (4) (1.6)  =  3.2   units ^2