CPhill

Bertie's "double and add the next digit" method is exactly the same thing we do in base 10.....consider  the number 1267

Starting from the left.....

Multiply the 1 by 10  = 10........add the next digt = 12 .........multiply this by 10   = 120.....add the next digit  = 126........multiply this by 10  = 1260 .......add the last digit = 1267......!!!

This method might be rightly called ...."multiply by the base, add the next digit"

Here's a way to understand why this "works"

(1 x 10) + 2

(1x10  + 2)10 + 6

(1x10^2 + 2*10 + 6)10  + 7

(1x 10^3 + 2x10^2 + 6x10) + 7  =   1267

Hit the "2nd" key........"asin" is the sine inverse function

asin(15/17) =  about 61.9°    or about 118.1°   if we're in the 2nd quadrant

(3/4) x 7 = 21/4   = 5 + 1/4  pints in seven days

2(L + W)  = 98

L + W  = 49

L = 49 - W

L^2 + W^2  = 41^2

(49 - W)^2 + W^2  = 1681

2W^2 - 98W + 2401 = 1681

2W^2 - 98W + 720  = 0

W^2 - 49W + 360  = 0

(W - 40) (W - 9)  = 0

W  = 40 and L = 9     or W = 9 and L = 40        [ depending upon your orientation ]

And the area = L * W   = 40 * 9   = 360  units^2

5 !   = 5 * 4 * 3 * 2 * 1  =  20 * 6   = 120

1000 ( 1 + .015)  = 1000 + 15  = $1015

If this is supposed to be   {  }  ......it indicates the "empty set"........it usually means that no solution to a problem exists........

Similar polygons are similar in all respects.....

If we divide each polygon into 10 equal triangles...the base of each triangle will be formed by a side.....and the side [base] of a triangle in the larger polygon will be  3+1/3  = 10/3   that of the base of a triangle in the smaller one.......and the height of each corresponding triangle in both figures will have the same relationship....[10/3].......so the ratio of the area of a triangle in the larger polygon to a corresponding triangle in the smaller polygon will just be the square of the ratio of the sides = [10/3]^2 = 100/9

And......the ratio of the area of one triangle in the larger polygon to the area of a triangle in the smaller polygon will have the same relationship as the area of the larger polygon to the area of smaller polygon  = [10/3]^2  = 100/9

Let the point where the one traveling 10 mph be (0,0).......let the other start at (0,150)

So

!0T + 5T  = 150         where T is the total time traveled.........simplify

15T = 150                divide both sides by 15

T = 10                      They meet after 10 hours.......so  using the faster train's rate......

10(10)  = 100......      They meet at the point (0, 100)......in other words, at the point 100 miles north of where the faster train began

Very nice, geno.....!!!!!