A63 inch long string is cut into pieces so that their lengths form a sequence. first, a 1 inch piece is cut from the string, and each successive piece that is cut is twice as long as the previous piece cut. in how many pieces can the original length of string be cut in this way?

There are fancy ways of doing this sort of thing, however from my experience in exams you are frankly best working this out by inspection and not worrying about doing it in a fancy way - the examiners want to test your mathematical aptitude not your ability to memorise complex methods.

The sequence of doubling can be done at light speed in your head, 1 2 4 8 16 32 etc.
Let's have a stab and go for 32:
32+16+8+4+2+1 = 63 - magic, inspection works better than whatever the formula book says (that's my pro tip tbh)
Remember however since the answer is exact, the last 32 inch piece does not need to be cut, therefore the string is cut 5 times.

40 inches by 55 inches.

Step-by-step explanation:

The question lacks the required diagram. Find the diagram in the attachment.

The flag is rectangular in nature and the formula for calculating the perimeter of a rectangle is P = 2(L+W) where;

L is the length of the flag

W is the width of the flag

From the diagram, L = y and W = 11y/8

Since the perimeter = 190 inches, we will substitute the given parameters into the formula and firs calculate the value of y.

190 = 2(y+11y/8)

190 = 2y + 11y/4

multiply through by 4

760 = 8y+11y

760 = 19y

y = 760/19

y = 40

The length of the flag is 40 inches and the Width of the flag is 11(40)/8 = 55inches.

Hence, the dimensions of the flag in inches is 40 inches by 55 inches.