Question 125554: A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he tries to add one more row and column, he finds that he has 25 too few characters. Find the number of characters used by the designer.
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! A designer attempts to arrange the characters of his artwork in the form of a square grid with equal numbers of rows and columns, but finds that 24 characters are left out. When he tries to add one more row and column, he finds that he has 25 too few characters. Find the number of characters used by the designer.
:
Looking at the first statement:
"A designer, attempting to arrange the characters of his artwork in the form of a square grid with equal number of row and columns, found that 24 characters were left out."
:
If x = number of characters in each row
we translate this statement as
(x*x) + 24
or
x^2 + 24 = total number of characters
:
"When he tried to add one more row and column, he found that he was short of 25 characters."
:
Now, the number of rows is "x + 1", and the number of columns is "x + 1", so we have
(x + 1)(x + 1) - 25 = total number of characters.
:
FOIL above and make it equal to the 1st equation:
:
x^2 + 2x + 1 - 25 = x^2 + 24
:
x^2 + 2x - 24 = x^2 + 24
:
x^2 - x^2 + 2x - 24 - 24 = 0
:
2x - 48 = 0
:
2x = +48
:
x = 48/2
:
x = 24
:
:
Grid is 24 by 24 which contains 576 character + 24 left out = 600
:
Grid of 25 by 25 = 625 character so he would have 25 blanks
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Did this make sense? Hope so.
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