Question 1205903: A square patchwork quilt is made of many small squares. The quilt is too small to place on designated furniture. Quilt is to be enlarged square one row each way. 177 more squares are made and are just enough to increase to desired size. Determine number of small squares in original.
Not sure how to solve.
Found 3 solutions by ikleyn, greenestamps, MathTherapy:
Answer by ikleyn(52776)   (Show Source):
You can put this solution on YOUR website! .
A square patchwork quilt is made of many small squares. The quilt is too small
to place on designated furniture. Quilt is to be enlarged square one row each way.
177 more squares are made and are just enough to increase to desired size.
Determine number of small squares in original.
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Let n be the number of small squares in each way, originally.
After adding one row each way, there are (n+1) small squares in each way.
The area of the original patchwork quilt is n^2;
The area of the patchwork quilt after enlarging is (n+1)^2;
the difference of the two areas is 177. So, we can write this equation
(n+1)^2 - n^2 = 177.
Simplify this equation and find n
n^2 + 2n + 1 - n^2 = 177
2n + 1 = 177
2n = 177 - 1 = 176
n = 176/2 = 88.
ANSWER. The number of small squares originally is 88^2 = 7744.
Solved.
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
The response from the other tutor shows a good formal algebraic solution.
For general problem solving, it is useful to know the result that she used to solve the problem:
THE DIFFERENCE BETWEEN THE SQUARES OF CONSECUTIVE INTEGERS IS THE SUM OF THE TWO INTEGERS
This is easy to see by drawing simple diagrams. For example, consider the difference between 4^2 and 5^2.
Draw a picture of a square 4 small squares on a side.
Add 4 small squares along one side to make a 4x5 rectangle.
Then add 5 small squares in the other direction to make the 5x5 square.
So in this problem, where adding one small square on each side increases the total number of small squares by 177, we know that 177 is the sum of two consecutive integers, of which the smaller is the number of small squares on the side of the original quilt.
177 = 88+89, so the original quilt was 88 squares on a side.
So the number of squares in the original quilt was 88^2 = 7744.
Answer by MathTherapy(10551)  (Show Source):
You can put this solution on YOUR website!
A square patchwork quilt is made of many small squares. The quilt is too small to place on designated furniture. Quilt is to be enlarged square one row each way. 177 more squares are made and are just enough to increase to desired size. Determine number of small squares in original.
Not sure how to solve.
Let number of small squares in each row, be S
Then, S will also be the number of small squares in each column
Area of ORIGINAL quilt would then be S x S = S2
With 1 small square being added to each row, number of small squares in each row would increase to
S + 1. Likewise, number of small squares in each COLUMN would also increase to S + 1, thereby making
the quilt's new area, (S + 1)2.
The difference in the original area and the new area is a total 177 small squares.
Therefore, we get the following: (S + 1)2 - S2 = 177
S2 + 2S + 1 - S2 = 177
2S + 1 = 177
2S = 176
Original number of small squares in each row/column, or 
Original number of small squares on quilt = 88 x 88, or 882 = 7,744
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