The length of the original piece of cloth was <u>20 meters</u>.
The cost of per meter cloth initially was <u>birr 10 per meter</u>.
In the question, we are given that a piece of cloth costs birr 200.
Also, we are told that if the piece was 5m longer and the cost of each meter of cloth was birr 2 less, the cost of the piece would have remained unchanged.
We are asked to find the length of the cloth piece and what is its original price per meter.
We assume the initial length of the cloth to be x meters.
The cost of the cloth is given to be birr 200.
Thus, the per meter cost of the cloth initially was birr 200/x per meter,
In the new case, we are given that:
The piece was 5 meters longer.
The initial length = x m.
The new length = x + 5 m.
The cost of each meter of cloth was birr 2 less.
The initial cost of each meter of cloth = birr 200/x per meter.
Thus, the new per meter cost = birr 200/x - 2 per meter = birr (200 - 2x)/x per meter.
Thus, the cost of the piece = birr (x + 5)(200 - 2x)/x.
Now, we are given that the cost of the piece in both cases is the same, which can be shown as:
(x + 5)(200 - 2x)/x = 200,
or, 200x + 1000 - 2x² - 10x = 200x,
or, x² + 5x - 500 = 0,
or, x² + 25x - 20x - 500 = 0,
or, x(x + 25) - 20(x + 25) = 0,
or, (x - 20)(x + 25) = 0, which gives that:
Either x - 20 = 0 ⇒ x = 20,
or, x + 25 = 0 ⇒ x = -25.
Since x cannot be negative, x = 20.
Thus, the length of the original piece of cloth was x meters = <u>20 meters</u>.
The cost of per meter cloth initially was birr 200/x per meter = <u>birr 10 per meter</u>.
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