Problem:
A carpenter cuts off from a plank of wood 7/13 m and then cuts 2/3 of what remains. If the remaining piece is 4 1/2 m, then find out the original length of the plank of wood?

Solution:
To find the original length of the plank of wood, we need to follow a step-by-step approach.

Step 1: Finding the length after the first cut
Let's assume the original length of the plank of wood to be x meters.
After cutting off 7/13 m from the plank, the remaining length becomes (x - 7/13) meters.

Step 2: Finding the length after the second cut
The carpenter then cuts 2/3 of what remains, which means 2/3 of (x - 7/13) meters.
So, the length after the second cut becomes (x - 7/13) - (2/3)(x - 7/13).

Step 3: Setting up the equation
According to the problem, the remaining length after the second cut is 4 1/2 m, which is equivalent to 9/2 m.
So, we can set up the equation as follows:
(x - 7/13) - (2/3)(x - 7/13) = 9/2

Step 4: Solving the equation
To solve the equation, we need to simplify it first.
(x - 7/13) - (2/3)(x - 7/13) = 9/2
Multiplying through by the least common denominator (13 * 3 = 39) to eliminate the fractions:
39(x - 7/13) - 26(x - 7/13) = 9/2
39x - 273/13 - 26x + 182/13 = 9/2
Combining like terms:
13x - 91/13 = 9/2
Multiplying through by the least common denominator (13 * 2 = 26) to eliminate the fractions:
26(13x - 91/13) = 26(9/2)
338x - 182 = 117
338x = 299
x = 299/338

Step 5: Simplifying the result
The original length of the plank of wood is 299/338 meters. However, we can simplify this fraction further.
Dividing both the numerator and denominator by their greatest common divisor, which is 13, we get:
x = 23/26

Answer:
The original length of the plank of wood is 23/26 meters.