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Question
Joe has a blue rope and a green rope; both ropes are the same length. He cuts the blue rope into 5 pieces of equal length. He cuts the green rope into 12 pieces of equal length.

Each piece of blue rope is 119cm longer than each piece of green rope.

How long was the blue rope before it was cut into pieces?

Answer
We are trying to find the original length of the blue rope, this is the unknown.

We will let the letter r stand for the original length of the blue rope (in centimetres). This means that the original length of the green rope must also be rcm because the ropes were the same length before they were cut.

The blue rope was cut into 5 pieces, so the length of each piece is r/5 cm.

The green rope was cut into 12 pieces, so the length of each piece is r/12 cm.

We also know that each piece of green rope is 119cm shorter than each piece of blue rope. This can be written:

Length of one piece of blue rope - Length of one piece of green rope = 119cm

Remember that the length of each piece of blue rope is r/5 cm and the length of each piece of green rope is r/12 cm.

So we can write:

r/5 cm - r/12cm = 119cm

We will use 60 as a common denominator to allow us to do the subtraction.

12r/60 cm - 5r/60 cm = 119cm

7r/60 cm = 119cm

Multiply both sides of the equation by 60/7.

rcm = 1020cm

Therefore, the original length of the blue rope was 1020cm.