Answer :
To solve the problem of dividing the three cables into equal pieces:
a) Maximum Length of Each Piece:
To find the maximum possible length of each piece, we need to determine the greatest common divisor (GCD) of the lengths of the three cables. The GCD is the largest number that can divide all the numbers without leaving a remainder.
1. Cable lengths are 120 cm, 160 cm, and 180 cm.
2. Find the GCD of these three numbers:
- Find the GCD of 120 and 160 first.
- Then, find the GCD of the result with 180.
After doing these calculations, we find that the GCD is 20 cm. So, each piece can be a maximum of 20 centimeters long.
b) Number of Pieces Generated:
Now, to find out how many pieces are generated for each cable, we divide the length of each cable by the length of each piece:
1. For the 120 cm cable:
[tex]\[
\frac{120}{20} = 6 \text{ pieces}
\][/tex]
2. For the 160 cm cable:
[tex]\[
\frac{160}{20} = 8 \text{ pieces}
\][/tex]
3. For the 180 cm cable:
[tex]\[
\frac{180}{20} = 9 \text{ pieces}
\][/tex]
To find the total number of pieces, simply add up all the pieces from each cable:
[tex]\[
6 + 8 + 9 = 23
\][/tex]
So, the cables are divided into a total of 23 pieces.
a) Maximum Length of Each Piece:
To find the maximum possible length of each piece, we need to determine the greatest common divisor (GCD) of the lengths of the three cables. The GCD is the largest number that can divide all the numbers without leaving a remainder.
1. Cable lengths are 120 cm, 160 cm, and 180 cm.
2. Find the GCD of these three numbers:
- Find the GCD of 120 and 160 first.
- Then, find the GCD of the result with 180.
After doing these calculations, we find that the GCD is 20 cm. So, each piece can be a maximum of 20 centimeters long.
b) Number of Pieces Generated:
Now, to find out how many pieces are generated for each cable, we divide the length of each cable by the length of each piece:
1. For the 120 cm cable:
[tex]\[
\frac{120}{20} = 6 \text{ pieces}
\][/tex]
2. For the 160 cm cable:
[tex]\[
\frac{160}{20} = 8 \text{ pieces}
\][/tex]
3. For the 180 cm cable:
[tex]\[
\frac{180}{20} = 9 \text{ pieces}
\][/tex]
To find the total number of pieces, simply add up all the pieces from each cable:
[tex]\[
6 + 8 + 9 = 23
\][/tex]
So, the cables are divided into a total of 23 pieces.
