Farhan has three pieces of rope with lengths of 140 cm, 168 cm, and 210 cm. He wishes to cut all three pieces of ropes into smaller pieces of equal length such that there is no leftover rope. We need to determine the greatest possible length of each smaller piece and the total number of smaller pieces Farhan can obtain.
To find the greatest possible length of each smaller piece, we need to find the highest common factor (HCF) or greatest common divisor (GCD) of the given rope lengths. The HCF represents the largest length that can evenly divide all three rope lengths without any remainder.
(i) Finding the Greatest Possible Length:
To find the HCF, we can use various methods such as prime factorization, division method, or Euclidean algorithm. Here, we will use the Euclidean algorithm, which is an efficient method for finding the HCF.
Step 1: Find the HCF of 140 cm and 168 cm.
Using the Euclidean algorithm:
168 = 140 × 1 + 28
140 = 28 × 5 + 0
The remainder becomes zero, so the HCF of 140 cm and 168 cm is 28 cm.
Step 2: Find the HCF of 28 cm and 210 cm.
Using the Euclidean algorithm:
210 = 28 × 7 + 14
28 = 14 × 2 + 0
Again, the remainder becomes zero, so the HCF of 28 cm and 210 cm is 14 cm.
Therefore, the greatest possible length of each smaller piece is 14 cm.
(ii) Finding the Total Number of Smaller Pieces:
To determine how many smaller pieces Farhan can obtain altogether, we need to divide each original rope length by the HCF and sum up these divisions.
For the first rope with a length of 140 cm, the number of smaller pieces is 140 cm / 14 cm = 10 pieces.
For the second rope with a length of 168 cm, the number of smaller pieces is 168 cm / 14 cm = 12 pieces.
For the third rope with a length of 210 cm, the number of smaller pieces is 210 cm / 14 cm = 15 pieces.
Therefore, Farhan can obtain a total of 37 Step-by-step smaller pieces altogether.