Question

Three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length. What is the greatest possible length of each plank? How many planks are formed?

Answer 1

The lengths of three pieces of timber are 42 m, 49 m and 63 m, respectively.
We have to divide the timber into equal length of planks.
∴ Greatest possible length of each plank = HCF(42, 49, 63)
Prime factorisation:
42 = $2\times 3\times 7$
49 = $7\times 7$
63 = $3\times 3\times 7$
∴ HCF = Product of smallest power of each common prime factor in the numbers = 7
Therefore, the greatest possible length of each plank is 7 m.
Now, to find the total number of planks formed by each of the piece, we divide the length of each piece by the HCF, i.e. by 7.
We know that;
$$\begin{gathered} 7\times 6=42 \\ 7\times 7=49 \\ 7\times 9=63 \end{gathered}$$
Therefore, total number of planks formed$=6+7+9=22$
Hence, total 22 planks will be formed.