Question

How many planks of size 2m\[ \times \]25cm\[ \times \]8cm can be prepared from a wooden block 5m long, 70m broad and 32cm thick, assuming that there is no wastage?

Answer 1

Hint: We use conversion of meters into centimeters and calculate the volume of a single plank using the formula of volume of cuboid. Similarly calculate volume of wooden block. Use a unitary method to find number of planks formed by dividing total volume of wooden block by volume of single plank.
* Unitary method helps us to calculate the number of units by dividing the total value of multiple units by the value of a single unit.
* Volume of a cuboid having length ‘l’, breadth ‘b’ and height ‘h’ is \[l \times b \times h\] cubic units.
* 1 meter has 100 centimeters in it.

Complete step-by-step solution:
We are given dimensions of plank and wooden block. We calculate their volumes separately.
Volume of plank:
We have the dimensions of wooden plank as 2m\[ \times \]25cm\[ \times \]8cm
Let length ‘l’ of plank be 2m
Then using conversion of meters into centimeters we can write 2m\[ = \]200cm
Let breadth ‘b’ of plank be 25cm
Let height ‘h’ of plank be 8cm
\[ \Rightarrow l = 200,b = 25,h = 8\]
Then volume of a plank\[ = l \times b \times h\]
Substitute the values of ‘l’, ‘b’ and ‘h’ in the formula
\[ \Rightarrow \]Volume of a plank\[ = \left( {200 \times 25 \times 8} \right)\]\[c{m^3}\]...................(1)
Volume of wooden block:
We have the dimensions of wooden block as 5m\[ \times \]70cm\[ \times \]32cm
Let length ‘L’ of wooden block be 5m
Then using conversion of meters into centimeters we can write 5m\[ = \]500cm
Let breadth ‘B’ of wooden block be 70cm
Let height ‘H’ of wooden block be 32cm
\[ \Rightarrow L = 500,B = 70,H = 32\]
Then volume of a wooden block\[ = L \times B \times H\]
Substitute the values of ‘L’, ‘B’ and ‘H’ in the formula
\[ \Rightarrow \]Volume of a wooden block\[ = \left( {500 \times 70 \times 32} \right)\]\[c{m^3}\].............… (2)
Now we use a unitary method to calculate the number of planks that can be formed from wooden blocks.
Number of wooden planks that can be formed from wooden block is given by dividing volume of wooden block by volume of a plank
Divide volume in equation (2) from volume in equation (1)
\[ \Rightarrow \] Number of wooden planks that can be formed from wooden block\[ = \dfrac{{\left( {500 \times 70 \times 32} \right)}}{{\left( {200 \times 25 \times 8} \right)}}\]
Cancel same factors from numerator and denominator
\[ \Rightarrow \] Number of wooden planks that can be formed from wooden block\[ = 28\]

\[\therefore \] Number of wooden planks that can be formed from wooden block is 28

Note: Many students make the mistake of solving the volume of both plank and block without conversion which is wrong, we should have all the dimensions in the same SI unit. Also, keep in mind while using the unitary method we divide the block by volume of 1 plank because we are forming planks from that wooden block.