Question

A man wants to cut three lengths from a single piece of board of length $91 \,cm$. The second length is to be $3 \,cm$ longer than the shortest and the third length is to be twice as long as the shortest. The possible length of the shortest board, if the third piece is to be atleast $5 \,cm$ longer than the second, is

• A. less than $8 \, cm$
• B. greater than or equal to $8 \, cm$ but less than or equal to $22 \,cm$
• C. less than $22 \, cm$
• D. greater than $22 \,cm$

Answer 1

Answer: (B)

Let the shortest side be $x cm$.
Then, by given condition, second length $=x+3 cm$
Third length $=2 x cm$
Also given, total length $=91$
Hence, sum of all the three lengths should be less than equal to 91 .
$ x+x+3+2 x \leq 91 $
$\Rightarrow 4 x+3 \leq 91$
Subtracting $(-3)$ to each term,
$-3+4 x+3 \leq 91-3$
$\Rightarrow 4 x \leq 88$
$\Rightarrow \frac{4 x}{4} \leq \frac{88}{4} $
$\Rightarrow x \leq \frac{88}{4}$
$\Rightarrow x \leq 22 cm...(i)$
Third length $\geq$ second length $+5$
Again, given that $ 2 x \geq(x+3)+5$
$2 x \geq x+(3+5)$
Transferring the term $x$ to LHS,
$2 x-x \geq 8 \Rightarrow x \geq 8....(ii)$
From Eqs. (i) and (ii), length of shortest board should be greater than or equal to 8 but less than or equal to 22 i.e., $8 \leq x \leq 22$