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. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second?

Answer 1

Hint: Assume a variable for the shortest length. Obtain the value of the other two pieces in terms of the shortest length. Then use the two inequalities given to find the range of values of the shortest length.

Complete step-by-step solution -
Let the shortest length be x. It is given that the second piece is 3 cm longer than the shortest piece. Hence, the length of the second piece is (x+3) cm.
The third piece is twice as long as the shortest length. Hence, the length of the third piece is 2x.
Observe that the sum of the lengths of the three pieces can not exceed the total length of 91 cm. Hence, we have the following inequality:
\[x + x + 3 + 2x \leqslant 91\]
Simplifying, we have:
\[4x \leqslant 91 - 3\]
\[4x \leqslant 88\]
Solving for x, we have:
\[x \leqslant \dfrac{{88}}{4}\]
\[x \leqslant 22...........(1)\]
Now, it is given that the length of the third piece should be at least 5 cm greater than the second piece. Hence, we have:
\[2x \geqslant x + 3 + 5\]
Simplifying, we have:
\[2x \geqslant x + 8\]
\[2x - x \geqslant 8\]
\[x \geqslant 8.............(2)\]
From equations (1) and (2), the possible length of the shortest piece lies in the range of 8 cm to 22 cm.

Note: It is not given that the sum of the three pieces is equal to the total length of the board. Hence, you should consider it as an inequality rather than equality to obtain the correct answer. If in this type of problem some words are given like “atleast”, ”minimum” and “maximum” then the linear inequality concept is to be applied for solving.