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?
Solution and Explanation
Let the length of the shortest piece be x cm.
Then, length of the second piece and the third piece are (x+3) cm and 2x cm respectively.
Since the three lengths are to be cut from a single piece of board of length 91 cm,
x cm + (x+ 3) cm + 2x cm ≤ 91 cm
⇒ 4x+ 3 ≤ 91
⇒ 4x ≤ 91-3
⇒ 4x ≤ 88
\(⇒ \frac{4x}{4} ≤\frac{ 88}{4}\)
⇒ x ≤ 22 .......(i)
Also, the third piece is at least 5 cm longer than the second piece.
∴ 2x ≥(x + 3) + 5
⇒ 2x ≥ x + 8
⇒ x ≥8 ….. (2)
From (1) and (2), we obtain 8 ≤ x ≤ 22
Thus, the possible length of the shortest board is greater than or equal to 8 cm but less than or equal to 22 cm.
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Concepts Used:
Inequalities
In mathematics, inequality is a relationship that compares two numbers or other mathematical expressions in a non-equal fashion. It is most commonly used to compare the size of two numbers on a number line.
Specifically, a linear inequality is a mathematical inequality that integrates a linear function. One of the symbols of inequality is observed in a linear inequality: In graph form, it represents data that is not equal.
Some of the linear inequality symbols are given below:
- < less than
- > greater than
- ≤ less than or equal to
- ≥ greater than or equal to
- ≠ not equal to
- = equal to
Inequalities can be demonstrated as questions that are solved using alike procedures to equations, or as statements of fact in the form of theorems. It is used to contrast numbers and find the range or ranges of values that pleases a variable's criteria.