Problem 101 - Precalculus Essentials

Question

Will help you prepare for the material covered in the next section. Write an algebraic expression for the fare increase if a 200 dollars plane ticket is increased to \(x\) dollars.

Step-by-Step Solution

Answer

The algebraic expression for the fare increase is \(x - 200\)

Step 1: Interpret the problem

First, it's important to understand what the problem is asking us to do. We are given that there is a plane ticket which originally costs 200 dollars and it is increased to \(x\) dollars. We are to write an algebraic expression that represents this fare increase.

Step 2: Translate the problem into an algebraic expression

In the problem, 'increase' denotes an addition operation. The increase of the price from 200 dollars to \(x\) dollars can be represented by the algebraic expression \(x - 200\). Here the unknown \(x\) is the new increased price of the ticket, and we subtract the original price of 200 dollars. This expression will give us the amount of increase.

Key Concepts

Writing Algebraic Expressions

Writing algebraic expressions is a foundational skill in algebra that allows students to represent real-world situations mathematically. When tackling such problems, it's important to identify the numerical values and the variable, which stands for an unknown quantity. For instance, in the exercise provided, we have a plane ticket that originally costs $200, and its price increases to an unknown value, represented as x.

To write an algebraic expression for the fare increase, we must understand how the numerical value and the variable interact. The increase is the difference between the new price and the original price. Thus, the algebraic expression x - 200 captures this relationship. It's essential to maintain consistency with the use of variables and ensure the expression is as simplistic as possible, while accurately representing the situation.

Interpreting Word Problems

When interpreting word problems, the reader must carefully parse the text to identify the mathematical operations involved. Key terms often hint at these operations; for example, 'increase' typically signifies addition, while 'decrease' could suggest subtraction. In our given problem, the word 'increase' prompted us to use subtraction to find out by how much the ticket's price has risen.

It's a common misconception to directly associate 'increase' with addition, but the context dictates that we must subtract the original price from the new price to find the increase. Interpreting word problems involves understanding these nuances, as well as switching between the real-world context and its mathematical model. Critical reading skills are crucial here, as well as the ability to translate verbal descriptions into mathematical language.

Translating Words to Algebra

Translating words to algebra is about converting the language of words into the language of mathematics. This process involves identifying quantities and operations described in words and rewriting them using symbols and numbers. To improve clarity in translating words to algebra, start by highlighting or underlining key terms that indicate specific operations.

For the exercise at hand, 'increased to' suggests a comparison, leading us to use subtraction (x - 200) to find the difference between the new price and the original price. It is important to recognize that different wordings can imply the same mathematical operation. Practicing with a variety of word problems can help develop the skill to recognize these patterns and translate them into clear, concise algebraic expressions.