Problem 46 - Elementary Algebra

Question

The temperature in the desert today was ninety-five degrees. Represent this temperature by a rational number.

Step-by-Step Solution

Answer

Question: Represent the temperature of ninety-five degrees as a rational number. Answer: The temperature of ninety-five degrees can be represented as the rational number \(\frac{95}{1}\).

Step 1: Convert the temperature to a numeral form

The temperature given is ninety-five degrees. We need to convert it to a numeral form, which is 95 degrees.

Step 2: Represent the temperature as a rational number

Now, we need to represent the temperature as a rational number. Since 95 is already an integer, we can represent it as a fraction by simply dividing it by 1. This gives us the rational number 95/1. So, the temperature of ninety-five degrees can be represented by the rational number \(\frac{95}{1}\).

Key Concepts

Representing Temperatures in Algebra

When we think about representing real-world quantities in algebra, temperature is a common example. Algebra allows us to use numbers and symbols to describe quantities that can change, as temperatures often do. In this context, being able to represent a temperature as a rational number is crucial for solving algebraic problems involving temperatures.

To represent a temperature in the algebraic form, we first identify it as a numerical value. For instance, 'ninety-five degrees' is a way of expressing a temperature verbally. However, in algebra, we would represent this as the numeral '95'.

But temperatures can also be below zero, which in algebraic terms would be represented by negative numbers. For example, 'negative fifteen degrees' would be written as '-15'.

Whether the temperature is above or below zero, it can be expressed as a rational number in algebra, providing a standardized form for mathematical calculations. Using rational numbers for temperatures allows us to perform operations like addition and subtraction to determine temperature differences, averages, or even to use temperatures in more complex algebraic equations.

Converting Words to Numerals

While solving math problems, especially in word problems, it's often necessary to translate words into numerals. This is a critical step as it paves the way for numerical calculations. The process involves understanding the numerical value that a word represents and then writing it down as a figure.

For example, the word 'ninety-five' translates to the numeral '95'. Other examples include 'thirty' to '30', 'two hundred' to '200', and so on. When converting words to numerals, it's also important to consider the context; words can signify ordinal numbers (like 'first' or 'second'), which represent position rather than quantity and would be converted differently.

Once the conversion is completed, the numerals can then be used in mathematical operations. This skill is essential not only for simple arithmetic but also for higher-level math where the understanding of functions, algebra, and statistics comes into play. Being proficient at converting words to numerals ensures clarity and helps avoid mistakes in mathematical problem-solving.

Fractions as Rational Numbers

The concept of fractions being rational numbers is foundational in understanding algebra and arithmetic. A rational number is any number that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, with the numerator 'p' and the non-zero denominator 'q'.

All fractions are rational numbers because they represent a part of a whole, and they can be written in a form where both the numerator and the denominator are whole numbers. For instance, the fraction \( \frac{3}{4} \) represents three parts out of four equal parts of a whole and is thus rational.

Even whole numbers and integers are rational. This might seem counterintuitive, but every integer 'n' can be expressed as \( \frac{n}{1} \)—an integer over 1. This is exactly what we saw in our original exercise with the temperature: 'ninety-five degrees' corresponds to the integer '95', which as a rational number is \( \frac{95}{1} \).

Understanding that fractions embody the concept of rationality in numbers is crucial for working with algebraic expressions and equations. It allows for consistent mathematical operations like addition, subtraction, multiplication, and division among numbers of different forms, and is a cornerstone of numerical representation.