Problem 125 - Prealgebra and Introductory Algebra

Question

Evaluate the expression \(x \div y\) for the given values of \(x\) and \(y.\) $$x=1.173, y=-0.69$$

Step-by-Step Solution

Answer

-1.7

Step 1: Substitute the Given Values

Replace \(x\) and \(y\) with their respective values in the expression \(x \div y\). So, the expression becomes \(1.173 \div -0.69\).

Step 2: Perform the Division

Now, perform the division calculation. Divide 1.173 by -0.69 to get the result.

Key Concepts

Arithmetic Operations

Arithmetic operations are basic calculations that help us understand relationships between numbers. They include addition, subtraction, multiplication, and division. Each operation has its own rules and characteristics that define how numbers interact with one another.

• Addition combines numbers to find a total sum.
• Subtraction finds the difference between numbers by taking one away from another.
• Multiplication is like repeated addition, combining equal groups of numbers.
• Division determines how many times one number is contained within another.

In the given exercise, division is used. This operation involves finding how many times the divisor ( -0.69 in this case) fits into the dividend (1.173). Division is essential for distributing quantities and measuring ratios. To perform division accurately, you have to align numbers properly depending on whether they are integers, decimals, or fractions.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction or ratio, meaning they can be written in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers, and \(b\) is not zero. These numbers include both positive and negative numbers, as well as zero, and encompass fractions and decimals.

• Integers like -1, 0, and 2 are rational because they can be written as fractions (-1/1, 0/1, and 2/1).
• Decimals like 1.173 and -0.69 are rational because they can be represented as fractions (1173/1000 and -69/100).

In our division problem, both 1.173 and -0.69 are rational numbers. This means they can be expressed as fractions, which makes performing division straightforward. When you divide one rational number by another, the result will also be a rational number, unless the division involves irrational factors, which is not the case here.

Negative Numbers

Negative numbers appear on the left side of zero on a number line. They play a crucial role in arithmetic calculations, especially when they represent values that decrease, like debt or temperature below zero. Recognizing how negative numbers impact arithmetic operations aids in accurate problem-solving.

• Division involving a negative number results in a change of sign. For example, dividing a positive number by a negative one, as in this exercise (1.173 divided by -0.69), will result in a negative outcome.
• Understanding negative numbers is vital for handling contexts where decrease or loss is discussed.

When dealing with division of negative numbers, the sign rule (where the division of a positive by a negative results in a negative) must be applied. This reflects real-world situations like going over budget or a below-zero drop in temperature. In calculations, managing signs correctly is key to getting accurate results.