Problem 69 - Algebra 2

Question

Find the reciprocal of each number. \(-1.25\)

Step-by-Step Solution

Answer

The reciprocal of \(-1.25\) is \(-\frac{4}{5}\).

Step 1: Understanding the Concept of a Reciprocal

The reciprocal of a number is essentially "one divided by that number". It's like flipping the number over; for example, the reciprocal of 2 is \( \frac{1}{2} \). This also applies to negative numbers.

Step 2: Convert the Number to a Fraction

First, we need to express \(-1.25\) as a fraction. \(-1.25\) can be rewritten as \(-\frac{125}{100}\).

Step 3: Simplify the Fraction

Now, simplify \(-\frac{125}{100}\) by finding the greatest common divisor (GCD) of 125 and 100, which is 25. Dividing both the numerator and the denominator by 25 gives \(-\frac{5}{4}\).

Step 4: Find the Reciprocal

To find the reciprocal of \(-\frac{5}{4}\), we simply flip the fraction, which gives us \(-\frac{4}{5}\).

Step 5: Verify the Reciprocal

To ensure correctness, multiply the original number \(-\frac{5}{4}\) by its reciprocal \(-\frac{4}{5}\). This results in \(1\), thus confirming that \(-\frac{4}{5}\) is indeed the reciprocal.

Key Concepts

Fractions

Fractions are a way to represent numbers that are not whole. They consist of two parts: the numerator, on top, which tells how many parts we have; and the denominator, on the bottom, which explains into how many parts the whole is divided. For instance, in the fraction \( \frac{3}{4} \), we have 3 parts out of a total of 4.Fractions can be found everywhere, from splitting a pizza among friends to calculating distances. Understanding them is crucial, as they form the basis of many mathematical concepts. They allow us to express values precisely, especially when dealing with parts of a whole or comparing relative sizes.It's important to keep in mind:

• Fractions can be more than one whole when the numerator is larger than the denominator.
• They can be equal to one whole when the numerator equals the denominator, e.g., \( \frac{4}{4} \).

Using fractions often involves performing operations like addition, subtraction, multiplication, and division. Making connections with everyday situations can help solidify your understanding of fractions.

Negative Numbers

Negative numbers are numbers less than zero, and they come with their own set of rules in math. They're often used to represent debts, sub-zero temperatures, or below-sea-level elevations. In math, they're essential for understanding quantities that decrease or deficit values.When working with negative numbers, remember:

• The reciprocal of a negative number is also negative. So, flipping a fraction with a negative sign before it maintains the negative sign.
• Negative numbers on a number line are those that appear to the left of zero.
• Multiplying two negative numbers yields a positive result, explaining how \(-1 \times -1 = 1\).

Mastering negative numbers allows you to navigate mathematical problems involving decreases, losses, or measurements below ground or standard baselines.

Simplifying Fractions

Simplifying fractions means making them as simple as possible, often to save time and effort when performing calculations. It involves reducing a fraction to its lowest terms by finding the greatest common divisor (GCD) of the numerator and denominator.Take, for instance, the fraction \( \frac{125}{100} \). The GCD of 125 and 100 is 25. When we divide both the numerator and denominator by 25, we get the simplified fraction \( \frac{5}{4} \).Here’s how to simplify a fraction:

• List the factors of the numerator and denominator.
• Identify the largest number that divides both evenly.
• Divide the numerator and denominator by this number.

Simplifying not only makes fractions easier to work with but also helps in comparing fractions and solving equations more efficiently. It’s a fundamental part of math that ensures clarity and simplicity.