Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-7,\) write the reciprocal (multiplicative inverse) of each given number. $$ \frac{7}{12} $$

Step-by-Step Solution

Answer

The reciprocal of \( \frac{7}{12} \) is \( \frac{12}{7} \).

Step 1: Understanding Reciprocals

The reciprocal (or multiplicative inverse) of a number is a value that, when multiplied by the original number, results in 1. For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).

Step 2: Applying to the Given Fraction

The given fraction is \( \frac{7}{12} \). To find its reciprocal, we swap the numerator and the denominator, thus the reciprocal is \( \frac{12}{7} \).

Step 3: Verifying the Reciprocal

To ensure correctness, multiply the original fraction \( \frac{7}{12} \) by its reciprocal \( \frac{12}{7} \) and check the result: \[ \frac{7}{12} \times \frac{12}{7} = \frac{7 \times 12}{12 \times 7} = \frac{84}{84} = 1 \]. This confirms that \( \frac{12}{7} \) is the correct reciprocal.

Key Concepts

Understanding the Multiplicative Inverse

The concept of a multiplicative inverse is core to understanding reciprocals. Every number has a multiplicative inverse that, when multiplied by the original number, results in one. This is similar to hitting the reset button on multiplication. For example, if you have a fraction \(\frac{a}{b}\), its multiplicative inverse is found by flipping the fraction to \(\frac{b}{a}\).

Why does this work? Imagine multiplying something by its reciprocal: you effectively restore the process back to the identity of 1. This is because \(a \times \frac{1}{a} = 1\), and the same logic applies to fractions. Flipping the places of the numerator and the denominator achieves this intriguing balance.

This concept is widely used not only in mathematics but also in real-life applications such as solving equations and dividing fractions in more complex calculations.

Exploring Fractions

Fractions represent a part of a whole. They are expressed as \(\frac{a}{b}\) where \(a\) is the numerator, and \(b\) is the denominator. Consider fractions as slices of a cake: if the whole cake is cut into equal slices and you take some slices, that amounts to your fraction.

An important aspect of fractions is their reciprocal, which reverses their roles. For example, \(\frac{7}{12}\) becomes \(\frac{12}{7}\) when you look at its reciprocal.

One critical point to remember is that while any fraction \(\frac{a}{b}\) can find its reciprocal by flipping it, the fraction itself and its reciprocal will always result in a product of 1 when multiplied with each other. This leads us to properly dividing numbers when calculations are involved, ensuring that operations are reversible.

Verifying Reciprocals

After finding the reciprocal of a fraction, it's crucial to be sure it's correct.

Here's how you do it: multiply the original fraction by its reciprocal. If you get 1, you've done everything right! For instance, if your starting point was \(\frac{7}{12}\), you'd flip it to \(\frac{12}{7}\) to get the reciprocal. Next, multiply them:

• \(\frac{7}{12} \times \frac{12}{7}\) = \(\frac{84}{84}\)

This simplifies directly to 1, confirming that \(\frac{12}{7}\) is indeed the correct reciprocal.

This check is more than just confirmation; it's an integral part of ensuring mathematical accuracy. Imagine adjusting the volume back to the original level after turning it down and up again. The result should precisely match the starting point, ensuring that the transformative process hasn't altered the intended outcome.