Problem 62 - Prealgebra

Question

Find the reciprocal of each number. $$\frac{3}{5}$$

Step-by-Step Solution

Answer

The reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\).

Step 1: Understanding Reciprocals

The reciprocal of a number is simply 1 divided by that number. For a fraction like \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).

Step 2: Identify the Numerator and Denominator

For the fraction \(\frac{3}{5}\), identify the numerator as 3 and the denominator as 5.

Step 3: Swap Numerator and Denominator

To find the reciprocal, swap the numerator with the denominator. So, the reciprocal of \(\frac{3}{5}\) becomes \(\frac{5}{3}\).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole or a division of quantities. They consist of two key components: the numerator and the denominator.
Fractions are written in the form \(\frac{a}{b}\), where \(a\) represents the number of parts we're considering, and \(b\) represents the total number of equal parts.

• Proper Fractions: These fractions have numerators smaller than their denominators, like \(\frac{3}{5}\). They represent a quantity less than one.
• Improper Fractions: Here, the numerator is equal to or greater than the denominator, such as \(\frac{5}{3}\). These can be converted into mixed numbers.
• Mixed Numbers: Combination of an integer and a proper fraction, like 1\(\frac{2}{3}\).

Understanding fractions is fundamental before diving into concepts like reciprocals, as you need to know which numbers to work with when manipulating the fraction.

Numerators

Numerators are at the top of a fraction. They show you what's being counted.
In the fraction \(\frac{3}{5}\), the numerator is 3. It's telling us we're considering 3 parts out of 5 total parts.

• Counting Parts: The numerator indicates how many parts of the whole are being focused on. So, in terms of a pie, if the pie is cut into 5 slices, choosing "3" as our numerator means we are looking at 3 slices.
• Affect on Value: Increasing or decreasing the numerator changes the overall value of the fraction, directly relating to how much of the whole you're looking at. For instance, \(\frac{3}{5}\) is less than \(\frac{4}{5}\); each step up in the numerator increases the fraction's value.

Remember when finding the reciprocal, the numerator becomes the denominator.

Denominators

The denominator is the bottom part of a fraction. It represents the total number of equal parts the whole is divided into.
In \(\frac{3}{5}\), 5 is the denominator, showing there are 5 equal parts in total.

• Determining Unit Size: The denominator helps you understand how small or large each "part" is; larger denominators mean smaller parts. For example, \(\frac{3}{10}\) means smaller parts than \(\frac{3}{5}\) since one part of \(\frac{3}{10}\) is smaller than one part of \(\frac{3}{5}\).
• Impact on Fraction Value: A smaller denominator means larger parts, leading to a bigger fraction. For example, \(\frac{3}{4}\) is larger than \(\frac{3}{5}\) because 1 of 4 parts is larger than 1 of 5 parts.

When finding the reciprocal, swap the roles of the numerator and denominator.