Problem 94 - Algebra 1

Question

RECIPROCALS Find the reciprocal. $$ \frac{2}{7} $$

Step-by-Step Solution

Answer

The reciprocal of the fraction \( \frac{2}{7} \) is \( \frac{7}{2} \).

Step 1: Understand the concept of reciprocal

The reciprocal of a number or a fraction is found by switching the numerator with the denominator. In simple words, if a given fraction is \(\frac{a}{b}\), the reciprocal will be \(\frac{b}{a}\).

Step 2: Apply the concept to the given fraction

The provided fraction is \(\frac{2}{7}\), so by following the concept, the reciprocal will be reverse of the fraction, i.e \(\frac{7}{2}\).

Step 3: Simplify the resulting fraction

The resulted reciprocal \(\frac{7}{2}\), when simplified, remains \(\frac{7}{2}\) as there is no common factor between 7 and 2 that could simplify it further.

Key Concepts

Understanding Fractions

Fractions represent parts of a whole. The term fraction comes from the Latin word fractus, meaning broken; essentially, a fraction breaks down a whole into smaller parts. A fraction consists of two components: a numerator and a denominator.

The numerator, positioned above the fraction bar, indicates the number of equal parts being considered. The denominator, below the fraction bar, represents the total number of equal parts into which the whole is divided.

Consider the fraction \(\frac{2}{7}\). Here, 2 is the numerator, telling us there are two parts being observed. Meanwhile, 7 is the denominator, showing that the whole is divided into seven parts, and we are interested in two of those parts.

How to Simplify Fractions

To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by this number to make the fraction as simple as possible without changing its value.

For example, to simplify \(\frac{8}{12}\), we find that the GCD of 8 and 12 is 4. Dividing both the numerator and denominator by 4 yields \(\frac{8 \div 4}{12 \div 4} = \frac{2}{3}\). The simplified fraction \(\frac{2}{3}\) is equivalent in value to the original fraction, \(\frac{8}{12}\), but it's in its simplest form.

A fraction is considered fully simplified when there is no number, except for 1, that can divide evenly into both the numerator and the denominator.

Numerators and Denominators

Every fraction contains two key parts: the numerator and the denominator. They have distinct roles in defining the value of the fraction.

The numerator represents how many parts you have, while the denominator tells you how many parts make up a whole. To truly understand a fraction's value, you need to look at both numbers in conjunction.

For instance, if you have \(\frac{3}{4}\), the numerator is 3 indicating that you have three parts, and the denominator is 4 telling you that the whole contains four equal parts. If either the numerator or the denominator were different, the value of the fraction would change.