Problem 57 - Prealgebra

Question

Find the reciprocal of each number. $$4$$

Step-by-Step Solution

Answer

The reciprocal of 4 is \( \frac{1}{4} \).

Step 1: Understand the Concept of Reciprocal

The reciprocal of a number is essentially 1 divided by that number. For example, the reciprocal of a number \( x \) is \( \frac{1}{x} \). The idea is that when you multiply a number by its reciprocal, you get 1. This is the key concept behind finding reciprocals.

Step 2: Identify the Given Number

The exercise asks us to find the reciprocal of the number 4. Therefore, the given number in this problem is 4.

Step 3: Apply the Reciprocal Formula

Since we need the reciprocal of 4, we apply the formula for the reciprocal. This involves placing 1 over the number. Thus, the reciprocal of 4 is \( \frac{1}{4} \).

Step 4: Verify the Result

To verify the result, multiply the original number by its reciprocal and check if the product is 1. In this case, \( 4 \times \frac{1}{4} = 1 \). Therefore, \( \frac{1}{4} \) is indeed the reciprocal of 4.

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation that represents the process of adding a number to itself a certain number of times. Consider it as repeated addition. For example, when we multiply 4 by 3, it is the same as adding 4 three times:

• 4 + 4 + 4 = 12
• The operation is also written as: 4 × 3 = 12

The multiplier (in this case, 3) tells us how many times to take the number (4). Multiplication is important for understanding reciprocals because, when a number is multiplied by its reciprocal, the result is always 1. This property helps confirm that a fraction is correctly simplified in problems involving reciprocals.

Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers: a numerator and a denominator. The numerator is the top number, and it indicates how many parts we consider. The denominator is the bottom number and shows the total number of equal parts the whole is divided into.

• For instance, in the fraction \( \frac{1}{4} \), 1 is the numerator, and 4 is the denominator.
• This indicates one out of four equal parts.

Fractions are essential in understanding reciprocals. The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). This simply means you swap the numerator and the denominator. Knowing this property allows you to find the reciprocal of any fraction effortlessly, thus aiding in solving problems involving division and multiplication by fractions.

Division

Division is the process of splitting a number into equal parts. It is one of the basic operations in arithmetic and is closely related to multiplication.

• When you divide, you determine how many times a number can be subtracted from another number.
• For example, dividing 12 by 4 means asking how many times you can subtract 4 from 12, which is 3 times.

Division is essential to understanding reciprocals because finding a reciprocal involves dividing 1 by the given number. This operation reverses the multiplication process. Whenever you need to divide involving fractions, like finding the reciprocal, you're essentially asking how many parts of the divisor fit into the dividend, reinforcing the understanding of reciprocal relationships.