Problem 42 - Fundamentals of Mathematics

Question

What part of 3 is \(\frac{1}{5} ?\)

Step-by-Step Solution

Answer

\( \frac{1}{5} \) of 3 is \( \frac{1}{15} \).

Step 1: Understand the Problem

We need to find out what part of the number 3 is represented by \( \frac{1}{5} \). To find this, we need to determine what fraction of 3 corresponds to \( \frac{1}{5} \).

Step 2: Set Up the Equation

We are looking for a number \( x \) such that \( x \times 3 = \frac{1}{5} \). The number \( x \) will represent the part of 3 that is \( \frac{1}{5} \).

Step 3: Solve for x

To solve for \( x \), rearrange the equation \( x \times 3 = \frac{1}{5} \) to get \( x = \frac{1}{5} \div 3 \). This is equivalent to multiplying \( \frac{1}{5} \) by the reciprocal of 3: \( x = \frac{1}{5} \times \frac{1}{3} \).

Step 4: Calculate the Value

Now calculate \( x = \frac{1}{5} \times \frac{1}{3} = \frac{1}{15} \). This tells us that \( \frac{1}{15} \) is the part of 3 that corresponds to \( \frac{1}{5} \).

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. Think of a fraction as a cut-up pizza, where each slice is one piece of the whole pizza. A fraction has two parts:

• The numerator, which is the top number and represents the number of pieces you have.
• The denominator, which is the bottom number and tells you how many equal pieces the whole is divided into.

Consider the fraction \( \frac{1}{5} \). Here, '1' is the numerator and denotes one part, while '5' is the denominator, indicating the whole is divided into five parts. So, \( \frac{1}{5} \) represents one out of five equal parts of something. Understanding fractions is crucial because they allow you to express quantities less than one, as well as to multiply and divide different numbers effectively. When performing operations with fractions, like multiplication, maintain the denominator while adjusting the numerator as required.

Introduction to Reciprocal

The reciprocal of a number is just flipping it over. For any whole number \( a \), the reciprocal is \( \frac{1}{a} \). For example, the reciprocal of \( 3 \) is \( \frac{1}{3} \).

• Reciprocals are useful in division, as dividing by a number is the same as multiplying by its reciprocal.
• In the context of fractions, reciprocals play a key role in division as you transform the problem into a multiplication problem.

When you need to divide by a fraction, say \( \frac{1}{5} \div 3 \), you multiply by the reciprocal of \( 3 \), which is \( \frac{1}{3} \). This changes the division into multiplication: \( \frac{1}{5} \times \frac{1}{3} \). Reciprocals add simplicity to complex arithmetic operations, especially when dealing with fractional numbers.

Decoding Mathematical Expressions

Mathematical expressions are combinations of numbers, symbols, and operators that represent a mathematical relationship or problem. When dealing with expressions like finding part of one number that corresponds to another, it's crucial to set up the relationship clearly.

• Use variables like \( x \) to represent unknown parts of the expression.
• Structure equations to reflect the problem, such as \( x \times 3 = \frac{1}{5} \) for our original exercise.
• Manipulating the expression involves rearranging it to isolate the variable and making calculations simpler.

In our exercise, solving the equation \( x \times 3 = \frac{1}{5} \) leads us to find \( x \) by dividing \( \frac{1}{5} \) by 3, or equivalently multiplying by the reciprocal of 3. Mathematical expressions help in organizing and simplifying problems into solvable steps. They allow us to find relationships among numbers and use operations like multiplication and division efficiently to find solutions.