Problem 34 - Fundamentals of Mathematics

Question

\(\frac{2}{3}\) of what number is \(1 ?\)

Step-by-Step Solution

Answer

The number is \(\frac{3}{2}\).

Step 1: Define the Problem

First, we need to find the number such that \(\frac{2}{3}\) of it equals 1. This means if \(x\) is the number we are looking for, then we have the equation \(\frac{2}{3}x = 1\).

Step 2: Solve the Equation

To solve \(\frac{2}{3}x = 1\), we can eliminate the fraction by multiplying both sides by the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). Thus, \(x = 1 \times \frac{3}{2}\).

Step 3: Perform the Multiplication

Calculate \(x = \frac{3}{2}\). Since \(1\) times any number is that number itself, \(x = \frac{3}{2}\). This is the value of \(x\).

Key Concepts

Solving Equations

Solving equations involves finding the value of a variable that makes the equation true. In our exercise, the equation is \(\frac{2}{3}x = 1\). Here, \(x\) represents the unknown number we are trying to find.

To solve this equation, we need to "isolate" the variable \(x\). This means we perform operations that undo what is being done to \(x\) on both sides of the equation. Our goal is to have \(x\) by itself on one side of the equation with a number or expression on the other side. This helps us find the exact value of \(x\).

In many cases, like this one, equations have fractions. Fractions tell us that the number is being divided by or multiplied by something specific. Dealing with these fractions correctly is key to solving the equation efficiently.

Reciprocals

A reciprocal of a number or fraction is what you multiply it by to get a product of 1. For a simple fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).

In our equation \(\frac{2}{3}x = 1\), the number \(\frac{2}{3}\) is being multiplied by \(x\). To isolate \(x\), we use the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\).

• Multiplying both sides by \(\frac{3}{2}\) allows us to eliminate the fraction, making it easier to solve for \(x\).
• The process of multiplying by the reciprocal is critical for moving beyond fractions when solving equations.

Multiplication of Fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. Suppose you have two fractions: \(\frac{a}{b}\) and \(\frac{c}{d}\).

The product is \(\frac{ac}{bd}\). This straightforward step is crucial when dealing with equations involving fractions. In our example, we multiplied \(1\) by the reciprocal \(\frac{3}{2}\), since \(1\) can be thought of as \(\frac{1}{1}\).

• The multiplication of \(1 \times \frac{3}{2}\) simplifies to \(\frac{3}{2}\).
• Understanding multiplication of fractions is not only pivotal for solving equations but also for operations involving fractions in general.