Problem 31 - Fundamentals of Mathematics

Question

\(\frac{3}{4}\) of what number is \(\frac{3}{4} ?\)

Step-by-Step Solution

Answer

The number is 1.

Step 1: Understanding the Problem

We need to find a number such that when we take \( \frac{3}{4} \) of it, we get \( \frac{3}{4} \). We can call this number \( x \).

Step 2: Setting Up the Equation

The problem can be written in equation form as \( \frac{3}{4} \times x = \frac{3}{4} \).

Step 3: Solving for the Unknown Number

To solve for \( x \), we divide both sides of the equation by \( \frac{3}{4} \): \[ x = \frac{3}{4} \div \frac{3}{4}\] This simplifies to \( x = 1 \), because any non-zero number divided by itself equals 1.

Key Concepts

Equation Solving

Equation solving is a key part of many math problems. In simple terms, solving an equation means finding the value of the unknown variable. The unknown variable is represented by a letter, commonly called \(x\).
To set up the equation, you need to translate the problem into a mathematical sentence. This sentence often involves multiplication, division, addition, or subtraction.
For our given problem, the equation was \(\frac{3}{4} \times x = \frac{3}{4}\). Here, we are asked to solve for \(x\) such that when \(\frac{3}{4}\) is multiplied by \(x\), the result is \(\frac{3}{4}\). This leads us to the next important step: solving the equation itself.

Mathematical Problem Solving

Mathematical problem-solving involves a systematic process to understand and tackle mathematical challenges. It starts by carefully reading the problem, ensuring that you know what is being asked.
Often, mathematical problem-solving requires setting up an equation and solving it, just like in this problem where we had to find a number \(x\) such that \(\frac{3}{4}\) of \(x\) gives us \(\frac{3}{4}\). Translating words into an equation is a crucial skill. It involves identifying the operation (like multiplication or division) inherent in the problem.
In this case, understanding that "\(\frac{3}{4}\) of what number is \(\frac{3}{4}\)?" translates to multiplying \(\frac{3}{4}\) by a number to get \(\frac{3}{4}\). Mathematically arranging these relationships accurately leads to a successful problem-solving process. It's also important to verify your answer to see if it makes sense in the context of the problem.

Division of Fractions

Division of fractions is a handy operation that is easier than it might sound. To divide fractions, you multiply by the reciprocal. The reciprocal of a fraction is simply switching its numerator and denominator.
Consider our example, where we needed to solve \(x = \frac{3}{4} \div \frac{3}{4}\). Here, \(\frac{3}{4}\) divided by itself consolidates to multiplying by its reciprocal, which is \(\frac{4}{3}\), though in this specific instance, division or reciprocal isn't necessary for calculation since it's dividing a number by itself resulting in 1.
Generally, to solve \(\frac{a}{b} \div \frac{c}{d}\), you would compute \(\frac{a}{b} \times \frac{d}{c}\). This often results in simplifying and sometimes results in a whole number or a simplified fraction. Understanding the division of fractions is crucial for successfully handling equations and other operations involving fractions.