Problem 32 - Fundamentals of Mathematics

Question

\(\frac{8}{11}\) of what number is \(\frac{8}{11} ?\)

Step-by-Step Solution

Answer

The number is 1.

Step 1: Set up the equation

Translate the problem into a mathematical equation. We know that \( \frac{8}{11} \) of a certain number is equal to \( \frac{8}{11} \). Let's call that number \( x \). So, the equation becomes: \[ \frac{8}{11} \times x = \frac{8}{11} \]

Step 2: Simplify the equation

To solve for \( x \), we need to isolate it on one side of the equation. Divide both sides of the equation by \( \frac{8}{11} \). This results in: \[ x = \frac{8}{11} \div \frac{8}{11} \] This simplifies to: \[ x = 1 \] because any number divided by itself equals 1, given it's not zero.

Key Concepts

Equation Solving

In the world of mathematics, solving equations is a process of finding the value of unknown variables that make the equation true. An equation consists of two mathematical expressions set equal to each other, often involving a variable, which is the unknown number we aim to solve for.

To solve an equation like the one in this exercise, \[\frac{8}{11} \times x = \frac{8}{11},\]we want to figure out the value of \(x\) that makes the statement true. The term \(\frac{8}{11}\) is being multiplied by \(x\). To solve for \(x\), we utilize inverse operations, which in this case involves division to isolate the variable.

Ultimately, solving equations helps us translate real-world problems into mathematical language, allowing for structured analysis and solution.

Simplification Techniques

Simplification is an essential skill in mathematics, and it involves reducing expressions or equations to their simplest form. This process makes it easier to work with complex expressions.

In the given equation, to solve for \(x\), we divide both sides by \(\frac{8}{11}\):\[x = \frac{8}{11} \div \frac{8}{11}.\]The division of a number by itself is a fundamental simplification rule, leading the expression to equal \(1\), provided the number is not zero. Thus, \(x = 1\) is the simplified result, showing us that the number we were seeking in the equation precisely matches the value needed to balance the equation.

By applying simplification techniques, complex mathematical processes become manageable, often reducing cumbersome numbers and expressions to their most basic form.

Multiplication and Division of Fractions

Understanding how to multiply and divide fractions is crucial for tackling fraction problems effectively. Let's briefly explore these operations.

**Multiplication of Fractions:**

• To multiply fractions, multiply the numerators (top numbers) together to get the new numerator.
• Then multiply the denominators (bottom numbers) together to get the new denominator.
• For example, \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\).

**Division of Fractions:**

• Dividing fractions involves multiplying by the reciprocal of the divisor. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
• For instance, dividing \(\frac{a}{b}\) by \(\frac{c}{d}\) means you multiply \(\frac{a}{b}\) by \(\frac{d}{c}\).
• So, \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\).

In the original exercise, this technique is used when dividing \(\frac{8}{11}\) by itself to solve the equation and find that \(x\) equals 1. Recognizing and applying multiplication and division of fractions can significantly simplify and solve many problems involving fractions.