Problem 24 - Fundamentals of Mathematics

Question

\(\frac{5}{12}\) of what number is \(\frac{5}{6} ?\)

Step-by-Step Solution

Answer

The number is 2.

Step 1: Define the Unknown Number

Let the unknown number be represented by \( x \). We are trying to find the value of \( x \) such that \( \frac{5}{12} \times x = \frac{5}{6} \).

Step 2: Set Up the Equation

Using the given information, we set up the equation \( \frac{5}{12} \times x = \frac{5}{6} \).

Step 3: Solve for x

To clear the fraction on the left side, multiply both sides of the equation by the reciprocal of \( \frac{5}{12} \), which is \( \frac{12}{5} \). This gives us \( x = \frac{5}{6} \times \frac{12}{5} \).

Step 4: Simplify the Expression

Simplify the expression: \( x = \frac{5 \times 12}{6 \times 5} \). Cancel the \( 5 \) in the numerator and denominator, which reduces to \( x = \frac{12}{6} \).

Step 5: Simplify the Fraction

Simplify \( \frac{12}{6} \) to its simplest form, resulting in \( x = 2 \).

Key Concepts

Equations with Fractions

When dealing with equations involving fractions, you are often asked to find a missing value. This missing value is typically represented by a variable, such as \(x\). In these problems, the fraction is multiplied by this unknown number, leading to an equation. The key is to isolate the variable.

• Step 1: Identify the Unknown. Start by defining what you need to find. In the exercise, the unknown number is \(x\).
• Step 2: Set up the Equation. Form an equation that represents the problem statement, e.g., \(\frac{5}{12} \times x = \frac{5}{6}\).
• Step 3: Solve for the variable. This typically involves clearing the fraction by performing similar operations on both sides.

Understanding how to manipulate such an equation is crucial for solving it successfully. Remember that the main goal is to have \(x\) by itself on one side of the equation.

Simplifying Fractions

Simplifying fractions is crucial in making calculations easier and more understandable. It involves reducing a fraction to its simplest form while maintaining the same value.Let's see how you simplify a fraction:

• Check for common factors in the numerator and the denominator.
• Divide both the numerator and the denominator by their greatest common divisor (GCD).

For example, in solving \(x = \frac{12}{6}\), both 12 and 6 are divisible by 6, which is their GCD. Dividing both by 6 simplifies it to \(x = 2\).Simplifying makes fractions easier to work with and understand, especially in complex equations or when you need to perform further calculations.

Reciprocal of a Fraction

The reciprocal of a fraction is what you get when you swap the numerator and the denominator. It is particularly useful in solving equations involving fractions.Here's why:

• Multiplying a fraction by its reciprocal results in 1, which simplifies the equation significantly.
• To clear fractions in an equation, you can multiply both sides by the reciprocal. For example, multiply by \(\frac{12}{5}\) to isolate \(x\) in \(\frac{5}{12} \times x = \frac{5}{6}\).

Remember, multiplying by the reciprocal clears the fraction on one side, effectively isolating the variable. This makes solving equations straightforward and ensures accuracy in your calculations.