Finding the Value of x in Equivalent Fractions

The problem asks us to find the value of 'x' in the equation where two fractions are stated to be equivalent: \( \frac{60}{75} = \frac{4}{x} \). This type of problem involves solving a proportion.

Understanding Equivalent Fractions and Proportions

Equivalent fractions represent the same part of a whole, even though they have different numerators and denominators. A proportion is an equation stating that two ratios (fractions) are equal. To solve for an unknown variable in a proportion, we can use methods like cross-multiplication or simplifying one of the fractions first.

Solving the Equivalent Fraction Problem

We are given the equation: \( \frac{60}{75} = \frac{4}{x} \)

Method 1: Using Cross-Multiplication

Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal to each other. In this case:

• Multiply 60 by x: \( 60 \times x \)
• Multiply 75 by 4: \( 75 \times 4 \)

Setting these products equal gives us the equation:

\( 60x = 75 \times 4 \)

Calculate the product on the right side:

\( 75 \times 4 = 300 \)

So the equation becomes:

\( 60x = 300 \)

To find 'x', divide both sides of the equation by 60:

\( x = \frac{300}{60} \)

Performing the division:

\( x = 5 \)

Method 2: Simplifying the Fraction First

We can simplify the fraction \( \frac{60}{75} \) before solving. Find the greatest common divisor (GCD) of 60 and 75. Both numbers are divisible by 5. \( 60 \div 5 = 12 \) and \( 75 \div 5 = 15 \). So, \( \frac{60}{75} = \frac{12}{15} \). Both 12 and 15 are divisible by 3. \( 12 \div 3 = 4 \) and \( 15 \div 3 = 5 \). So, \( \frac{12}{15} = \frac{4}{5} \).

Thus, the original equation \( \frac{60}{75} = \frac{4}{x} \) simplifies to:

\( \frac{4}{5} = \frac{4}{x} \)

In this simplified form, we can see that the numerators are equal (both are 4). For the fractions to be equivalent, their denominators must also be equal.

Therefore, \( x = 5 \).

Both methods yield the same result, confirming that the value of x is 5.

Summary of Calculation Steps

Revision Table: Key Concepts

Additional Information: Solving Proportions

Proportions are widely used in various fields, including scaling recipes, map reading, calculating ratios in science, and determining exchange rates. Understanding how to solve for an unknown in a proportion is a fundamental skill in mathematics.

Besides cross-multiplication and simplification, another way to think about proportions is finding the scaling factor. In the equation \( \frac{60}{75} = \frac{4}{x} \), we can see how the numerator changed from 60 to 4. \( 60 \div 15 = 4 \). This means the numerator was divided by 15. For the fractions to be equivalent, the denominator must also be divided by the same factor. So, \( x = 75 \div 15 \). Calculating this division, \( 75 \div 15 = 5 \). This also gives \( x = 5 \).

This scaling factor method is particularly useful when the numbers are easily divisible.