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Cross-multiplication is a powerful tool for checking if two fractions are equivalent. It involves multiplying the numerator (the top number) of one fraction by the denominator (the bottom number) of the other fraction, and vice versa. This method allows us to quickly determine whether two ratios, or proportions, are equal.

For example, consider the proportions \( \frac{50}{35} \) and \( \frac{10}{7} \). Through cross-multiplication, we multiply the numbers diagonally: 50 times 7 and 35 times 10.

• Calculate \( 50 \cdot 7 \) which gives 350.
• Calculate \( 35 \cdot 10 \) which also results in 350.

When these two products are equal, as they are in this case (350 = 350), it shows that the fractions \( \frac{50}{35} \) and \( \frac{10}{7} \) are equivalent. Thus, the given proportion is true.

Equivalent fractions are different fractions that represent the same value or portion of a whole. They have different numerators and denominators, but when simplified, they express the same number. For example, \( \frac{50}{35} \) and \( \frac{10}{7} \) are equivalent fractions.

To find equivalent fractions, you can simplify or reduce a fraction to its lowest terms. Alternatively, you can expand a fraction by multiplying the numerator and the denominator by the same number.

Let's dive into the equivalence of \( \frac{50}{35} \):

• First, divide both the numerator and the denominator by their greatest common divisor (GCD), which is 5.
• Simplify to get \( \frac{50 \div 5}{35 \div 5} = \frac{10}{7} \).

Both \( \frac{50}{35} \) and \( \frac{10}{7} \) express the same amount, showing their equivalence without altering their inherent value.

Ratio comparison involves examining two ratios to see if they express the same relationship. This is especially useful in situations where you are comparing parts, quantities, or measurements in mathematics or everyday situations.

When given two ratios in the form of fractions, such as \( \frac{50}{35} \) and \( \frac{10}{7} \), the goal is to determine if these fractions represent the same relationship.

There are two primary methods to compare these ratios:

• Simplification: Simplify both ratios to see if they reduce to the same fraction.
• Cross-Multiplication: Use cross-multiplication to check the equality of the cross products, as demonstrated previously.

By finding a common simplified form or by confirming that the cross products are equal, you can confidently state that two ratios are equivalent. Such comparisons are vital in understanding proportional relationships and ensuring consistency across various applications.