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Cross-multiplication is one of the most powerful tools for solving proportions. A proportion is an equation that shows two ratios are equal. It typically appears in the form \( \frac{a}{b} = \frac{c}{d} \). To solve such an equation, cross-multiply by multiplying the numerator of the first ratio by the denominator of the second, and vice versa.

For example, given a proportion \( \frac{m}{9} = \frac{10}{45} \), cross-multiplying would involve:

• Multiplying \( m \times 45 \)
• And multiplying \( 9 \times 10 \)

This results in the equation \( 45m = 90 \). Cross-multiplying helps clear out the denominators, transforming the original proportion into a more manageable linear equation.

It's crucial to remember that cross-multiplication is applicable only when you are dealing with a true proportion, and it helps in eliminating the denominator to solve for the unknown variable.

Simplifying fractions is a key mathematical skill that involves reducing a fraction to its simplest form. This means rewriting the fraction so that the numerator and denominator are as small as possible, without changing the fraction's value.

Consider the fraction \( \frac{90}{45} \) that results from solving \( 45m = 90 \). To simplify, we look for the greatest common factor (GCF) that divides both 90 and 45 evenly. The GCF here is 45.

• Divide both 90 and 45 by their GCF, 45.
• This gives \( \frac{90 \div 45}{45 \div 45} = \frac{2}{1} \).

So, \( m = 2 \).

Simplifying fractions makes calculations easier and results clearer. Practice by always looking for common factors. Reducing fractions is all about making them as straightforward as possible.

An algebraic equation forms the bridge between arithmetic operations and algebraic thinking. When solving proportions, translating the given statement into an algebraic equation is crucial.

In the problem \( \frac{m}{9} = \frac{10}{45} \), after cross-multiplying, we encounter the equation \( 45m = 90 \). This is a linear equation because it involves only one variable \( m \).

• The objective is to isolate \( m \).
• Divide both sides by 45.
• This results in \( m = \frac{90}{45} = 2 \).

The goal is to rearrange the equation to solve for the unknown variable. Understanding how to handle algebraic equations enables you to approach complex problems systematically, breaking them down into smaller, solvable parts.