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Proportions are equations that express the equality of two ratios or fractions. When faced with a question like 'What percent is 21 out of 60?', we are comparing two quantities to find their ratio in terms of 100, as percentage is a ratio per 100. This is essentially a proportional relationship between the part (21) and the whole (60).

Proportion problems typically have four terms: the numerator of the first ratio, the denominator of the first ratio, the numerator of the second ratio, and the denominator of the second ratio (often 100 in percentage problems). It's crucial to place these terms in the correct order to maintain the consistency of the relationship, ensuring that the numerators and denominators represent the same types of quantities respectively.

Cross-multiplication is a key step in solving proportion problems. This method involves multiplying the numerator of one ratio by the denominator of the other ratio across the equation, effectively 'crossing' over.

The power of cross-multiplication lies in its ability to clear fractions and isolate the variable, making it easier to solve the equation. After setting up your proportion like \frac{x}{100} = \frac{21}{60}\, cross-multiplication would yield \(60x = 21 \times 100\). This simple step translates our proportion into a standard linear equation with the variable, \(x\), on one side, ready to be solved.

Benefits of Cross-Multiplication

• It simplifies the equation to solve for the unknown.
• Eliminates fractions, which are often a source of errors.
• Provides a visual and logical method to understand the relationship between the ratios.

The final step after cross-multiplication is solving the equation for the unknown variable, which, in our case, is the percentage value we are trying to find. To do this, we perform operations that will isolate the variable on one side of the equation.

From our cross-multiplication step, we obtained \(60x = 2100\). To solve for \(x\), we divide both sides of the equation by 60, giving us \(x = 35\). Here, we've successfully isolated \(x\), determining that 21 is 35% of 60.

Tips for Efficient Equation Solving

• Always perform the same operation on both sides of the equation to maintain equality.
• Break down complex operations into simpler steps if necessary.
• Check your solution by substituting the value back into the original equation to verify the correctness.