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Cross multiplication is a key method in solving proportions effectively. If you have a proportion in the form of \( \frac{a}{b} = \frac{c}{d} \), cross multiplication involves multiplying across the equals sign. Specifically, you multiply the numerator of the first fraction by the denominator of the second one, and the denominator of the first fraction by the numerator of the second one. This gives us:

• \( a \times d = b \times c \)

After cross multiplying, you're left with an equation that no longer has fractions, making it simpler to solve.
In the original problem, \( \frac{x}{3} = \frac{9}{10} \), cross multiplying gives \( x \times 10 = 3 \times 9 \) or \( 10x = 27 \). This equation is much easier to handle without the fractions, guiding us smoothly towards finding the solution for \( x \).

Once you have an equation like \( 10x = 27 \), the goal is to solve for the unknown variable \( x \). Isolating the variable means getting \( x \) alone on one side of the equation. This involves performing operations that "undo" any actions applied to the variable.
For \( 10x = 27 \), \( x \) is being multiplied by 10. To isolate \( x \), we divide both sides of the equation by 10:

• \( 10x \div 10 = 27 \div 10 \)
• This simplifies to \( x = \frac{27}{10} \)

This step is crucial because once \( x \) is isolated, you've found the value of \( x \) that satisfies the original proportion.

After isolating \( x \), you might end up with a fractional solution, as we did with \( \frac{27}{10} \). Simplifying fractions is about making sure your final answer is in the simplest form.
A fraction is considered simplified when the numerator and the denominator have no common factors other than 1. In our case, 27 and 10 do not share any common divisors besides 1, which means \( \frac{27}{10} \) is already simplified.
By always checking for common factors, you ensure that your solution is as straightforward and clean as possible, making it easier for anyone to understand.