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Understanding percentages is crucial to solving many real-world problems, including those involving discounts, interest rates, and many types of calculations in various fields. A percentage is a way of expressing a number as a fraction of 100. It is denoted using the percent sign (%). For instance, 60% represents 60 out of 100, or simply 60/100 when written as a fraction.

When solving percentage problems, it’s important to remember that the percentage tells us how many parts per hundred we're talking about. Relating this to the given problem where Karen puts 60% of her paycheck into her savings, we can understand it as for every 100 dollars earned, she is saving 60 dollars. In mathematical terms, this is written as \( \frac{60}{100} \) of her paycheck. By turning the percentage into a decimal or fraction, it allows for further mathematical operations, like calculating specific amounts or adjusting proportions.

When it comes to personal finance, being able to calculate savings is a fundamental skill. Saving a fraction of earnings or income is a common practice, which can be easily understood using basic proportion concepts. Let's apply this to the Karen's scenario: she saves 60% of her paycheck. This percentage can be converted into a proportion, illustrating the part of the paycheck that is saved versus the whole paycheck amount.

To find out the total amount from the known savings, as seen in the exercise, we look at the amount saved (\( \$120 \) in this case) as part of the whole paycheck. This is expressed in algebraic terms by setting up a proportion, as \( \frac{60}{100} = \frac{120}{x} \) where \( x \) represents the total paycheck amount. Here, the 60 represents the savings, 100 is the whole paycheck expressed as 100%, and \( x \) is the variable we need to solve for. It's a practical real-world application that emphasizes the importance of percentages in budgeting and financial planning.

At its heart, algebra involves using letters and symbols to represent numbers and quantities in formulae and equations. The exercise with Karen's paycheck is a simple illustration of how algebra is used to find an unknown value through the use of variables and proportions.

To solve for \( x \) which in this case is Karen's total paycheck, we use cross-multiplication to transform the proportion into an equation: \( 60x = 100 \cdot 120 \) followed by straightforward arithmetic operations—multiplying and dividing—to isolate \( x \) and find its value. In the final step, by dividing 12000 by 60, we determine that \( x = \$200 \) which completes the algebraic solution. This exercise shows how basic algebra is essential to solve for unknowns and is applicable in various everyday contexts, not just in academic settings.