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Simplifying fractions is a key concept in solving many mathematical problems. It involves reducing a fraction to its simplest form. For example, let's take a fraction like \(\frac{\frac{1}{5}}{\frac{1}{10}}\). To simplify this, you multiply the numerator (top part) by the reciprocal of the denominator (bottom part). The reciprocal of \(\frac{1}{10}\) is 10 because reciprocals are what you multiply by to get 1. So, \(\frac{1}{5} \times 10 = 2\). The simplified fraction is therefore 2.

This step-by-step simplification can be applied to any fractions. Always start by finding the reciprocal of the denominator and multiply it with the numerator. This technique helps make the fractions easier to work with and is a crucial skill in algebra and higher mathematics.

The concept of a reciprocal is essential in many areas of math. A reciprocal of a number is simply one divided by that number. For example, the reciprocal of 5 is \(\frac{1}{5}\), and the reciprocal of \(\frac{1}{10}\) is 10. This is because when a number and its reciprocal are multiplied together, the result is always 1.

In the given exercise, we used the reciprocal to simplify the fraction. Specifically, we needed the reciprocal of \(\frac{1}{10}\) to solve \(\frac{\frac{1}{5}}{\frac{1}{10}}\). By multiplying \(\frac{1}{5}\) by 10 (the reciprocal of \(\frac{1}{10}\)), we simplified the expression down to 2. Understanding reciprocals can make many math problems more manageable and is a foundational concept in algebra.

Solving for a variable means finding the value of the unknown that makes the equation true. In our exercise, we rearranged and simplified our given equation to isolate the variable \(\text{x}\).

Start with the simplified equation: \2 = \frac{\frac{1}{10}}{\text{x}}\. To solve for \(\text{x}\), multiply both sides of the equation by \(\text{x}\) to get \2 \text{x} = \frac{1}{10}\. Next, isolate \(\text{x}\) by dividing both sides by 2. This gives us \ \text{x} = \frac{1}{20} \. Always remember to perform the same operation on both sides of the equation when solving for a variable. This keeps the equation balanced and helps you find the correct value for the variable.