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When working with fractions, finding the Least Common Multiple (LCM) of the denominators is crucial. The LCM is the smallest number that is a multiple of two or more numbers. It helps us combine fractions with different denominators by creating a common ground, so they can be easily added or subtracted.

For example, with the fractions \(\frac{1}{8}\) and \(\frac{1}{10}\), we need to find the LCM of 8 and 10. We look for the smallest number that both 8 and 10 can divide without leaving a remainder. Listing multiples can help:

• Multiples of 8: 8, 16, 24, 32, 40, 48...
• Multiples of 10: 10, 20, 30, 40, 50...

The smallest common multiple is 40. Hence, the LCM of 8 and 10 is 40. This allows us to write both fractions with a common denominator of 40 for easier addition.

Adding fractions requires a common denominator to combine the numerators effectively. Once we have the Least Common Multiple (LCM), as found above, we convert each fraction to an equivalent fraction with this denominator.

For example, converting \(\frac{1}{8}\) and \(\frac{1}{10}\) to have the LCM of 40, we get:

\[ \frac{1}{8} = \frac{1 \times 5}{8 \times 5} = \frac{5}{40} \]

and

\[ \frac{1}{10} = \frac{1 \times 4}{10 \times 4} = \frac{4}{40}\]

Now that both fractions have a common denominator, we can easily add them:

\[ \frac{5}{40} + \frac{4}{40} = \frac{5 + 4}{40} = \frac{9}{40} \]
Thus, the sum of \(\frac{1}{8}\) and \(\frac{1}{10}\) is \(\frac{9}{40}\).

The reciprocal of a number is simply flipping its numerator and denominator. For any fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This concept is particularly useful when solving equations involving fractions.

In our exercise, after adding the fractions, we have the equation:

\[ \frac{9}{40} = \frac{1}{t} \]

To solve for \ t \ in this equation, we take the reciprocal of both sides. This changes the equation to:

\[ t = \frac{40}{9}\]
By understanding the concept of reciprocals, we can easily isolate and solve for unknown variables in fraction equations.