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When working with fractions, finding a common denominator is often essential. A common denominator is a shared multiple of the denominators of two or more fractions. For example, for the fractions \(\frac{7}{8}\) and \(\frac{1}{16}\), the denominators are 8 and 16. Their least common multiple is 16.
By converting fractions to have the same denominator, they become easier to compare and compute. In our exercise, we converted \(\frac{7}{8}\) to \(\frac{14}{16}\) to match \(\frac{1}{16}\).
Remember, always multiply both the numerator and the denominator by the same number to find the equivalent fraction. For instance, \(\frac{7}{8}\) becomes \(\frac{7 \times 2}{8 \times 2} = \frac{14}{16}\). This keeps the fraction's value the same while having a common denominator.

Dividing fractions might seem complicated, but a neat trick makes it simple: multiply by the reciprocal. The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).
In our exercise, we needed to divide \(\frac{14}{16}\) by \(\frac{1}{16}\). This is done by multiplying \(\frac{14}{16}\) by the reciprocal of \(\frac{1}{16}\), which is \(\frac{16}{1}\):
\( \frac{14}{16} \times \frac{16}{1} = 14 \).
This operation demonstrates that dividing by a fraction is the same as multiplying by its reciprocal. This simple rule transforms complex division into straightforward multiplication.

When solving math problems, following a structured approach helps immensely. Let's break down our exercise:

• Understand the Problem: Troy needs to hammer a \(\frac{7}{8}\)-inch nail entirely into the wood, with each strike pushing the nail \(\frac{1}{16}\) inch deeper.
• Convert to Common Denominator: Change \(\frac{7}{8}\) to \(\frac{14}{16}\) so it matches the denominator of \(\frac{1}{16}\).
• Set Up the Division: Divide the total depth (\