Problem 6 Work each problem.Only one of th... [FREE SOLUTION] (2024)

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Fractions often need to be simplified before they can be added or subtracted. This is where the concept of a common denominator comes into play.
A common denominator is a common multiple of the denominators of two or more fractions. It allows us to convert the fractions so they have the same denominator, making it easier to perform operations on them.
For example, to add \(\frac{1}{3}\) and \(\frac{1}{12}\), we find their common denominator. The denominators are 3 and 12, so the smallest common denominator is 12. We convert \(\frac{1}{3}\) to \(\frac{4}{12}\), making the fractions \(\frac{4}{12}\) and \(\frac{1}{12}\). Now, addition is straightforward: \(\frac{4}{12} + \frac{1}{12} = \frac{5}{12}\).
Remember:

• Find the least common multiple (LCM) of the denominators.
• Convert each fraction to an equivalent fraction with the common denominator.
• Perform the intended operation (add, subtract, etc.).

Understanding the parts of a fraction is crucial. Every fraction consists of a numerator and a denominator.
The numerator is the number above the fraction line. It represents the 'part' of the whole. For instance, in the fraction \(\frac{3}{4}\), 3 is the numerator, meaning we have 3 parts out of a total of 4.
The denominator is the number below the fraction line. It signifies the total number of equal parts the whole is divided into. In \(\frac{3}{4}\), 4 is the denominator, indicating the whole is divided into 4 smaller parts.
Fractions can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, consider \(\frac{20}{36}\). The GCD of 20 and 36 is 4, so we divide both by 4:
\(\frac{20 ÷ 4}{36 ÷ 4} = \frac{5}{9}\). This gives us the simplified fraction \(\frac{5}{9}\). Remember:

• Numerator: Part of the whole (top number).
• Denominator: Total parts of the whole (bottom number).
• Simplify by dividing by the GCD.

The reciprocal of a fraction is what you get when you flip the numerator and the denominator. Knowing how to find and use reciprocals is important, especially when dividing fractions.
To divide by a fraction, you multiply by its reciprocal. For example, to divide \(\frac{5}{12}\) by \(\frac{3}{4}\), you multiply \(\frac{5}{12}\) by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\). So, \(\frac{5}{12} ÷ \frac{3}{4} = \frac{5}{12} × \frac{4}{3} = \frac{20}{36}\). Simplify \(\frac{20}{36}\) by dividing by the GCD of 20 and 36, which is 4:
\(\frac{20 ÷ 4}{36 ÷ 4} = \frac{5}{9}\).The steps are:

• Find the reciprocal of the divisor fraction.
• Multiply the dividend fraction by this reciprocal.
• Simplify the resulting fraction, if possible.

This process helps streamline operations with fractions and makes dividing them straightforward.