Problem 5 - Fundamentals of Mathematics

Question

Find the reciprocal of each number. $$2 \frac{2}{7}$$ HINT: Write this number as an improper fraction first.

Step-by-Step Solution

Answer

The reciprocal of \(2 \frac{2}{7}\) is \(\frac{7}{16}\).

Step 1: Convert Mixed Number to Improper Fraction

To find the reciprocal of a mixed number, first convert it into an improper fraction. The mixed number given is \(2 \frac{2}{7}\). Convert it by multiplying the whole number part (2) by the denominator of the fractional part (7) and then add the numerator of the fractional part (2). This gives: \(2 \times 7 + 2 = 14 + 2 = 16\). Therefore, the improper fraction is \(\frac{16}{7}\).

Step 2: Find the Reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator. So, for the fraction \(\frac{16}{7}\), the reciprocal is \(\frac{7}{16}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are useful for representing quantities that are more than whole but less than the next whole number. For example, the mixed number \(2 \frac{2}{7}\) represents 2 whole parts and \(\frac{2}{7}\) of another part. This is read as 'two and two-sevenths'.

When working with mixed numbers, it’s often necessary to convert them to improper fractions for calculations such as multiplication, division, or finding reciprocals. The whole number and the fractional part combined make it easier to visualize quantities, but for precise calculations, the improper fraction form is more practical.

Improper Fractions

An improper fraction is a type of fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). This means the value of the fraction is equal to or greater than one. In our exercise, the mixed number \(2 \frac{2}{7}\) was converted to the improper fraction \(\frac{16}{7}\).

To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator.
• Add the result to the numerator of the fractional part.
• Place this sum over the original denominator.

For example, \(2 \frac{2}{7}\) becomes \(\frac{16}{7}\) because \(2 \times 7 + 2 = 16\). Understanding improper fractions is essential for advanced fraction operations and understanding how to take reciprocals.

Fractions

Fractions are a way to represent any number as parts of a whole. They consist of a numerator and a denominator. The numerator indicates how many parts you have, while the denominator shows the total number of equal parts that make up a whole. For instance, in \(\frac{16}{7}\), '16' is the numerator indicating parts, while '7' is the denominator indicating that the whole is divided into 7 equal parts.

Fractions fall into three categories:

• Proper Fractions: The numerator is less than the denominator, resulting in a value less than one. E.g., \(\frac{3}{4}\).
• Improper Fractions: The numerator is equal to or greater than the denominator, resulting in a value equal to or greater than one. E.g., \(\frac{16}{7}\).
• Mixed Numbers: A combination of a whole number and a fraction. E.g., \(2 \frac{2}{7}\).

Understanding these basics is crucial for successfully performing operations like addition, subtraction, multiplication, division, and finding reciprocals of fractions.