Problem 45 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{9}{5} \div 3$$

Step-by-Step Solution

Answer

The value is \( \frac{3}{5} \).

Step 1: Understand Division by a Fraction

To divide a fraction by a whole number, recognize that division can be expressed as multiplication by the reciprocal. This means that \( \frac{9}{5} \div 3 \) can be rewritten as \( \frac{9}{5} \times \frac{1}{3} \).

Step 2: Multiply Fractions

Multiply the fractions obtained in the previous step: \( \frac{9}{5} \times \frac{1}{3} \). This involves multiplying the numerators together and the denominators together: \( \frac{9 \times 1}{5 \times 3} = \frac{9}{15} \).

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{9}{15} \) by finding the greatest common divisor (GCD) of 9 and 15, which is 3. Then divide the numerator and the denominator by 3 to get \( \frac{3}{5} \).

Step 4: Confirm the Simplification

Ensure that the fraction \( \frac{3}{5} \) is in its simplest form. Since the GCD of 3 and 5 is 1, no further simplification is possible.

Key Concepts

Multiplication by Reciprocal

When dividing a fraction by a number, it's often easier to think of the process as multiplying by the reciprocal instead. The reciprocal of a number is simply 1 divided by that number. For instance, the reciprocal of 3 is \( \frac{1}{3} \).
So, when you have a problem like \( \frac{9}{5} \div 3 \), it can be reinterpreted as \( \frac{9}{5} \times \frac{1}{3} \). This change transforms the division into a straightforward multiplication problem, which is normally simpler to calculate.

• Find the reciprocal of the divisor.
• Convert the division into multiplication using the reciprocal.

Simplifying Fractions

After multiplying the fractions together, you might end up with a result that needs simplifying. Simplifying fractions involves reducing fractions to their smallest possible terms while maintaining equivalence.
For example, after multiplying \( \frac{9}{5} \times \frac{1}{3} \), you get \( \frac{9}{15} \). This fraction can be reduced to a simpler form for ease of interpretation. Simplifying not only makes the fractions prettier but also easier to use in further calculations.

• Check if both the numerator and the denominator have common factors.
• Divide both by the greatest common factor to simplify.

Greatest Common Divisor

The greatest common divisor (GCD), is vital for simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. For the fraction \( \frac{9}{15} \), the GCD is 3.
To simplify, divide both 9 and 15 by their GCD of 3, which gives you the fraction \( \frac{3}{5} \). Always check for the GCD first to ensure you simplify the fraction correctly.

• Identify common factors shared by the numerator and denominator.
• Choose the largest factor common to both as the GCD.

This GCD approach ensures that the fraction is reduced to its simplest form effectively and efficiently.