Problem 1 - Fundamentals of Mathematics

Question

Perform the following multiplications. $$ \frac{2}{5} \cdot \frac{1}{6} $$

Step-by-Step Solution

Answer

\( \frac{1}{15} \)

Step 1: Multiply the Numerators

When multiplying fractions, start by multiplying the numerators (the top numbers of the fraction). For the fractions \( \frac{2}{5} \) and \( \frac{1}{6} \), multiply \(2\) by \(1\) to get the new numerator. \[2 \times 1 = 2\]

Step 2: Multiply the Denominators

Next, multiply the denominators (the bottom numbers of the fraction). For the fractions \( \frac{2}{5} \) and \( \frac{1}{6} \), multiply \(5\) by \(6\) to get the new denominator. \[5 \times 6 = 30\]

Step 3: Write the Resulting Fraction

Combine the results of Steps 1 and 2 to form the fraction. The product of the numerators is the new numerator and the product of the denominators is the new denominator. \[\frac{2}{30}\]

Step 4: Simplify the Fraction

Simplify the fraction \( \frac{2}{30} \). Find the greatest common divisor (GCD) of 2 and 30, which is 2. Divide both the numerator and the denominator by their GCD. \[\frac{2 \div 2}{30 \div 2} = \frac{1}{15}\]

Key Concepts

Understanding Numerators

When we talk about fractions, the numerator is the number on the top. It's the part of the fraction that tells us how many parts we have out of the whole. For example, in the fraction \( \frac{2}{5} \), the numerator is 2, meaning you have 2 parts out of 5.In fraction multiplication, you start by multiplying the numerators together. For the exercise \( \frac{2}{5} \times \frac{1}{6} \), you multiply the numerators, 2 and 1. This gives you a new numerator:

• Multiply: \( 2 \times 1 \)
• Result: \( 2 \)

This step is simple but crucial. Always start with the numerators to begin forming your resulting fraction.

The Role of Denominators

Denominators are the bottom numbers of a fraction, and they tell us into how many parts the whole is divided. In the fraction \( \frac{2}{5} \), 5 is the denominator, indicating the whole is divided into 5 parts.When multiplying fractions, the second step involves multiplying the denominators. In our example, this means taking 5 from \( \frac{2}{5} \) and multiplying it by 6 from \( \frac{1}{6} \):

• Multiply: \( 5 \times 6 \)
• Result: \( 30 \)

The importance of this step is in determining the size of the resulting parts from the fraction multiplication. The larger the denominator, the smaller each part is.

Simplifying Fractions: Making It Easy to Interpret

Simplifying fractions means reducing the fraction to its simplest form. It's a crucial part of fraction multiplication as it makes the result cleaner and easier to understand.In the given exercise, after multiplying the numerators and the denominators, we got \( \frac{2}{30} \). To simplify this, find the greatest common divisor (GCD) of 2 and 30. The GCD in this case is 2. Then, divide both the numerator and the denominator by this GCD:

• Divide numerator: \( 2 \div 2 = 1 \)
• Divide denominator: \( 30 \div 2 = 15 \)

This gives us the simplified fraction \( \frac{1}{15} \). Simplifying ensures that the fraction is presented in its most reduced form, making it more straightforward to use in further calculations or comparisons.