Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{2}{5} \times \frac{1}{6}=\) ______

Short answer

The product is \( \frac{1}{15} \).

Step-by-step solution

Multiply the Numerators

To multiply fractions, start by multiplying the numerators (the top numbers of each fraction) together. Here it's 2 and 1. So, calculate: \[ 2 \times 1 = 2 \]

Multiply the Denominators

Next, multiply the denominators (the bottom numbers of each fraction) together. Here it's 5 and 6. So, calculate: \[ 5 \times 6 = 30 \]

Form the New Fraction

Now, take the result of Step 1 and Step 2 to form a new fraction. The result will be the fraction with the product of numerators as its numerator and the product of denominators as its denominator.\[ \frac{2}{30} \]

Reduce to Lowest Terms

We need to simplify the fraction \( \frac{2}{30} \). To do this, find the greatest common divisor (GCD) of 2 and 30, which is 2. Divide both the numerator and the denominator by their GCD to reduce.\[ \frac{2 \div 2}{30 \div 2} = \frac{1}{15} \]

Key concepts

Understanding the Numerator

In a fraction, the numerator is the number above the line. It tells you how many parts of the whole you're dealing with. When multiplying fractions, the numerators are combined by straightforward multiplication. In the example

• Given fractions: \( \frac{2}{5} \) and \( \frac{1}{6} \)
• Numerators: 2 and 1

To begin, multiply the numerators: \( 2 \times 1 = 2 \). This new value becomes the numerator of the product fraction. Simple, right? Just focus on the top numbers and multiply!

Decoding the Denominator

The denominator in a fraction is found under the line. It tells you into how many equal parts the whole is divided. Like the numerator, when multiplying fractions, the denominators are combined by multiplication. Considering our example:

• Given fractions: \( \frac{2}{5} \) and \( \frac{1}{6} \)
• Denominators: 5 and 6

Multiply them: \( 5 \times 6 = 30 \). This product becomes the new fraction's denominator. Remember, while the numbers seem to get bigger, the job of the denominator is only to tell us about the partition of the whole, not necessarily the value itself.

Simplifying Fractions: A Key Step

Simplifying fractions means making the numbers smaller without changing the value of the fraction. This is essential to give you the simplest form of a fraction, making it easily readable or comparable to others. Once you’ve found the fraction by multiplying numerators and denominators, it’s time to clean up:

• Initial fraction: \( \frac{2}{30} \)

By simplifying, you can now express this as a simpler fraction. But how? You do this by eliminating any common factors between the numerator and the denominator.

The Role of the Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving any remainder. When simplifying fractions, finding the GCD is vital as it helps bring the fraction to its simplest form. For the fraction \( \frac{2}{30} \):

• Determine the factors of 2 (1, 2) and 30 (1, 2, 3, 5, 6, 10, 15, 30)
• Common factor: 2

Use the GCD to divide both the numerator and the denominator:\[\frac{2 \div 2}{30 \div 2} = \frac{1}{15}\] This results in the simplest form of the product fraction. Simplifying fractions helps in calculations and makes results neater and easier to understand.