Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{1}{3} \times \frac{4}{12}=\) ______

Short answer

\(\frac{1}{9}\)

Step-by-step solution

Multiply the Numerators

To multiply fractions, begin by multiplying the numerators together. Here, multiply 1 by 4 to get:\[1 \times 4 = 4\]

Multiply the Denominators

Next, multiply the denominators together. Here, multiply 3 by 12 to get:\[3 \times 12 = 36\]

Form the Product Fraction

Combine the results from Steps 1 and 2 to form the fraction:\[\frac{4}{36}\]

Reduce the Fraction to Lowest Terms

To reduce the fraction \(\frac{4}{36}\), find the greatest common divisor (GCD) of 4 and 36, which is 4. Divide both the numerator and the denominator by 4:\[\frac{4 \div 4}{36 \div 4} = \frac{1}{9}\]

Confirm the Solution

Review the reduction to ensure all operations are correct. After confirming, you get the final answer:\[\frac{1}{9}\]

Key concepts

Numerator and Denominator

In any fraction, two essential parts make the whole: the numerator and the denominator. The numerator is the number on top, representing how many parts of the whole or set you are considering. On the other hand, the denominator, the number at the bottom, tells you into how many equal parts the whole is divided.

For instance, in the fraction \(\frac{1}{3}\), the numerator is 1, indicating one part is considered, while the denominator is 3, representing that the whole is divided into three equal parts. This structure of fractions is crucial when multiplying them.

• **Numerator:** Top part, indicating parts considered.
• **Denominator:** Bottom part, representing division of the whole.

Understanding these roles helps in the multiplication of fractions, where you multiply the numerators with each other and the denominators with each other to form a new fraction.

Reduction to Lowest Terms

Reduction to lowest terms, or simplifying a fraction, is the process of making a fraction as simple as possible. This involves finding a fraction equivalent to the original but with a smaller numerator and denominator. The smaller the numbers, the easier for us to work with them.

For example, if you have the fraction \(\frac{4}{36}\), you can reduce it by dividing both the numerator and denominator by their common divisor. In this case, dividing by 4 gives you the reduced fraction \(\frac{1}{9}\).

• **Why reduce?** It makes fractions easier to compare, add, subtract, and understand.
• **How to reduce?** Find the common factor and divide.

When fractions are reduced to their lowest terms, it brings clarity and simplicity to arithmetic operations involving fractions.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a key concept in reducing fractions to their lowest terms. It is the largest positive integer that divides each of the numbers in question without leaving a remainder. For fractions, finding the GCD of the numerator and denominator helps you determine how far you can simplify the fraction.

In the case of \(\frac{4}{36}\), the GCD is 4, and it helps you to see that both 4 and 36 can be equally divided by this number.

• **Finding GCD:** Look for the largest shared factor of both numbers.
• **Use of GCD:** Divide both the numerator and the denominator by the GCD to simplify.

Reducing fractions using the GCD ensures that you are left with the simplest form of the fraction, making it easier to understand and work with in further calculations.