Question

Find \(\frac{3}{11} \times \frac{x}{y}\)

Short answer

Answer: The product of \(\frac{3}{11}\) and \(\frac{x}{y}\) is \(\frac{3x}{11y}\).

Step-by-step solution

Identify the numerators and denominators

The numerators are 3 and x, and the denominators are 11 and y.

Multiply the numerators

Multiply 3 and x together: \(3 \times x = 3x\)

Multiply the denominators

Multiply 11 and y together: \(11 \times y = 11y\)

Combine numerators and denominators

Now we combine the numerators and denominators to get the product of the two fractions: \(\frac{3x}{11y}\) So, \(\frac{3}{11} \times \frac{x}{y} = \frac{3x}{11y}\).

Key concepts

Numerators and Denominators

Understanding numerators and denominators is key to mastering fraction multiplication. A fraction consists of two main components:

• The **numerator**, which is the number on top of the fraction. It signifies how many parts of the whole are being considered.
• The **denominator**, which is the number at the bottom. This indicates how many parts the whole is divided into.

In the expression \(\frac{3}{11} \times \frac{x}{y}\), the **numerators** are \(3\) and \(x\), because they are the numbers positioned above the division line in each fraction. On the other hand, \(11\) and \(y\) are the **denominators**. This categorization is significant as it helps us perform specific operations—like multiplication—correctly with fractions.
Recognizing the numerators and denominators allows you to separate the problem into manageable pieces when multiplying fractions.

Multiplying Numerators

To multiply fractions, the first main step involves the numerators. These are the numbers situated at the top of each fraction. When you multiply fractions, you multiply the numerators together to get the new numerator of the resulting fraction.
In our exercise, we have numerators \(3\) and \(x\). To find the product of these numerators, we multiply them directly:

• The multiplication across the top gives us: \(3 \times x = 3x\).

This straightforward multiplication of numerators is crucial for forming the top part of the resulting fraction. By handling the numerators, we ensure the fraction accurately represents the multiplication process at this stage.

Multiplying Denominators

The second step in multiplying fractions is to focus on the denominators. These numbers sit at the bottom of the fraction and indicate into how many parts the whole is divided.
Just like with the numerators, you will multiply the denominators to form a new denominator in the resulting fraction. In the example given, we have denominators \(11\) and \(y\). Calculate their product as follows:

• The multiplication across the bottom gives us: \(11 \times y = 11y\).

This calculation defines the bottom part of your new fraction, which ensures the whole remains correctly partitioned even after multiplication.
By multiplying the denominators, we finalize the process, maintaining the integrity and meaning of the original fractions while creating a new, multiplied fraction. This method allows for fractions to be combined directly and efficiently.