Question

Evaluate each expression. $$ \frac{1}{3} \cdot \frac{4}{7}+\frac{17}{21} $$

Short answer

1

Step-by-step solution

- Multiply Fractions

First, multiply the fractions \(\frac{1}{3} \text{ and } \frac{4}{7}\). To multiply two fractions, multiply the numerators together and the denominators together. So, \(\frac{1}{3} \times \frac{4}{7} = \frac{1 \times 4}{3 \times 7} = \frac{4}{21}\).

- Add the Fractions

Now, add the fractions \(\frac{4}{21} \text{ and } \frac{17}{21}\). Since the denominators are already the same, simply add the numerators: \(\frac{4 + 17}{21} = \frac{21}{21} = 1\).

Key concepts

Multiplying Fractions

To multiply fractions, you simply need to multiply the numerators (the top parts) together and the denominators (the bottom parts) together. It is straightforward, but let's look at a quick example for added clarity.

Suppose you have the fractions \(\frac{1}{3}\) and \(\frac{4}{7}\). To multiply these:

• Multiply the numerators: \(1 \times 4 = 4\)
• Multiply the denominators: \(3 \times 7 = 21\)

So, \( \frac{1}{3} \times \frac{4}{7} = \frac{4}{21} \). Easy, right? Following this method ensures you'll always get the correct product when working with fractions.

Adding Fractions

Adding fractions might seem a bit tricky at first, but once you understand the steps, it becomes quite simple. The most important thing to remember is that you can only add fractions when their denominators are the same. Let's use an example:

Suppose you have \(\frac{4}{21}\) and \(\frac{17}{21}\). Since the denominators (21) are already the same, you can directly add the numerators:

• Add the numerators: \(4 + 17 = 21\)

So, \( \frac{4}{21} + \frac{17}{21} = \frac{21}{21} = 1 \). By following these steps, you can easily add fractions together when their denominators match.

Common Denominators

A common denominator is a shared multiple of the denominators of two or more fractions. It is essential when adding or subtracting fractions because you can only perform these operations when the denominators are the same.

If you have different denominators, like \(\frac{1}{3}\) and \(\frac{1}{4}\), you will first need to find a common denominator. The easiest way to find one is to multiply the two denominators together:

• In this case, \ 3 \times 4 = 12 \

Then, convert both fractions to have this common denominator. Here’s how:

• For \(\frac{1}{3}\), multiply the numerator and denominator by 4: \( \frac{1 \times 4}{3 \times 4} = \frac{4}{12}\)
• For \( \frac{1}{4}\), multiply the numerator and denominator by 3: \( \frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)

Now you can add the two fractions because they have a common denominator: \( \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \). Understanding and finding common denominators is important for successfully managing fraction operations.