Problem 116 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Write the answer as an algebraic expression. $$\frac{2}{3} \div \frac{a}{7}$$

Step-by-Step Solution

Answer

\(\frac{14}{3a}\)

Step 1: Write the Division Expression as Multiplication of Reciprocal

To subtract fractions, one of the methods is to multiply by the reciprocal of the second fraction. This means that \(\frac{2}{3} \div \frac{a}{7}\) will turn into \(\frac{2}{3} \times \frac{7}{a}\)

Step 2: Multiply the Fractions

Upon conversion, the next step is to multiply the fractions directly. So, \( \frac{2}{3} \times \frac{7}{a} = \frac{2 \times 7}{3 \times a}\) which simplifies to \( \frac{14}{3a}\)

Step 3: Final Simplification

The expression \(\frac{14}{3a}\) is the final result since there are no common factors in the numerator and denominator for further simplification.

Key Concepts

Division of Fractions

Division of fractions might seem tricky at first, but it can be made simple by using a basic rule. The division of one fraction by another can be transformed into a multiplication problem by taking the reciprocal of the fraction you’re dividing by. A reciprocal of a fraction is just flipping the numerator and the denominator. So, when you see

• \( \frac{2}{3} \div \frac{a}{7} \)

all you need to do is write it as

• \( \frac{2}{3} \times \frac{7}{a} \).

This step turns division into something more familiar and manageable. Thus, remember:

• The division of fractions can always be converted into a multiplication problem by using the reciprocal.

Multiplication of Fractions

Once you've transformed the division problem into a multiplication problem, tackling the multiplication of fractions becomes straightforward. To multiply two fractions:

• Multiply the numerators (the top numbers) together.
• Multiply the denominators (the bottom numbers) together.

In our original exercise, after converting the division to multiplication, you have:

• \( \frac{2}{3} \times \frac{7}{a} \).

When you multiply them, follow these steps:

• The numerators: \( 2 \times 7 = 14 \)
• The denominators: \( 3 \times a = 3a \)

Thus, you get the product:

• \( \frac{14}{3a} \).

This method ensures we combine the fractions into a single, simplified form.

Simplification of Fractions

Simplification of fractions is the process of making the fraction as simple as possible. This means reducing it so there's no common factor between the numerator and the denominator. Always check if there's a number (besides 1) that evenly divides both parts. If there is, divide both the numerator and denominator by that number.
In this case, the result from our multiplication was \( \frac{14}{3a} \). Look at both parts:

• The numerator is 14 and the denominator is \(3a\).

Check for common factors:

• 14 and 3 have no common factors other than 1.
• 14 and \(a\) are different types, so no numeric simplification exists.

Since they don’t share any common factors, \( \frac{14}{3a} \) is already simplified to its simplest form. It’s always a good practice to ask yourself if further simplification is possible when you solve such problems.