Problem 160 - Intermediate Algebra

Question

In the following exercises, perform the indicated operation. $$ \frac{2}{5} \div \frac{y}{9} $$

Step-by-Step Solution

Answer

\( \frac{18}{5y} \)

Step 1 - Write the Division as Multiplication

To divide by a fraction, multiply by its reciprocal. For the given problem, we can write: \[ \frac{2}{5} \div \frac{y}{9} = \frac{2}{5} \times \frac{9}{y} \]

Step 2 - Multiply the Numerators

Multiply the numerators of the fractions: \[ 2 \times 9 = 18 \]

Step 3 - Multiply the Denominators

Multiply the denominators of the fractions: \[ 5 \times y = 5y \]

Step 4 - Combine the Results

Now put the results from Steps 2 and 3 together: \[ \frac{2}{5} \times \frac{9}{y} = \frac{18}{5y} \]

Key Concepts

Fraction Division

When we talk about dividing fractions, it might sound a little tricky at first. However, with a bit of understanding, it becomes quite straightforward. The key thing to remember is that 'dividing by a fraction' is the same as 'multiplying by its reciprocal'. This concept simplifies the process significantly and is central to performing division operations on fractions.

For example, dividing \(\frac{2}{5}\) by \(\frac{y}{9}\) involves transforming the division into a multiplication. Specifically, we multiply \(\frac{2}{5}\) by the reciprocal of \(\frac{y}{9}\), which is \(\frac{9}{y}\). Thus: \[ \frac{2}{5} \times \frac{9}{y} = \frac{18}{5y} \]

By following this approach, we can handle fraction division problems with greater ease, ensuring the solution process remains clear and understandable. Breaking it down into steps prevents confusion and helps us keep track of what we're doing.

Reciprocal

Understanding the concept of a reciprocal is crucial for fraction operations. The reciprocal of a fraction is obtained by flipping the numerator and the denominator. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\). This simple flip of the fraction plays a pivotal role in converting a division problem into a multiplication one.

Let's see how this applies to our exercise: the given problem involves \(\frac{2}{5} \ \div \ \frac{y}{9}\). To solve it, we first found the reciprocal of \(\frac{y}{9}\), which is \(\frac{9}{y}\). This changes our division problem into: \[ \frac{2}{5} \times \frac{9}{y} \]

Using the reciprocal makes the process much simpler and more manageable. This approach is applicable to all fraction division problems, allowing us to tackle them using a consistent method.

Multiplication of Fractions

When we multiply fractions, the process involves two straightforward steps: multiplying the numerators (top numbers) and multiplying the denominators (bottom numbers). By doing this, we combine the fractions into one single fraction.

For example, in our problem: \[ \frac{2}{5} \times \frac{9}{y} \] we multiply the numerators: \(2 \times 9 = 18\), and then the denominators: \(5 \times y = 5y\). After performing these multiplications, we combine them: \[ \frac{18}{5y} \]

By breaking the multiplication process into these simple steps, we can determine the final result accurately and efficiently. This method applies to any fractions we need to multiply, making it an essential skill for mastering fraction operations in algebra.