Problem 10 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Example \(1 .\) $$ \frac{\frac{3}{5}}{\frac{2}{7}} $$

Step-by-Step Solution

Answer

The simplified form of \( \frac{\frac{3}{5}}{\frac{2}{7}} \) is \( \frac{21}{10} \).

Step 1: Identify the complex fraction

The given expression is a complex fraction: \( \frac{\frac{3}{5}}{\frac{2}{7}} \). This means that the fraction itself has a fraction in the numerator and a fraction in the denominator.

Step 2: Rewrite the complex fraction as a division problem

A complex fraction like \( \frac{\frac{3}{5}}{\frac{2}{7}} \) is equivalent to dividing the numerator by the denominator: \( \frac{3}{5} \div \frac{2}{7} \).

Step 3: Apply the rule for dividing fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \( \frac{2}{7} \) is \( \frac{7}{2} \). Thus, the division becomes: \( \frac{3}{5} \times \frac{7}{2} \).

Step 4: Multiply the fractions

Multiply the numerators together and the denominators together: \( \frac{3 \times 7}{5 \times 2} = \frac{21}{10} \).

Step 5: Simplify the result

The fraction \( \frac{21}{10} \) cannot be simplified further as it is already in its simplest form.

Key Concepts

Division of Fractions

Dividing fractions might seem tricky at first, but it becomes simple once you get the hang of it! When you need to divide one fraction by another, the process is straightforward: you convert the division into a multiplication problem. This might sound strange since division and multiplication are different operations, but here's the magic twist.
The rule for dividing fractions involves multiplying by the reciprocal of the divisor (the fraction you are dividing by). Think of it like this:

• The first step is identifying your divisor, which in our exercise is \(\frac{2}{7}\).
• Then, find the reciprocal of the divisor (more on that later), which is simply flipping the numerator and the denominator.
• Finally, change the division sign between the two fractions to a multiplication sign.

So, dividing \(\frac{3}{5}\) by \(\frac{2}{7}\) becomes multiplying \(\frac{3}{5}\) by the reciprocal of \(\frac{2}{7}\). Easy peasy, right? This step clears up the "divide" part in our complex fraction problem efficiently.

Simplifying Fractions

Simplifying fractions might sound boring, but it's super helpful in math. Simplification makes fractions easier to understand by reducing them to their simplest form.
For example, in our problem, we ended up with the fraction \(\frac{21}{10}\) as our solution. Simplification is about seeing if we can make that fraction shorter or simpler:

• First, check if there's a number that divides both the numerator (top number) and the denominator (bottom number) evenly.
• Next, divide both the numerator and denominator by this common number, if it exists.
• If no common number exists other than 1, then the fraction is already in its simplest form.

In our case, \(\frac{21}{10}\) doesn't have any common divisors except 1, so it's already as simple as it gets! Remember, always aim for simplicity, as simpler fractions are easier to work with.

Reciprocal of a Fraction

The reciprocal of a fraction flips the numerator and the denominator. This means:

• For \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).
• The key part here is that a fraction multiplied by its reciprocal equals 1 (\(\frac{a}{b} \times \frac{b}{a} = 1\)).

Finding the reciprocal is crucial when dividing fractions. It's the secret behind transforming a tricky division into a breeze of multiplication.
In the exercise we tackled, the reciprocal of \(\frac{2}{7}\) is \(\frac{7}{2}\). This little flipping trick allows us to say goodbye to division and hello to multiplication, making our calculations much friendlier. So next time, think of the reciprocal as your math tool to make division simpler.