Problem 1 - Elementary Algebra

Question

Describe how to divide two fractions.

Step-by-Step Solution

Answer

\( \frac{a}{b} \div \frac{c}{d} = \frac{a \times d}{b \times c} \).

Step 1 - Identify the fractions

First, identify the two fractions that need to be divided. For example, consider the fractions \( \frac{a}{b} \) and \( \frac{c}{d} \).

Step 2 - Find the reciprocal of the second fraction

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \( \frac{c}{d} \) is \( \frac{d}{c} \).

Step 3 - Multiply the fractions

Now, multiply the first fraction by the reciprocal of the second fraction: \( \frac{a}{b} \times \frac{d}{c} \). This involves multiplying the numerators together and the denominators together: \( \frac{a \times d}{b \times c} \).

Step 4 - Simplify the fraction

Finally, simplify the resulting fraction if possible. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by the GCD.

Key Concepts

Reciprocal

When dividing fractions, the concept of the reciprocal is essential. The reciprocal of a fraction is simply switching the numerator and the denominator. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). This transformation is vital because dividing by a fraction is the same as multiplying by its reciprocal.

• If you have the fraction \( \frac{3}{4} \), its reciprocal is \( \frac{4}{3} \).
• To divide by \( \frac{3}{4} \), you'd multiply by \( \frac{4}{3} \).

It's that simple! Understanding reciprocals helps make fraction division much easier.

Multiplying Fractions

After finding the reciprocal of the second fraction, the next step is to multiply fractions. This process is straightforward. You multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example, if you are working with \( \frac{a}{b} \times \frac{d}{c} \), you do:

• Numerator: \( a \times d \)
• Denominator: \( b \times c \)

Therefore, the result is \( \frac{a \times d}{b \times c} \).
Always multiply straight across the numerators and denominators.

Simplifying Fractions

Simplifying fractions is the final step after multiplying. This involves making the fraction as simple as possible. To simplify, you need to find the greatest common divisor (GCD) of the numerator and the denominator. Here's how to do it:

• Identify the GCD of both the numerator and the denominator.
• Divide both by their GCD.

For example, if the fraction is \( \frac{8}{12} \), the GCD of 8 and 12 is 4. Dividing both by 4 results in \( \frac{2}{3} \). A simplified fraction is always easier to understand and work with.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is crucial in simplifying fractions. The GCD is the largest number that divides both the numerator and denominator without leaving a remainder. There are various methods to find the GCD:

• Prime Factorization: Break down both numbers into prime factors and multiply the common factors.
• Euclidean Algorithm: This involves repeated division and finding remainders until reaching zero.

For example, for 56 and 98, you can use the Euclidean algorithm:

• 98 ÷ 56 = 1 remainder 42
• 56 ÷ 42 = 1 remainder 14
• 42 ÷ 14 = 3 remainder 0

So, the GCD is 14. Dividing the original fraction by the GCD helps in simplifying it effectively.