Problem 36 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{15}{4} \div \frac{27}{8}$$

Step-by-Step Solution

Answer

\(\frac{10}{9}\)

Step 1: Understand the Division of Fractions

To divide fractions, you need to multiply by the reciprocal of the fraction you're dividing by. In this problem, the expression \( \frac{15}{4} \div \frac{27}{8} \) can be rewritten by multiplying by the reciprocal of \( \frac{27}{8} \), which is \( \frac{8}{27} \).

Step 2: Rewrite the Expression

Rewriting the division problem as a multiplication problem gives us: \( \frac{15}{4} \times \frac{8}{27} \). Now we will multiply these two fractions.

Step 3: Multiply the Fractions

To multiply two fractions, multiply the numerators together and the denominators together. So, we calculate: \( \frac{15 \times 8}{4 \times 27} \). This simplifies to \( \frac{120}{108} \).

Step 4: Simplify the Fraction

The fraction \( \frac{120}{108} \) can be simplified by finding the greatest common divisor (GCD) of 120 and 108, which is 12. Dividing both the numerator and the denominator by 12, we get: \( \frac{120 \div 12}{108 \div 12} = \frac{10}{9} \).

Step 5: Final Answer

Thus, the value of the expression \( \frac{15}{4} \div \frac{27}{8} \) simplifies to \( \frac{10}{9} \).

Key Concepts

Reciprocal

To understand division of fractions, the key idea is using the **reciprocal**. The reciprocal of a fraction is what you get when you flip the fraction upside down. If you have a fraction like \( \frac{27}{8} \), its reciprocal will be \( \frac{8}{27} \). This means that the numerator (the top number) becomes the denominator (the bottom number) and vice versa.

When you need to divide by a fraction, you actually multiply by its reciprocal. In our original problem with \( \frac{15}{4} \div \frac{27}{8} \), you take the reciprocal of the fraction after the division sign, which is \( \frac{27}{8} \rightarrow \frac{8}{27} \). Then, you multiply \( \frac{15}{4} \) by \( \frac{8}{27} \). This method makes division simpler and easier.

Remember:

• Flipping is the key — just swap the top and bottom numbers.
• Multiplying by the reciprocal turns the division into something more familiar and solvable.

Simplification

Once you have multiplied two fractions together, you often need to simplify the result. **Simplification** means making a fraction as simple as possible by ensuring the numerator and the denominator have no common factors other than 1.

In the step-by-step solution, after multiplying \( \frac{15}{4} \) by \( \frac{8}{27} \), we ended up with \( \frac{120}{108} \). This fraction might look intimidating, but simplification can help make it more understandable.

To simplify, try:

• Looking for common factors in the numerator and denominator.
• Dividing both the top and bottom by their greatest common divisor (GCD).

In this example, that meant dividing both 120 and 108 by their GCD, which made it simpler: \( \frac{120}{108} \rightarrow \frac{10}{9} \). Always simplify fractions if possible to ensure your answer is in its simplest form.

Greatest Common Divisor

The **Greatest Common Divisor (GCD)** is a key concept for simplifying fractions. It refers to the largest number that can evenly divide two numbers, breaking them down into simpler values. For fraction simplification, GCD is essential.

To find the GCD of two numbers like 120 and 108 in our problem, list factors for each and identify the largest common one. It’s a bit like finding the biggest puzzle piece that fits both numbers perfectly.

• For 120, the factors are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
• For 108, the factors are: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108.
• The largest common factor here is 12.

By using 12, you divide both the numerator and denominator, reducing \( \frac{120}{108} \) to \( \frac{10}{9} \) in our example. Understanding and finding the GCD carefully can slice your work in half when simplifying fractions!