Problem 91 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer in simplest form. $$ \frac{3}{4} \div \frac{9}{24} $$

Step-by-Step Solution

Answer

The simple form of the fraction \( \frac{3}{4} \div \frac{9}{24} \) is 2

Step 1: Write the problem

Begin by writing down the problem: \( \frac{3}{4} \div \frac{9}{24} \)

Step 2: Rewrite the division as multiplication

Next, rewrite the division as multiplication by the reciprocal. Remember that the reciprocal of a fraction is obtained by swapping its numerator and denominator. So, the reciprocal of \( \frac{9}{24} \) is \( \frac{24}{9} \). The problem becomes: \( \frac{3}{4} \times \frac{24}{9} \)

Step 3: Multiply the fractions

Now, you can multiply the fractions by multiplying the numerators together and the denominators together. That gives: \( \frac{3 \times 24}{4 \times 9} = \frac{72}{36} \)

Step 4: Simplify the fraction

Finally, simplify the resultant fraction. Both the numerator and denominator are divisible by 36, which gives: \( \frac{72}{36} = 2 \)

Key Concepts

Reciprocal of a Fraction

The reciprocal of a fraction is a fundamental concept when it comes to dividing fractions. It is also known as the multiplicative inverse of a fraction. To find the reciprocal, you simply swap the numerator (top number) with the denominator (bottom number).

For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). When you divide fractions, you actually multiply by the reciprocal of the divisor (the fraction you are dividing by). So in the exercise where we divided \( \frac{3}{4} \) by \( \frac{9}{24} \), we instead multiplied \( \frac{3}{4} \) by the reciprocal of \( \frac{9}{24} \), which is \( \frac{24}{9} \).

This operation changes the division of fractions into multiplication, simplifying the process considerably. It's important to get comfortable finding reciprocals because it is a key step in dividing fractions effectively.

Simplest Form

Reducing a fraction to its simplest form involves dividing both the numerator and the denominator by their greatest common divisor (GCD). A fraction is in simplest form when the numerator and the denominator have no common factors other than 1.

For example, after multiplying the fractions in our exercise, we get \( \frac{72}{36} \). To reduce this to its simplest form, we look for the highest number that divides into both 72 and 36. In this case, it's 36 itself. Dividing both by 36, we get \( \frac{72 \/ 36}{36 \/ 36} = \frac{2}{1} = 2 \).

It's essential to always present your final answer in simplest form because it is the most reduced version of the fraction, free from any reducible factors. This step is crucial to ensure the clarity and correctness of your results.

Multiplying Fractions

Multiplying fractions is straightforward. To multiply two fractions, you multiply the numerators together to get the new numerator, and then multiply the denominators together to get the new denominator.

In our practice problem, where we multiplied \( \frac{3}{4} \times \frac{24}{9} \), we multiplied 3 (the numerator of the first fraction) by 24 (the numerator of the second fraction), resulting in 72 as the new numerator. Similarly, we multiplied 4 (the denominator of the first fraction) by 9 (the denominator of the second fraction), giving us 36 as the new denominator. This gave us the fraction \( \frac{72}{36} \), which we then simplified to its simplest form.

Tips for Multiplying Fractions:

• Always multiply across: numerator with numerator, and denominator with denominator.
• Before multiplying, check if you can cross-cancel any common factors to simplify earlier, making your final simplifying step easier.
• Don't forget to simplify your answer to its simplest form.