Problem 95 - Algebra 1: Concepts and Skills

Question

Divide. Write the answer in simplest form. $$ \frac{29}{32} \div \frac{23}{24} $$

Step-by-Step Solution

Answer

The simple form of the division operation \( \frac{29}{32} \div \frac{23}{24} \) is \( \frac{87}{92} \).

Step 1: Find the Reciprocal of the Divisor Fraction

The reciprocal of a fraction is obtained by swapping - or 'inverting' - its numerator and denominator. Thus, the reciprocal of \( \frac{23}{24} \) is \( \frac{24}{23} \).

Step 2: Multiply the Given Fraction by the Reciprocal of the Divisor Fraction

Now, perform the multiplication operation between \( \frac{29}{32} \) and \( \frac{24}{23} \). This is done by multiplying the numerators together and denominators together: \( \frac{29 \times 24}{32 \times 23} \).

Step 3: Simplify the Resulting Fraction

The multiplication gives the fraction \( \frac{696}{736} \). This fraction simplifies to \( \frac{87}{92} \) when reduced to its simplest form. Reduction to simplest form is performed by dividing both the numerator and the denominator by their highest common factor (HCF), which in this case is 8.

Key Concepts

Simplifying Fractions

Understanding how to simplify a fraction is crucial when it comes to solving math problems. Simplifying, also known as reducing a fraction, means to make it as simple as possible. This is done by finding the greatest common divisor (GCD) of the numerator and the denominator—the largest number by which both can be divided evenly. For instance, consider a fraction such as \( \frac{696}{736} \). To simplify this fraction, we look for the highest number that divides both 696 and 736. When we find that number to be 8, we divide both the numerator and the denominator by 8, resulting in the simplified fraction \( \frac{87}{92} \).

Here's a step-by-step guide to simplifying fractions:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• The result is a simplified fraction.

When the numerator and the denominator have no common factors other than 1, the fraction is in its simplest form.

Reciprocal of a Fraction

To divide fractions, we need to understand the concept of the reciprocal of a fraction. The reciprocal is simply the flipped version of the original fraction – the numerator becomes the denominator and vice versa. For example, the reciprocal of \( \frac{23}{24} \) is \( \frac{24}{23} \).

To find the reciprocal, follow these steps:

• Write down the original fraction.
• Swap the numerator with the denominator to 'invert' the fraction.

When dividing fractions, such as \( \frac{29}{32} \div \frac{23}{24} \), we multiply the first fraction by the reciprocal of the second. This multiplication effectively performs the division operation. It's important to remember that the reciprocal of a whole number such as 5 is \( \frac{1}{5} \) since any whole number can be written as a fraction with 1 as the denominator.

Fraction Multiplication

Guidelines for Multiplying Fractions

Multiplying fractions is straightforward once you know the basics. Simply multiply the numerators (top numbers) to find the new numerator, and multiply the denominators (bottom numbers) to find the new denominator. Here's how to multiply fractions:

• Multiply the numerators.
• Multiply the denominators.
• Simplify the resulting fraction if possible.

For our given problem of \( \frac{29}{32} \times \frac{24}{23} \)—which started as a division problem—we multiply across to get \( \frac{696}{736} \). As we’ve seen, the next step is to simplify this fraction to its simplest form, \( \frac{87}{92} \).

Keep in mind that fraction multiplication doesn't require a common base like addition or subtraction of fractions does. It's one of the reasons why converting a division problem to multiplication via reciprocals is an effective strategy.