Problem 13 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{7}{8} \div \frac{8}{7}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{49}{64} \).

Step 1: Understand the Operation

To divide fractions, we actually multiply by the reciprocal of the divisor. Here, the operation involves dividing \( \frac{7}{8} \) by \( \frac{8}{7} \).

Step 2: Find the Reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator. The reciprocal of \( \frac{8}{7} \) is \( \frac{7}{8} \).

Step 3: Convert Division into Multiplication

Replace the division with multiplication by the reciprocal found in Step 2. Thus, \( \frac{7}{8} \div \frac{8}{7} \) becomes \( \frac{7}{8} \times \frac{7}{8} \).

Step 4: Multiply the Fractions

To multiply fractions, multiply the numerators together and the denominators together. Here, \(7 \times 7 = 49\) and \(8 \times 8 = 64\). The product is \( \frac{49}{64} \).

Step 5: Simplify the Result (if needed)

Check if the fraction \( \frac{49}{64} \) can be simplified. Since 49 and 64 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Reciprocal of a Fraction

Dividing fractions starts with finding the reciprocal of the divisor. But what exactly is a reciprocal? A reciprocal of a fraction is simply flipping its numerator and denominator.
For example, the reciprocal of \( \frac{8}{7} \) is \( \frac{7}{8} \).
Reciprocals play a crucial role in fraction division because they allow us to convert division problems into multiplication problems, which are generally easier to solve.

• To find the reciprocal, always make sure you switch the top and bottom numbers of the fraction.
• If a number is whole, like 5, you can think of it as \( \frac{5}{1} \). Its reciprocal would then be \( \frac{1}{5} \).
• Checking your reciprocal is easy: when you multiply a fraction by its reciprocal, the result is always 1. For instance, \( \frac{8}{7} \times \frac{7}{8} = 1 \).

Knowing how to find the reciprocal of a fraction can make the seemingly complex process of dividing fractions a lot more manageable.

Multiplication of Fractions

Once you have the reciprocal of the divisor, you can replace division with multiplication. Multiplying fractions is straightforward: just multiply across the numerators and then the denominators.
For instance, to multiply \( \frac{7}{8} \times \frac{7}{8} \), you do:

• Multiplying the numerators: \( 7 \times 7 = 49 \)
• Multiplying the denominators: \( 8 \times 8 = 64 \)
• Putting it together: The result is \( \frac{49}{64} \)

One tip to make multiplication easier is to look for any cancellations between a numerator of one fraction and the denominator of the other.
While this might not apply to our current exercise, it can simplify your work significantly in more complex problems.

Simplifying Fractions

After multiplying, it's important to simplify the fraction whenever possible to express your answer in its simplest form. Simplifying means reducing the fraction to the smallest numbers possible that maintain the same value.
The fraction \( \frac{49}{64} \) needs to be checked for simplification.

• List the factors of the numerator (49): 1, 7, 49.
• List the factors of the denominator (64): 1, 2, 4, 8, 16, 32, 64.
• Since the only common factor between 49 and 64 is 1, the fraction doesn't need any further simplification.

If you find common factors other than 1, divide both the numerator and the denominator by the greatest common factor to simplify the fraction. Remember, a fraction is in its simplest form when the numerator and the denominator share no common factors other than 1.