Problem 31 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{2}{3} \cdot \frac{3}{4} \div \frac{5}{8}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{4}{5} \).

Step 1: Identify the Expression

We are given the expression \( \frac{2}{3} \cdot \frac{3}{4} \div \frac{5}{8} \). To solve it, we will start with the division part by replacing the divisor with its reciprocal.

Step 2: Replace Division with Multiplication

Replace the division \( \div \frac{5}{8} \) by multiplying with its reciprocal, which is \( \frac{8}{5} \). Thus, the expression becomes \( \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{8}{5} \).

Step 3: Multiply the Numerators

Multiply all the numerators together: \( 2 \times 3 \times 8 = 48 \).

Step 4: Multiply the Denominators

Multiply all the denominators together: \( 3 \times 4 \times 5 = 60 \).

Step 5: Form the New Fraction

Place the multiplied numerators over the multiplied denominators to form the fraction: \( \frac{48}{60} \).

Step 6: Simplify the Fraction

Find the greatest common divisor (GCD) of 48 and 60. The GCD is 12. Divide both the numerator and the denominator by 12 to simplify the fraction: \( \frac{48 \div 12}{60 \div 12} = \frac{4}{5} \).

Key Concepts

Reciprocals

Understanding reciprocals is crucial when working with division of fractions. A reciprocal of a number is essentially what you multiply that number by to get a product of 1. For a fraction, you find its reciprocal by swapping its numerator and denominator.

For example, the reciprocal of \( \frac{5}{8} \) is \( \frac{8}{5} \). Notice how the numerator and denominator have exchanged places. Reciprocals are useful because they transform division problems into multiplication problems.

In the given exercise, instead of dividing by \( \frac{5}{8} \), you multiply by its reciprocal, \( \frac{8}{5} \). This makes calculations easier, as multiplication is often more straightforward to perform than division.

Multiplication of Fractions

Multiplying fractions is easier than it looks. When multiplying fractions, you follow a simple rule: multiply the numerators together to get your new numerator and multiply the denominators together to get your new denominator.

Using the modified expression from the exercise, \( \frac{2}{3} \times \frac{3}{4} \times \frac{8}{5} \), you perform the multiplication in a straightforward manner:

• Multiply the numerators: \( 2 \times 3 \times 8 = 48 \)
• Multiply the denominators: \( 3 \times 4 \times 5 = 60 \)

This gives you the new fraction \( \frac{48}{60} \).

Multiplying fractions can sometimes increase the complexity of the numbers involved. This is why simplifying fractions is a valuable skill to master.

Simplifying Fractions

Once you've multiplied fractions, you often need to simplify the result. Simplifying involves reducing the fraction to its smallest possible form, where the numerator and denominator have no common factors other than 1.

For the fraction \( \frac{48}{60} \), find the greatest common divisor (GCD), which is the largest number that evenly divides both the numerator and the denominator. Here, the GCD of 48 and 60 is 12.

To simplify, divide both the numerator and the denominator by their GCD:

• Divide the numerator: \( 48 \div 12 = 4 \)
• Divide the denominator: \( 60 \div 12 = 5 \)

Now you have the simplified fraction \( \frac{4}{5} \).

This process ensures that the result is as simple as possible, making it easier to understand and use in further calculations.