Problem 34 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{20}{72} \cdot \frac{42}{18} \div \frac{20}{16}$$

Step-by-Step Solution

Answer

The quotient is \(\frac{14}{27}\).

Step 1: Identify the Operation with a Division

The expression given is \(\frac{20}{72} \cdot \frac{42}{18} \div \frac{20}{16}\). Notice that there is a division operation involving \(\frac{20}{16}\). To solve this, we'll replace the division by multiplication using the reciprocal of \(\frac{20}{16}\).

Step 2: Replace Division with Multiplication using Reciprocal

To replace the division by \(\frac{20}{16}\), we multiply by its reciprocal. The reciprocal of \(\frac{20}{16}\) is \(\frac{16}{20}\). The expression becomes \(\frac{20}{72} \cdot \frac{42}{18} \cdot \frac{16}{20}\).

Step 3: Simplify Each Fraction if Possible

Simplify each fraction in the expression. \(\frac{20}{72}\) simplifies to \(\frac{5}{18}\) since both numerator and denominator are divisible by 4. The fraction \(\frac{42}{18}\) simplifies to \(\frac{7}{3}\) since both numerator and denominator are divisible by 6. The fraction \(\frac{16}{20}\) simplifies to \(\frac{4}{5}\) since both numerator and denominator are divisible by 4.

Step 4: Multiply the Simplified Fractions Together

Multiply the simplified fractions: \( \frac{5}{18} \cdot \frac{7}{3} \cdot \frac{4}{5} \). First, \( \frac{5}{18} \cdot \frac{7}{3} = \frac{35}{54} \), and then multiply \( \frac{35}{54} \cdot \frac{4}{5}\).

Step 5: Multiply and Simplify the Final Expression

Continuing from the previous step, multiply \(\frac{35}{54} \cdot \frac{4}{5}\). This results in \(\frac{35 \times 4}{54 \times 5} = \frac{140}{270}\). Simplify \(\frac{140}{270}\) by dividing both numerator and denominator by 10 to get \(\frac{14}{27}\).The final simplified expression is \(\frac{14}{27}\).

Key Concepts

Multiplying Fractions

When you multiply fractions, you multiply the numerators (the top numbers) together and then the denominators (the bottom numbers) together. This process is straightforward because you don’t need to make the denominators the same like you do with addition or subtraction.
When you have two or more fractions, the multiplication follows this pattern: if you have two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), their product is \( \frac{a \cdot c}{b \cdot d} \). Let's apply this to the simplified fractions from the example:

• First, multiply \( \frac{5}{18} \) by \( \frac{7}{3} \): this gives \( \frac{5 \times 7}{18 \times 3} = \frac{35}{54} \).
• Next, multiply this result by \( \frac{4}{5} \), yielding \( \frac{35 \times 4}{54 \times 5} = \frac{140}{270} \).

This approach keeps it simple and follows a consistent method to find the product of fractions.

Simplifying Fractions

Simplifying fractions, sometimes called reducing, is the process of making the fraction as simple as possible. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In the exercise provided, each fraction was simplified before multiplication.

• For instance, \( \frac{20}{72} \) simplifies to \( \frac{5}{18} \), dividing both by 4.
• \( \frac{42}{18} \) simplifies to \( \frac{7}{3} \), since 6 divides evenly into both numbers.
• Finally, \( \frac{16}{20} \) becomes \( \frac{4}{5} \), with 4 as their GCD.

In every case, simplifying makes computation easier because you are working with smaller numbers. This step is crucial for accuracy and efficiency.

Reciprocal of a Fraction

The reciprocal of a fraction flips the numerator and the denominator. So, for a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
Reciprocals are useful in fraction division.
When dividing fractions, you multiply by the reciprocal of the divisor. In the exercise, the divisor \( \frac{20}{16} \) was replaced with its reciprocal \( \frac{16}{20} \) to change the division into multiplication.

• This conversion is essential because it simplifies the division process into an easy multiplication task.
• By flipping the divisor to use multiplication, the operation follows the rules of multiplying fractions which are often simpler to handle.

Reciprocals are easy to find and absolutely vital for dividing fractions.