Problem 49 - Prealgebra

Question

Find the quotients. (Divide.) $$\frac{8}{5} \div \frac{14}{5}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{4}{7} \).

Step 1: Understand Division of Fractions

To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction can be found by swapping its numerator and denominator.

Step 2: Find the Reciprocal

The reciprocal of the second fraction, \( \frac{14}{5} \), is \( \frac{5}{14} \). This means we will replace the division operation with multiplication by this reciprocal.

Step 3: Multiply the Fractions

Now, multiply the two fractions: \( \frac{8}{5} \times \frac{5}{14} \). Multiply the numerators together and the denominators together: \( 8 \times 5 = 40 \) and \( 5 \times 14 = 70 \). So, the result is \( \frac{40}{70} \).

Step 4: Simplify the Resulting Fraction

Simplify \( \frac{40}{70} \) by finding the greatest common divisor (GCD) of 40 and 70, which is 10. Divide both the numerator and the denominator by 10: \( \frac{40}{10} = 4 \) and \( \frac{70}{10} = 7 \). The simplified fraction is \( \frac{4}{7} \).

Key Concepts

Reciprocal of a Fraction

When we talk about the reciprocal of a fraction, we are referring to a simple yet powerful concept used widely in division involving fractions. A reciprocal essentially "flips" a fraction. This means that to find the reciprocal of a fraction, all you do is swap the numerator (the top number) and the denominator (the bottom number). For example, if you have a fraction \( \frac{a}{b} \), its reciprocal will be \( \frac{b}{a} \). The idea behind this is rooted in the property of multiplication. When a fraction is multiplied by its reciprocal, the result is always 1.

• This property is very useful for fraction division, because by multiplying by the reciprocal instead of dividing, calculations become straightforward.
• In the division exercise \( \frac{8}{5} \div \frac{14}{5} \), we needed the reciprocal of \( \frac{14}{5} \), which is \( \frac{5}{14} \).

Understanding the reciprocal is key to performing division of fractions effectively.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics that helps make numbers more manageable. It involves reducing the fraction to its smallest form where the numerator and the denominator have no common factors other than 1.
To simplify a fraction:\[ \frac{40}{70} \], you must first identify numbers that can divide both the numerator and the denominator evenly. This process not only makes calculations easier but also presents the answer in its simplest form, which is often required for final answers in tests and homework.

• After multiplying fractions, like in our example \( \frac{8}{5} \times \frac{5}{14} \ = \frac{40}{70} \), spotting common factors like 10 helps simplify.
• Using the greatest common divisor strategy, find the largest number that divides both the numerator and the denominator to simplify efficiently.
• For \( \frac{40}{70} \), dividing both top (40) and bottom (70) by their GCD which is 10, we get \( \frac{4}{7} \).

Simplifying makes working with fractions easier and often provides a cleaner end result.

Greatest Common Divisor

The greatest common divisor (GCD), sometimes known as the greatest common factor (GCF), is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD is pivotal when simplifying fractions or solving fraction-related problems.

To determine the GCD of two numbers, you can use various methods.

• Prime Factorization: Break down both numbers into their prime factors and multiply the common prime factors.


• List Method: Write down all the divisors of each number and find the greatest number that appears in both lists.


• Euclidean Algorithm: A more efficient method, especially for large numbers, based on the principle that GCD of two numbers also divides their difference.

For the exercise featuring \( \frac{40}{70} \), the GCD of 40 and 70 is 10. Dividing both numbers by 10 gives us the simplest form: \( \frac{4}{7} \). Mastery of the GCD helps streamline the simplification process by quickly reducing fractions to their simplest form.