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Chapter 2: Problem 100

Simplify. $$\frac{\frac{7}{24}}{\frac{3}{8}}$$

Short Answer

Expert verified
The simplified version of the given complex fraction is \(\frac{7}{9}\).

Step by step solution

01

Identify the problem

This is a complex fraction where the numerator is \(\frac{7}{24}\) and the denominator is \(\frac{3}{8}\). The problem is to simplify this complex fraction.
02

Convert division to multiplication

Change the division operation into a multiplication operation, and take the reciprocal (flip) of the second fraction. This changes the expression to \(\frac{7}{24} * \frac{8}{3}\).
03

Multiply the fractions

To multiply fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. This calculation gives \(\frac{7*8}{24*3} = \frac{56}{72}\).
04

Simplify the resulting fraction

Simplify the fraction to lowest terms by dividing the numerator and the denominator by their Greatest Common Divisor (GCD). The simplified expression is \(\frac{56 梅 8}{72 梅 8} = \frac{7}{9}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Fractions
Complex fractions can seem intimidating at first glance, but when you break them down, they're quite manageable. A complex fraction is simply a fraction where the numerator, the top part, or the denominator, the bottom part, is also a fraction or contains fractions. To simplify a complex fraction,
you turn the division of fractions into a multiplication problem by flipping the second fraction 鈥� this is known as finding the reciprocal. In the exercise given, we simplify \( \frac{\frac{7}{24}}{\frac{3}{8}} \) by using this technique, which transforms the complex fraction into a simpler form that is easier to handle.
Multiplication of Fractions

Steps to Multiply Fractions

To multiply fractions, follow these steps:
  • Multiply the numerators (top numbers) together to get the new numerator.
  • Multiply the denominators (bottom numbers) together to get the new denominator.
  • Place the product of the numerators over the product of the denominators.
When applied to our exercise, \( \frac{7}{24} \times \frac{8}{3} \) follows these steps, resulting in \( \frac{56}{72} \) before simplification. This straightforward multiplication is an essential skill for working with complex fractions.
Greatest Common Divisor
The Greatest Common Divisor (GCD), also known as the greatest common factor, is crucial for simplifying fractions. It is the highest number that divides evenly into two or more numbers. To simplify a fraction to its lowest terms, divide both the numerator and the denominator by the GCD.

Finding the GCD

There are multiple ways to find the GCD, such as listing out the factors or using the Euclidean algorithm. In our exercise, the GCD of 56 and 72 is 8. Dividing both parts of the fraction \( \frac{56}{72} \) by 8 gives us the simplified result of \( \frac{7}{9} \)鈥攁 fraction in its lowest terms, which indicates that we have accomplished our goal of simplification.

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