Problem 25 - Fundamentals of Mathematics

Question

Find each value. (Section 4.7) What part of \(\frac{9}{14}\) is \(\frac{6}{7} ?\)

Step-by-Step Solution

Answer

\( \frac{6}{7} \) is \( \frac{4}{3} \) of \( \frac{9}{14} \).

Step 1: Understand the Problem

We are asked to find what part of \( \frac{9}{14} \) is \( \frac{6}{7} \). This involves determining the fraction by which \( \frac{9}{14} \) must be multiplied to result in \( \frac{6}{7} \).

Step 2: Set up the Equation

To find what part \( x \) of \( \frac{9}{14} \) is \( \frac{6}{7} \), we set up the equation: \( x \times \frac{9}{14} = \frac{6}{7} \).

Step 3: Solve the Equation for x

To solve for \( x \), isolate \( x \) by dividing both sides by \( \frac{9}{14} \). This can also be done by multiplying both sides by the reciprocal of \( \frac{9}{14} \), resulting in the equation \( x = \frac{6}{7} \times \frac{14}{9} \).

Step 4: Simplify the Multiplication

Perform the multiplication \( \frac{6}{7} \times \frac{14}{9} \). First, simplify. The number 14 and 7 have a common factor: divide the 14 by 7 to get 2. The calculation now is \( \frac{6}{1} \times \frac{2}{9} \).

Step 5: Calculate the Result

Now calculate the product: \( \frac{6 \times 2}{1 \times 9} = \frac{12}{9} \). Simplify \( \frac{12}{9} \) by dividing both the numerator and denominator by 3, resulting in \( \frac{4}{3} \).

Step 6: Final Answer

Thus, \( \frac{4}{3} \) or 1\( \frac{1}{3} \) is the part of \( \frac{9}{14} \) that \( \frac{6}{7} \) represents.

Key Concepts

Multiplying Fractions

Multiplying fractions might seem challenging at first, but it becomes quite simple once you understand the concept. When faced with multiplying two fractions, such as \( \frac{6}{7} \times \frac{14}{9} \), you need to multiply the numerators (the top numbers of the fractions) together and the denominators (the bottom numbers of the fractions) together.

Here's how it works using our example:

• Multiply the numerators: \( 6 \times 14 = 84 \)
• Multiply the denominators: \( 7 \times 9 = 63 \)

This gives you a preliminary result of \( \frac{84}{63} \). However, before finalizing your answer, you always want to look at simplifying your fraction, which we will discuss in more detail next.

Simplifying Fractions

Simplifying fractions is an essential step that ensures your fractions are as concise as possible. To simplify a fraction, like our previous example \( \frac{84}{63} \), find the greatest common divisor (GCD) for the numerator and the denominator.

You can simplify by:

• Finding the GCD, which in this case is 21 for both 84 and 63.
• Dividing both the numerator and the denominator by this GCD: \( \frac{84 \div 21}{63 \div 21} = \frac{4}{3} \)

By simplifying, not only does the fraction become easier to read and understand, but it also becomes more practical to work with. In our case, \( \frac{4}{3} \) is the most reduced form after simplification, which means that indeed \( \frac{6}{7} \) constitutes \( \frac{4}{3} \) of \( \frac{9}{14} \).

Reciprocal of a Fraction

Understanding the concept of the reciprocal is pivotal when dividing fractions or solving equations involving fractions. The reciprocal of a fraction is simply the fraction flipped upside down. So, for the fraction \( \frac{9}{14} \), its reciprocal is \( \frac{14}{9} \).

The reciprocal comes into play when you divide fractions. Instead of dividing directly, you multiply by the reciprocal. In solving our equation \( x \times \frac{9}{14} = \frac{6}{7} \), we isolated \( x \) by multiplying both sides by the reciprocal of \( \frac{9}{14} \), transforming our equation into \( x = \frac{6}{7} \times \frac{14}{9} \). This makes the process straightforward and avoids the direct division of fractions which can be more complicated to handle. Always remember: to "divide" by a fraction, multiply by its reciprocal.