Problem 23 - Fundamentals of Mathematics

Question

Find \(\frac{5}{8}\) of \(\frac{9}{20}\) of \(\frac{4}{9}\).

Step-by-Step Solution

Answer

The answer is \( \frac{1}{8} \).

Step 1: Understand the Problem

We need to find a fraction of a fraction of a fraction. Specifically, we need to multiply these three fractions together: \( \frac{5}{8} \), \( \frac{9}{20} \), and \( \frac{4}{9} \).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions together. The numerators are 5, 9, and 4. So, we calculate: \[ 5 \times 9 \times 4 = 180 \]

Step 3: Multiply the Denominators

Multiply the denominators of the fractions together. The denominators are 8, 20, and 9. So, we calculate: \[ 8 \times 20 \times 9 = 1440 \]

Step 4: Form the New Fraction

Combine the results from the previous steps to form the new fraction: \[ \frac{180}{1440} \]

Step 5: Simplify the Fraction

Simplify \( \frac{180}{1440} \) by finding the greatest common divisor (GCD). The GCD of 180 and 1440 is 180. Divide both the numerator and the denominator by this number to simplify the fraction: \[ \frac{180}{180} = 1 \] \[ \frac{1440}{180} = 8 \] Thus, the simplified fraction is \( \frac{1}{8} \).

Step 6: Verify the Solution

Double-check the calculations to ensure there are no errors. The simplified fraction \( \frac{1}{8} \) is correct after confirming the multiplications and simplification steps are accurately followed.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics. It involves reducing a fraction to its simplest form, meaning you convert it into a fraction with the smallest possible numerator and denominator. This process is vital because it helps make fractions easier to understand and work with. To simplify a fraction, identify the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number.
For instance, from our exercise, the fraction \( \frac{180}{1440} \) is simplified by dividing the numerator and denominator by their GCD, which is 180. So, we have:

• \( \frac{180}{180} = 1 \)
• \( \frac{1440}{180} = 8 \)

Thus, the fraction simplifies to \( \frac{1}{8} \). This process not only makes calculations more manageable but also ensures that the fraction is presented in its most basic form.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. It plays a crucial role in simplifying fractions. By finding the GCD, you can reduce fractions to their simplest form, making calculations easier. To find the GCD, list the factors of each number and identify the largest factor they share.
For example, to simplify \( \frac{180}{1440} \), we find the GCD of 180 and 1440, which is 180. This means 180 is the biggest number that evenly divides both the numerator 180 and the denominator 1440.

• This concept ensures precise fraction simplification.
• It helps in avoiding errors during calculations.

Always remember, the GCD is your best friend when dealing with fractions!

Multiplying Numerators

When multiplying fractions, you start by focusing on the numerators, which are the top parts of the fractions. To do this, simply multiply the numerators of all the fractions involved together.
In our given problem, the numerators are 5, 9, and 4. Here’s how to multiply them:

• \( 5 \times 9 = 45 \)
• \( 45 \times 4 = 180 \)

These results are then used to form the numerator of the final fraction.
Multiplying numerators is straightforward and is the first step in multiplying fractions. Successfully executing this step is critical for solving fraction-related problems.

Multiplying Denominators

The next step after dealing with numerators is to focus on the denominators when multiplying fractions. Denominators are the bottom parts of fractions, and you multiply them in a similar fashion as numerators.
For our exercise, the denominators are 8, 20, and 9. Here’s how the multiplication proceeds:

• \( 8 \times 20 = 160 \)
• \( 160 \times 9 = 1440 \)

The result, 1440, becomes the denominator for the final fraction.
By ensuring accuracy in multiplying denominators, you maintain the integrity of the mathematical operation. Like numerators, getting this step right is crucial for correctly solving fraction multiplication.