Problem 9 - Fundamentals of Mathematics

Question

Find \(\frac{5}{8}\) of \(\frac{1}{10}\).

Step-by-Step Solution

Answer

\( \frac{5}{8} \) of \( \frac{1}{10} \) is \( \frac{1}{16} \).

Step 1: Understand the Problem

To find \( \frac{5}{8} \) of \( \frac{1}{10} \), we need to multiply these two fractions together.

Step 2: Multiply the Numerators

Multiply the numerators of both fractions: \( 5 \times 1 = 5 \).

Step 3: Multiply the Denominators

Multiply the denominators of both fractions: \( 8 \times 10 = 80 \).

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to form the new fraction: \( \frac{5}{80} \).

Step 5: Simplify the Fraction

Simplify \( \frac{5}{80} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This gives \( \frac{1}{16} \).

Key Concepts

Understanding Fractions

Fractions represent parts of a whole. They are used to define segments that are smaller than a whole unit. A fraction consists of two numbers: the numerator and the denominator. The numerator is the number above the line and it indicates how many parts of the whole we are considering. The denominator is the number below the line and it represents the total number of equal parts that the whole is divided into.
For example, in the fraction \( \frac{1}{2} \), the numerator is 1, which means one part out of two equal parts of the whole. Fractions like \( \frac{5}{8} \) or \( \frac{1}{10} \) can also represent specific parts of a whole.
In mathematics, it is common to perform operations with fractions, such as addition, subtraction, multiplication, and division. Multiplying fractions, like in our exercise, involves multiple steps to get the result. Understanding fractions is essential to manage such mathematical operations effectively.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. It is achieved when both the numerator and the denominator can no longer be divided by the same number, except for 1. This process helps make fractions easier to understand and work with.
To simplify a fraction, you need to find a number that divides evenly into both the numerator and the denominator. For instance, with \( \frac{5}{80} \), you realize that both 5 and 80 are divisible by 5.

• Divide the numerator (5) by 5 to get 1.
• Divide the denominator (80) by 5 to get 16.

This results in the simplified fraction \( \frac{1}{16} \). When a fraction is fully simplified, it is easier to compare, add, subtract, or multiply with other fractions. Always aim to write fractions in their simplest form for clarity and ease of calculation.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. Finding the GCD is a critical step in simplifying fractions, as it shows the greatest extent to which you can divide both the numerator and the denominator.
To find the GCD, you can use various methods, such as prime factorization, the Euclidean algorithm, or simply listing out factors. For the fraction \( \frac{5}{80} \), we look for the GCD of 5 and 80.

• List the factors of 5: 1, 5.
• List the factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80.

The highest number common in both lists is 5, which is the GCD. Using the GCD, you can then divide the numerator and the denominator to simplify the fraction to \( \frac{1}{16} \). Understanding and using the greatest common divisor ensures fractions are reduced accurately and efficiently to their simplest form.