Problem 49 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$\frac{1}{6}\( of \)\frac{12}{13}\( of \)\frac{26}{36}$$

Step-by-Step Solution

Answer

The value of the expression is \( \frac{1}{9} \).

Step 1: Simplify the Expression

Let's simplify the expression without using any diagrams. We have the expression \( \frac{1}{6} \times \frac{12}{13} \times \frac{26}{36} \). First, notice that multiplication of fractions involves multiplying the numerators across and denominators across.

Step 2: Simplify After Combining

Multiply the numerators: \( 1 \times 12 \times 26 = 312 \) and the denominators: \( 6 \times 13 \times 36 = 2808 \). This gives us the fraction \( \frac{312}{2808} \).

Step 3: Reduce the Fraction

To reduce the fraction \( \frac{312}{2808} \), find the greatest common divisor (GCD) of 312 and 2808. Upon calculation, the GCD is 24. Divide both the numerator and the denominator by 24: \( \frac{312}{24} = 13 \) and \( \frac{2808}{24} = 117 \). This simplifies to \( \frac{13}{117} \).

Step 4: Further Simplify if Possible

Check if \( \frac{13}{117} \) can be simplified further. The GCD of 13 and 117 is 13. Divide both by 13: \( \frac{13}{13} = 1 \) and \( \frac{117}{13} = 9 \). This simplifies the fraction to \( \frac{1}{9} \).

Key Concepts

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is a crucial concept when it comes to simplifying fractions. It refers to the largest number that divides two or more numbers without leaving a remainder.
To simplify fractions effectively, finding the GCD helps determine the largest factor common to both the numerator and the denominator. It acts as a tool to reduce fractions by dividing both components by this number.
Here's how it works:

• Identify the numbers you want to find the GCD for — in this case, the numerator (312) and the denominator (2808).
• Use a method like prime factorization, listing divisors, or the Euclidean algorithm to find the GCD. For 312 and 2808, the GCD is 24.
• Divide both the numerator and the denominator by this GCD to simplify the fraction, resulting in \( \frac{13}{117} \).

Applying the GCD is a reliable method to ensure your fractions are in their simplest form.

Numerator and Denominator

Understanding the roles of the numerator and denominator is fundamental to grasping any operations involving fractions.

• The numerator is the top number in a fraction and represents how many parts of the whole we are considering.
• The denominator is the bottom number that indicates into how many parts the whole is divided. It provides a scope or size of each part.

The simplification process revolves around these two elements. For instance, starting with the expression \( \frac{1}{6} \times \frac{12}{13} \times \frac{26}{36} \),

• First, multiply all the numerators together: \( 1 \times 12 \times 26 = 312 \).
• Next, multiply all denominators together: \( 6 \times 13 \times 36 = 2808 \).
• Finally, you obtain a new fraction: \( \frac{312}{2808} \).

To simplify comprehensively, focus on refining this new fraction as much as possible, leveraging concepts like the GCD.

Multiplication of Fractions

Multiplication of fractions is a straightforward operation once you understand the mechanism behind it. It involves directly multiplying the numerators and the denominators.
Here's how it works in simple steps:

• First, consider the fractions you wish to multiply. For instance, \( \frac{1}{6} \), \( \frac{12}{13} \), and \( \frac{26}{36} \).
• Multiply the numerators across: \( 1 \times 12 \times 26 = 312 \).
• Do the same for denominators: \( 6 \times 13 \times 36 = 2808 \).

This results in the fraction \( \frac{312}{2808} \). The beauty of this method is its simplicity and straightforwardness. Once the numerators and denominators are multiplied, simplification or reduction usually follows to ensure the fraction is in its simplest form. Utilizing the greatest common divisor, the result can be further simplified to \( \frac{1}{9} \), ensuring clarity and simplicity in the fraction.