Problem 15 - Fundamentals of Mathematics

Question

Perform the following multiplications. $$ \left(\frac{13}{7}\right)\left(\frac{14}{26}\right) $$

Step-by-Step Solution

Answer

The result of the multiplication is 1.

Step 1: Multiply the Numerators

To multiply two fractions, start by multiplying the numerators. For the fractions \( \frac{13}{7} \) and \( \frac{14}{26} \), multiply 13 and 14: \[13 \times 14 = 182\]

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions. For \( \frac{13}{7} \) and \( \frac{14}{26} \), multiply 7 and 26:\[7 \times 26 = 182\]

Step 3: Formulate the Product Fraction

Combine the results from Steps 1 and 2 to write the product fraction:\[\frac{182}{182}\]

Step 4: Simplify the Fraction

To simplify \( \frac{182}{182} \), notice that both the numerator and the denominator are the same. Therefore, this fraction simplifies to:\[1\]

Key Concepts

Numerators and Denominators

When working with fractions, understanding the role of numerators and denominators is crucial. In any fraction, the number above the line is the numerator, while the number below the line is the denominator.

For example, in the fraction \( \frac{13}{7} \), 13 is the numerator, and 7 is the denominator. The numerator indicates how many parts we have, while the denominator shows how many equal parts make up a whole.

This concept is key when multiplying fractions, as you multiply both the numerators and the denominators separately.

• Multiply the numerators: In our multiplication \( \frac{13}{7} \times \frac{14}{26} \), 13 and 14 multiply to give 182.
• Multiply the denominators: Similarly, 7 and 26 multiply to yield 182.

Thus, understanding numerators and denominators helps us form the new fraction, \( \frac{182}{182} \).

Simplifying Fractions

A fundamental step in dealing with fractions is simplifying them. Simplifying a fraction means rewriting it in its simplest form, where the numerator and denominator have no common factors other than 1.

In the multiplication problem \( \frac{182}{182} \), both the numerator and the denominator are the same number. This is a special case where the fraction simplifies directly to 1.

We simplify by dividing both the numerator and the denominator by their greatest common factor (GCF). In this case, the GCF is 182:

• Divide the numerator and denominator by 182: \( \frac{182 \div 182}{182 \div 182} = \frac{1}{1} \)
• This simplifies to the number 1.

Recognizing that any fraction with identical numerators and denominators simplifies to 1 is a helpful shortcut.

Equivalent Fractions

Equivalent fractions represent the same value, even though they may look different. They are different expressions of the same proportion.

For instance, if we consider the fraction \( \frac{14}{26} \), it could be simplified to \( \frac{7}{13} \). Both \( \frac{14}{26} \) and \( \frac{7}{13} \) are equivalent because multiplying \( \frac{7}{13} \) by \( \frac{2}{2} \) gives \( \frac{14}{26} \).

Finding equivalent fractions is an essential skill for fraction multiplication because it allows for easier computation and simplification.

• Identify common factors for simplifying fractions before multiplication to reduce complexity.
• Always multiply or divide the numerator and the denominator by the same number to maintain equivalence.

Understanding equivalent fractions can also aid in simplifying the product of fraction operations, ensuring a neat and easy solution.