Problem 13 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. [Examples 1–3] $$\left(1+\frac{5}{6}\right)\left(1-\frac{5}{6}\right)$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{11}{36}\).

Step 1: Parentheses First

Identify and calculate the operations inside the parentheses. For \( \left(1+\frac{5}{6}\right) \), the sum is \(1 + \frac{5}{6} = \frac{11}{6}\). For \(\left(1-\frac{5}{6}\right)\), the result is \(1 - \frac{5}{6} = \frac{1}{6}\).

Step 2: Multiplication of Results

Multiply the results from the parentheses: \( \frac{11}{6} \times \frac{1}{6} = \frac{11 \times 1}{6 \times 6} = \frac{11}{36} \).

Key Concepts

Prealgebra

Prealgebra is the foundation of algebra and involves understanding basic arithmetic and the properties of numbers to prepare for more advanced math concepts. This topic covers various operations such as addition, subtraction, multiplication, and division, and introduces the idea of variables and simple equations.

When solving mathematical problems involving multiple operations, the order in which you perform these operations makes a big difference. This is why learning the Order of Operations is crucial. The order to follow can often be remembered using the acronym PEMDAS:

• P for Parentheses - solve anything inside parentheses first.
• E for Exponents - solve any exponents or powers next.
• MD for Multiplication and Division - solve these from left to right.
• AS for Addition and Subtraction - likewise, solve from left to right.

In this exercise, the focus is on the operations inside the parentheses followed by multiplication, emphasizing the 'P' and 'M' of PEMDAS.

Fractions

Fractions represent parts of a whole and consist of a numerator and a denominator. Understanding fractions is key in many areas of mathematics. They allow us to express numbers that fall between integers, enabling a more precise depiction of many daily and scientific quantities.

For example, in the expression \( \left(1+\frac{5}{6}\right)\left(1-\frac{5}{6}\right) \), each fraction provides a precise representation of the operation being conducted. The sum \(1 + \frac{5}{6}\) results in the improper fraction \(\frac{11}{6}\), while the difference \(1 - \frac{5}{6}\) results in \(\frac{1}{6}\).

Adding and subtracting fractions require common denominators. Here, because 1 can be represented as a fraction with any denominator, it's easy to add or subtract fractions by converting 1 into a fraction with the same denominator, like \(\frac{6}{6}\). This is why \(1 - \frac{5}{6}\) becomes \(\frac{6}{6} - \frac{5}{6} = \frac{1}{6}\). Understanding and working with fractions becomes easier with practice and helps in numerous mathematical problems.

Multiplication of Fractions

Multiplying fractions is a straightforward process compared to adding or subtracting them. When multiplying two fractions together, you multiply the numerators to get the new numerator, and the denominators to get the new denominator. This is illustrated in our exercise by multiplying \( \frac{11}{6} \) by \( \frac{1}{6} \).

Here's how it works step by step:

• Multiply the numerators: \( 11 \times 1 = 11 \).
• Multiply the denominators: \( 6 \times 6 = 36 \).
• Combine the results: \( \frac{11 \times 1}{6 \times 6} = \frac{11}{36} \).

There is no need to find a common denominator as required in addition or subtraction. This makes multiplication a convenient operation with fractions. Additionally, always check if the result can be simplified by finding the greatest common divisor of the numerator and the denominator. In this case, \( \frac{11}{36} \) is already in its simplest form. Mastering this concept will greatly assist in solving problems involving fractions.