Problem 65 - Prealgebra

Question

Perform the indicated operations. $$3+17 \times \frac{1}{65}$$

Step-by-Step Solution

Answer

The result is \(\frac{212}{65}\).

Step 1: Identify the Operations

In the expression \(3 + 17 \times \frac{1}{65}\), there are two types of operations involved: addition and multiplication. According to the order of operations, multiplication precedes addition.

Step 2: Perform the Multiplication

First, perform the multiplication operation, which is \(17 \times \frac{1}{65}\). This means you are multiplying 17 by the fraction \(\frac{1}{65}\). The result of this is \(\frac{17}{65}\).

Step 3: Perform the Addition

Now, add the result of the multiplication to the number 3. So, you need to calculate \(3 + \frac{17}{65}\). To add these, convert 3 to a fraction with the same denominator, which is \(\frac{195}{65}\). You add: \(\frac{195}{65} + \frac{17}{65} = \frac{212}{65}\).

Key Concepts

Addition and Multiplication

In mathematics, understanding the order of operations is crucial when dealing with expressions containing different types of calculations, such as addition and multiplication. Order of operations dictates which operations to perform first to arrive at the correct result. This concept is commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division (from left to right), Addition, and Subtraction (from left to right). In the expression \(3 + 17 \times \frac{1}{65}\), multiplication is performed before addition as prescribed by PEMDAS.

Here's how it works:

• Step 1: Identify the operations: addition (\(+\)) and multiplication (\(\times\)).
• Step 2: Apply the multiplication first. Multiply 17 by \(\frac{1}{65}\) to get \(\frac{17}{65}\).
• Step 3: Move on to the addition by calculating \(3 + \frac{17}{65}\).

Applying the operations in the correct order ensures accuracy of the final result.

Fractions

Fractions represent parts of a whole or a division of any quantity into equal parts. They have a numerator and a denominator, where the numerator indicates how many parts are considered, and the denominator shows how many parts make up one whole. For example, \(\frac{1}{65}\) indicates that one part is being considered out of 65 equal parts.

When dealing with fractions in operations like in the expression \(3 + 17 \times \frac{1}{65}\), understanding how to handle them is key:

• Conversion: Converting a whole number into a fraction can simplify addition. In this case, the number 3 is converted to \(\frac{195}{65}\) so that it shares a common denominator with \(\frac{17}{65}\).
• Multiplication of Fractions: Multiply straight across the numerator and the denominator. So, \(17 \times \frac{1}{65} = \frac{17}{65}\).
• Addition of Fractions: Add fractions directly when they have a common denominator. Hence, \(\frac{195}{65} + \frac{17}{65} = \frac{212}{65}\).

Fractions can initially appear complicated, but with practice, they become easier to manage in calculations.

Step-by-Step Solution

Working through problems with a step-by-step approach is a helpful way to systematically understand and solve mathematical expressions. This method allows you to concentrate on one operation at a time, eliminating the confusion of handling complex expressions all at once.

Here is the breakdown of how to solve \(3 + 17 \times \frac{1}{65}\) step-by-step:

• Step 1: Identify the different operations present. Knowing what calculations you'll perform helps you apply the order of operations correctly.
• Step 2: Handle each mathematical operation one at a time. First, calculate the multiplication part to reduce the complexity of the expression. Here, multiply \(17\) by \(\frac{1}{65}\) to simplify to \(\frac{17}{65}\).
• Step 3: Prepare the expression for addition by ensuring a common denominator for fractions. Convert \(3\) into \(\frac{195}{65}\) to enable straightforward fraction addition.
• Final Step: Add the fractions. Now that the fractions have the same denominator, sum them to get the final result as \(\frac{212}{65}\).

This structured, stepwise manner makes it easier to tackle even seemingly complicated mathematical problems.