Problem 445 - Prealgebra

Question

Hiking Bill hiked 8 miles on the first day of his backpacking trip, 14 miles the second day, 11 miles the third day, and 17 miles the fourth day. What is the total number of miles Bill hiked?

Step-by-Step Solution

Answer

50 miles

Step 1 - Understand the Problem

Identify the distances Bill hiked on each day: 8 miles on the first day, 14 miles on the second day, 11 miles on the third day, and 17 miles on the fourth day.

Step 2 - Sum the Distances

Add the distances from each day to find the total number of miles. Use the formula for summation: \[ 8 + 14 + 11 + 17 \]

Step 3 - Calculate

Perform the addition step-by-step: First, add the first two distances: 8 + 14 = 22.Next, add the third distance: 22 + 11 = 33.Finally, add the fourth distance: 33 + 17 = 50.

Step 4 - Write Down the Answer

The total number of miles Bill hiked is 50 miles.

Key Concepts

addition

Addition is one of the fundamental arithmetic operations. It involves calculating the total of two or more numbers or amounts. When performing addition, it’s crucial to line up numbers by their place values. This alignment ensures that operations on each digit fall into proper columns. In an addition problem, the numbers being added together are called 'addends,' and the result is the 'sum'. For example, in the exercise provided, the distances hiked each day are the addends: 8, 14, 11, and 17. When we add these together, we get the sum of 50.
To carry out the addition step-by-step:

• Start with the first two numbers: 8 + 14 = 22.
  
• Then, add the next number: 22 + 11 = 33.
  
• Finally, add the last number: 33 + 17 = 50.
  

This step-by-step process ensures clarity and accuracy in addition problems.

word problems

Word problems are mathematical problems presented in a narrative form. They require you to extract numerical information and perform calculations based on it. Here are some key steps to solve word problems effectively:

• Read the problem carefully to understand what is being asked.
  
• Identify the key pieces of information and the numbers involved.
  
• Determine what operations are necessary to solve the problem (in this case, addition).
  
• Carry out the operations step-by-step.
  

In the example exercise, the narrative gives the distances Bill hiked over four days. By translating this information into mathematical operations (adding the distances), we can determine the total miles hiked. Word problems help improve problem-solving skills because they require understanding the context, picking out the relevant details, and applying appropriate mathematical operations.

summation

Summation is the process of adding a sequence of numbers to find their total. It is a concise way to represent the addition of multiple elements. In mathematics, summation is often denoted by the Greek letter sigma (Σ). An example of summation from the exercise is:
\[8 + 14 + 11 + 17 = 50\]\
This notation signifies that we want to add the values, leading to the total sum. Summation can be useful for simplifying expressions and efficiently performing repeated additions in a structured way. It is a core concept in arithmetic, frequently used in higher mathematics like calculus and statistics.
In everyday scenarios, summation acts as a practical tool for quickly finding totals, whether it be distances, quantities, or measurements. By mastering summation, students can better handle various real-life calculations and more complex mathematical problems.