Calculating Total Miles Walked Adding Fractions Explained

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A person walked a dog 2/3 of a mile on Thursday and 5/7 of a mile on Friday. What is the total number of miles the person walked the dog?

In this article, we will explore a common mathematical problem involving fractions and addition, specifically focusing on calculating the total distance a person walked a dog over two days. This is a fundamental concept in mathematics that applies to various real-life situations. Understanding how to solve such problems is crucial for developing strong math skills. Let’s break down the problem step by step.

The Problem: Walking a Dog Over Two Days

Our main problem involves a person who walked a dog 23{\frac{2}{3}} of a mile on Thursday and 57{\frac{5}{7}} of a mile on Friday. The question we aim to answer is: What is the total number of miles the person walked the dog? This problem requires us to add two fractions, a skill that is essential in many areas of mathematics and daily life. Before we dive into the solution, let’s understand the underlying concepts and why this type of problem is important.

Understanding Fractions and Addition

Fractions represent parts of a whole. In our case, 23{\frac{2}{3}} means two parts out of three, and 57{\frac{5}{7}} means five parts out of seven. Adding fractions is a way of combining these parts to find a total. However, we can't simply add the numerators (the top numbers) and the denominators (the bottom numbers) directly when the denominators are different. We need a common denominator.

A common denominator is a number that both denominators can divide into evenly. This allows us to express the fractions in terms of the same 'size' of parts, making addition straightforward. The least common denominator (LCD) is the smallest number that satisfies this condition, which simplifies our calculations.

Why This Problem Matters

This problem is more than just a math exercise; it's a practical scenario. Calculating distances, whether for a walk, a trip, or any other journey, is a common task. Moreover, the ability to work with fractions is crucial in various fields, including cooking, construction, finance, and more. Mastering fraction addition builds a solid foundation for more advanced mathematical concepts.

Step-by-Step Solution: Finding the Total Distance

To find the total distance the person walked the dog, we need to add the distances from Thursday and Friday. This involves several steps:

1. Identify the Fractions

The distances walked are given as fractions: 23{\frac{2}{3}} of a mile on Thursday and 57{\frac{5}{7}} of a mile on Friday. These fractions represent the portions of a mile walked on each day.

2. Find the Least Common Denominator (LCD)

To add fractions, they must have the same denominator. The denominators in our problem are 3 and 7. We need to find the least common multiple (LCM) of 3 and 7, which will be our least common denominator (LCD).

The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, ... The multiples of 7 are: 7, 14, 21, 28, 35, ...

The least common multiple of 3 and 7 is 21. Therefore, the LCD is 21.

3. Convert the Fractions to Equivalent Fractions with the LCD

Now, we need to convert both fractions to equivalent fractions with a denominator of 21. An equivalent fraction represents the same value but has a different numerator and denominator. To do this, we multiply both the numerator and the denominator of each fraction by the number that will make the denominator equal to the LCD.

For 23{\frac{2}{3}}, we multiply both the numerator and the denominator by 7:

23×77=2×73×7=1421{ \frac{2}{3} \times \frac{7}{7} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21} }

For 57{\frac{5}{7}}, we multiply both the numerator and the denominator by 3:

57×33=5×37×3=1521{ \frac{5}{7} \times \frac{3}{3} = \frac{5 \times 3}{7 \times 3} = \frac{15}{21} }

4. Add the Equivalent Fractions

Now that both fractions have the same denominator, we can add them by adding the numerators and keeping the denominator the same:

1421+1521=14+1521=2921{ \frac{14}{21} + \frac{15}{21} = \frac{14 + 15}{21} = \frac{29}{21} }

5. Simplify the Result

The result, 2921{\frac{29}{21}}, is an improper fraction because the numerator is greater than the denominator. To make it easier to understand, we can convert it to a mixed number. A mixed number consists of a whole number and a proper fraction.

To convert 2921{\frac{29}{21}} to a mixed number, we divide 29 by 21:

29÷21=1 with a remainder of 8{ 29 \div 21 = 1 \text{ with a remainder of } 8 }

This means that 2921{\frac{29}{21}} is equal to 1 whole and 821{\frac{8}{21}} of another whole. So, the mixed number is:

1821{ 1 \frac{8}{21} }

The Answer: Total Distance Walked

The total distance the person walked the dog is 1821{1 \frac{8}{21}} miles. This means the person walked more than one mile but less than two miles in total over the two days. This answer provides a clear and practical understanding of the distance covered.

Why Each Step Is Important

Each step in solving this problem is crucial for accuracy and understanding:

  1. Identifying the fractions correctly sets the stage for the entire calculation. Incorrectly identifying the fractions will lead to a wrong answer.
  2. Finding the least common denominator is essential for adding the fractions. Without a common denominator, the fractions cannot be added correctly.
  3. Converting the fractions to equivalent fractions with the LCD ensures that we are adding 'like' quantities. This step maintains the value of the fractions while making addition possible.
  4. Adding the equivalent fractions is the core of the calculation. This step combines the distances to find the total distance.
  5. Simplifying the result converts the improper fraction to a mixed number, making it easier to understand and interpret the answer.

Skipping or misunderstanding any of these steps can lead to errors in the final result. Therefore, a thorough understanding of each step is vital for mastering fraction addition.

Alternative Methods and Tips for Fraction Addition

While the step-by-step method described above is effective, there are alternative approaches and tips that can help in adding fractions more efficiently.

Cross-Multiplication Method

One alternative method is cross-multiplication. For two fractions ab{\frac{a}{b}} and cd{\frac{c}{d}}, you can add them using the formula:

ab+cd=ad+bcbd{ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} }

In our case, 23{\frac{2}{3}} and 57{\frac{5}{7}}, we can apply this method:

23+57=(2×7)+(5×3)3×7=14+1521=2921{ \frac{2}{3} + \frac{5}{7} = \frac{(2 \times 7) + (5 \times 3)}{3 \times 7} = \frac{14 + 15}{21} = \frac{29}{21} }

This method directly gives you the result without explicitly finding the LCD, but it's essential to simplify the fraction at the end if needed.

Tips for Efficient Fraction Addition

  1. Always simplify fractions before adding. If the fractions can be simplified, do so first to make the numbers smaller and easier to work with.
  2. Use prime factorization to find the LCD. If the denominators are large, finding the prime factors can help determine the LCD more efficiently.
  3. Practice regularly to build fluency. The more you practice, the faster and more accurate you will become at adding fractions.
  4. Estimate the answer before calculating. This can help you check if your final answer is reasonable. For example, in our problem, 23{\frac{2}{3}} is a bit more than 12{\frac{1}{2}}, and 57{\frac{5}{7}} is also more than 12{\frac{1}{2}}, so the total should be more than 1, which aligns with our answer.

Real-World Applications of Fraction Addition

Fraction addition is not just a theoretical concept; it has numerous real-world applications. Understanding how to add fractions can help in various daily tasks and professional fields.

Cooking and Baking

In cooking, recipes often involve fractional amounts of ingredients. For example, a recipe might call for 12{\frac{1}{2}} cup of flour and 14{\frac{1}{4}} cup of sugar. To double the recipe, you would need to add these fractions to find the total amount of each ingredient required.

Construction and Carpentry

In construction, measurements often involve fractions. For instance, a carpenter might need to add the lengths of several pieces of wood to determine the total length needed for a project. Accurate fraction addition is crucial to ensure precise cuts and fits.

Finance and Budgeting

When managing finances, you might need to add fractional parts of your income for different expenses. For example, if you allocate 13{\frac{1}{3}} of your income to rent and 14{\frac{1}{4}} to utilities, you can add these fractions to find the total portion of your income spent on housing-related expenses.

Time Management

Time can also be represented in fractions. If you spend 12{\frac{1}{2}} hour on one task and 13{\frac{1}{3}} hour on another, adding these fractions will tell you the total time spent on both tasks.

Practice Problems: Test Your Understanding

To solidify your understanding of fraction addition, try solving these practice problems:

  1. A baker used 34{\frac{3}{4}} cup of flour for cookies and 25{\frac{2}{5}} cup of flour for a cake. How much flour did the baker use in total?
  2. A student spent 13{\frac{1}{3}} of their day studying and 16{\frac{1}{6}} of their day doing chores. What fraction of the day did the student spend on these activities?
  3. A runner ran 25{\frac{2}{5}} of a mile on Monday and 13{\frac{1}{3}} of a mile on Tuesday. What total distance did the runner run?

Solving these problems will help you reinforce your skills and become more confident in adding fractions. Remember to follow the steps we discussed: find the LCD, convert the fractions, add the numerators, and simplify the result.

Conclusion: Mastering Fraction Addition

In conclusion, adding fractions is a fundamental mathematical skill with wide-ranging applications. By understanding the steps involved—identifying the fractions, finding the least common denominator, converting to equivalent fractions, adding the numerators, and simplifying the result—you can solve problems accurately and efficiently. Whether it’s calculating distances, measuring ingredients, or managing finances, the ability to add fractions is invaluable.

Remember to practice regularly and apply these skills to real-world scenarios to reinforce your understanding. With consistent effort, you can master fraction addition and confidently tackle more complex mathematical challenges.