Solving 2 7/15 + 9/20 - 3/10 A Step-by-Step Guide

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Are you struggling with fraction arithmetic? Don't worry, you're not alone! Many students find adding and subtracting fractions challenging, especially when the denominators are different. This article breaks down the process of solving the fraction problem 2 7/15 + 9/20 - 3/10 step by step, making it easy to understand and master. We'll cover the essential concepts, explain each step in detail, and provide helpful tips along the way. By the end of this guide, you'll be able to confidently tackle similar fraction problems and improve your math skills. Let's dive in!

Understanding Fractions

Before we tackle the problem, let's quickly review the basics of fractions. A fraction represents a part of a whole and is written in the form of a/b, where:

  • a is the numerator, representing the number of parts we have.
  • b is the denominator, representing the total number of equal parts the whole is divided into.

For example, in the fraction 1/2, 1 is the numerator, and 2 is the denominator, meaning we have one part out of two equal parts.

Types of Fractions

There are mainly three types of fractions:

  1. Proper Fractions: The numerator is less than the denominator (e.g., 1/2, 3/4).
  2. Improper Fractions: The numerator is greater than or equal to the denominator (e.g., 5/3, 7/7).
  3. Mixed Numbers: A whole number combined with a proper fraction (e.g., 2 1/2, 3 1/4).

In our problem, we have a mixed number (2 7/15) and two proper fractions (9/20 and 3/10). To solve this problem efficiently, we'll need to convert the mixed number into an improper fraction.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we use the following formula:

  • Improper Fraction = (Whole Number × Denominator + Numerator) / Denominator

For the mixed number 2 7/15:

  • Whole Number = 2
  • Denominator = 15
  • Numerator = 7

Improper Fraction = (2 × 15 + 7) / 15 = (30 + 7) / 15 = 37/15

So, 2 7/15 is equal to 37/15. Now, we can rewrite the original problem as:

37/15 + 9/20 - 3/10

Now that we have all fractions in the proper form, we can move on to the next step: finding the least common denominator.

Finding the Least Common Denominator (LCD)

Before we can add or subtract fractions, they must have the same denominator. The least common denominator (LCD) is the smallest multiple that all the denominators share. To find the LCD, we can use two methods: listing multiples or prime factorization.

Method 1: Listing Multiples

List the multiples of each denominator until you find a common multiple:

  • Multiples of 15: 15, 30, 45, 60, 75, ...
  • Multiples of 20: 20, 40, 60, 80, ...
  • Multiples of 10: 10, 20, 30, 40, 50, 60, ...

The smallest common multiple is 60, so the LCD is 60.

Method 2: Prime Factorization

Find the prime factorization of each denominator:

  • 15 = 3 × 5
  • 20 = 2 × 2 × 5 = 2² × 5
  • 10 = 2 × 5

To find the LCD, take the highest power of each prime factor present in the factorizations:

  • 2² (from 20)
  • 3 (from 15)
  • 5 (from 15, 20, and 10)

LCD = 2² × 3 × 5 = 4 × 3 × 5 = 60

Both methods give us the same LCD, which is 60. Now we can rewrite each fraction with the denominator of 60.

Rewriting Fractions with the LCD

To rewrite each fraction with the LCD of 60, we need to multiply both the numerator and the denominator by the same factor so that the denominator becomes 60.

  1. 37/15: To change 15 to 60, we multiply by 4 (15 × 4 = 60). So, we multiply both the numerator and the denominator by 4: (37 × 4) / (15 × 4) = 148/60
  2. 9/20: To change 20 to 60, we multiply by 3 (20 × 3 = 60). So, we multiply both the numerator and the denominator by 3: (9 × 3) / (20 × 3) = 27/60
  3. 3/10: To change 10 to 60, we multiply by 6 (10 × 6 = 60). So, we multiply both the numerator and the denominator by 6: (3 × 6) / (10 × 6) = 18/60

Now, our problem looks like this:

148/60 + 27/60 - 18/60

With all fractions having the same denominator, we can now perform the addition and subtraction.

Adding and Subtracting Fractions

To add and subtract fractions with the same denominator, we simply add or subtract the numerators and keep the denominator the same. Let's perform the operations:

  1. Addition: 148/60 + 27/60 = (148 + 27) / 60 = 175/60
  2. Subtraction: 175/60 - 18/60 = (175 - 18) / 60 = 157/60

So, our result is 157/60. This is an improper fraction, so let's convert it back to a mixed number to make it easier to understand.

Converting Improper Fractions to Mixed Numbers

To convert an improper fraction to a mixed number, we divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and the denominator stays the same.

  1. Divide 157 by 60: 157 ÷ 60 = 2 with a remainder of 37.
  2. Write the mixed number: The quotient is 2, the remainder is 37, and the denominator is 60. So, the mixed number is 2 37/60.

Therefore, 157/60 is equal to 2 37/60.

Final Answer

So, the solution to the problem 2 7/15 + 9/20 - 3/10 is 2 37/60.

Verification

To make sure our answer is correct, we can check our work using a calculator or by converting the mixed number back to an improper fraction and verifying the steps.

Key Takeaways

  • Fractions must have a common denominator before they can be added or subtracted.
  • The least common denominator (LCD) is the smallest multiple that all the denominators share.
  • To convert a mixed number to an improper fraction: (Whole Number × Denominator + Numerator) / Denominator.
  • To convert an improper fraction to a mixed number, divide the numerator by the denominator; the quotient is the whole number, the remainder is the numerator, and the denominator stays the same.

Practice Problems

To reinforce your understanding, try solving these similar problems:

  1. 1 1/2 + 3/4 - 2/5
  2. 3 2/3 - 1/6 + 5/9
  3. 4/5 + 7/10 - 1/4

By practicing, you'll become more confident in your ability to solve fraction problems quickly and accurately.

Tips for Success

  • Always simplify fractions before adding or subtracting, if possible. This makes the numbers smaller and easier to work with.
  • Double-check your work to avoid making mistakes, especially when rewriting fractions with the LCD.
  • Practice regularly to improve your skills and build confidence.
  • Use visual aids like fraction bars or diagrams to help you understand the concepts.
  • Don't be afraid to ask for help from teachers, tutors, or classmates if you're struggling.

Mastering fractions is a crucial step in mathematics. By understanding the concepts and practicing regularly, you can build a solid foundation for more advanced math topics. Remember, every problem is an opportunity to learn and improve. Keep practicing, and you'll become a fraction expert in no time!

This detailed guide should help you understand how to solve fraction problems effectively. Remember to practice regularly, and you'll master these concepts in no time. Good luck!