Here Are Two Calculations:${ \begin{array}{c} \frac{2}{15} + \frac{1}{3} \ \text{and} \ \frac{6}{7} - \frac{1}{3} \end{array} }$Which Of The Two Calculations Is Closer In Value To { \frac{1}{2}$}$?You Must Show Your Working And

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Introduction

When dealing with fractions, it's essential to understand how to compare and manipulate them to solve various mathematical problems. In this article, we will explore two calculations involving fractions and determine which one is closer in value to 1/2. We will show our working and provide a step-by-step explanation of the process.

The Calculations

We are given two calculations to evaluate:

  1. 215+13\frac{2}{15} + \frac{1}{3}
  2. 67−13\frac{6}{7} - \frac{1}{3}

Our goal is to determine which of these two calculations is closer in value to 12\frac{1}{2}.

Calculation 1: 2/15 + 1/3

To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 15 and 3 is 15. Therefore, we can rewrite 13\frac{1}{3} as 515\frac{5}{15}.

215+13=215+515=715\frac{2}{15} + \frac{1}{3} = \frac{2}{15} + \frac{5}{15} = \frac{7}{15}

Calculation 2: 6/7 - 1/3

To subtract these fractions, we need to find a common denominator. The LCM of 7 and 3 is 21. Therefore, we can rewrite 67\frac{6}{7} as 1821\frac{18}{21} and 13\frac{1}{3} as 721\frac{7}{21}.

67−13=1821−721=1121\frac{6}{7} - \frac{1}{3} = \frac{18}{21} - \frac{7}{21} = \frac{11}{21}

Comparing the Results

Now that we have evaluated both calculations, we can compare the results to 12\frac{1}{2}.

715≈0.4667\frac{7}{15} \approx 0.4667 1121≈0.5238\frac{11}{21} \approx 0.5238 12=0.5\frac{1}{2} = 0.5

Conclusion

Based on our calculations, we can see that 1121\frac{11}{21} is closer in value to 12\frac{1}{2} than 715\frac{7}{15}. This is because 1121\frac{11}{21} is approximately 0.5238, which is closer to 0.5 than 715\frac{7}{15}, which is approximately 0.4667.

Discussion

When comparing fractions, it's essential to find a common denominator to ensure accuracy. In this case, we used the least common multiple (LCM) to find a common denominator for both calculations. We also used approximation to compare the results to 12\frac{1}{2}.

Real-World Applications

Understanding how to compare and manipulate fractions is essential in various real-world applications, such as:

  • Cooking: When measuring ingredients, fractions are often used to ensure accuracy.
  • Finance: Fractions are used to calculate interest rates and investment returns.
  • Science: Fractions are used to represent proportions and ratios in scientific experiments.

Tips and Tricks

Here are some tips and tricks to help you compare and manipulate fractions:

  • Find a common denominator: Use the least common multiple (LCM) to find a common denominator for fractions.
  • Use approximation: Approximate fractions to decimal form to compare them easily.
  • Simplify fractions: Simplify fractions by dividing both the numerator and denominator by their greatest common divisor (GCD).

Conclusion

In conclusion, we have compared two calculations involving fractions and determined which one is closer in value to 12\frac{1}{2}. We have shown our working and provided a step-by-step explanation of the process. Understanding how to compare and manipulate fractions is essential in various real-world applications, and we hope this article has provided you with valuable insights and tips to help you master this skill.

Introduction

In our previous article, we explored two calculations involving fractions and determined which one is closer in value to 12\frac{1}{2}. In this article, we will answer some frequently asked questions related to comparing fractions.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 15 and 3 is 15, because 15 is the smallest number that both 15 and 3 can divide into evenly.

Q: How do I find the LCM of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that they have in common. Alternatively, you can use the following formula:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that two or more numbers have in common. For example, the GCD of 15 and 3 is 3, because 3 is the largest number that both 15 and 3 can divide into evenly.

Q: How do I compare fractions with different denominators?

A: To compare fractions with different denominators, you need to find a common denominator. You can do this by finding the least common multiple (LCM) of the two denominators.

Q: Can I compare fractions by converting them to decimals?

A: Yes, you can compare fractions by converting them to decimals. This can be a useful way to compare fractions, especially if you are not familiar with finding common denominators.

Q: How do I simplify fractions?

A: To simplify a fraction, you need to divide both the numerator and denominator by their greatest common divisor (GCD). For example, the fraction 68\frac{6}{8} can be simplified by dividing both the numerator and denominator by 2, resulting in 34\frac{3}{4}.

Q: What is the difference between adding and subtracting fractions?

A: When adding fractions, you need to find a common denominator and add the numerators. When subtracting fractions, you need to find a common denominator and subtract the numerators.

Q: Can I add or subtract fractions with different signs?

A: Yes, you can add or subtract fractions with different signs. When adding fractions with different signs, you need to find a common denominator and add the numerators. When subtracting fractions with different signs, you need to find a common denominator and subtract the numerators.

Q: How do I convert a fraction to a mixed number?

A: To convert a fraction to a mixed number, you need to divide the numerator by the denominator and write the remainder as the new numerator. For example, the fraction 174\frac{17}{4} can be converted to a mixed number by dividing 17 by 4, resulting in 4 with a remainder of 1, which can be written as 4144\frac{1}{4}.

Q: Can I convert a mixed number to a fraction?

A: Yes, you can convert a mixed number to a fraction by multiplying the whole number part by the denominator and adding the numerator. For example, the mixed number 4144\frac{1}{4} can be converted to a fraction by multiplying 4 by 4 and adding 1, resulting in 174\frac{17}{4}.

Conclusion

In conclusion, we have answered some frequently asked questions related to comparing fractions. We hope this article has provided you with valuable insights and tips to help you master this skill. Remember to always find a common denominator when comparing fractions, and to use approximation to compare fractions with different denominators.