Simplify:$\[ \left(\frac{1}{2} - \frac{1}{6}\right) \div \frac{7}{12} \\]

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Introduction

When dealing with mathematical expressions, simplification is a crucial step to ensure that the expression is in its most basic form. This simplification process involves combining like terms, canceling out common factors, and rearranging the expression to make it easier to work with. In this article, we will focus on simplifying the given expression (12−16)÷712\left(\frac{1}{2} - \frac{1}{6}\right) \div \frac{7}{12} using various mathematical techniques.

Understanding the Expression

The given expression involves fractions and division. To simplify it, we need to start by evaluating the expression inside the parentheses. The expression inside the parentheses is 12−16\frac{1}{2} - \frac{1}{6}. To subtract these fractions, we need to find a common denominator, which is 6 in this case.

Finding a Common Denominator

To find a common denominator, we need to express both fractions with the same denominator. In this case, we can multiply the numerator and denominator of 12\frac{1}{2} by 3 to get 36\frac{3}{6}. Now, we can subtract 16\frac{1}{6} from 36\frac{3}{6}.

Subtracting Fractions

Now that we have a common denominator, we can subtract the fractions. 36−16=26\frac{3}{6} - \frac{1}{6} = \frac{2}{6}. We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2.

Simplifying the Fraction

Dividing both the numerator and denominator by 2, we get 13\frac{1}{3}. So, the expression inside the parentheses simplifies to 13\frac{1}{3}.

Dividing by a Fraction

Now that we have simplified the expression inside the parentheses, we can focus on the division part of the expression. When dividing by a fraction, we can multiply by the reciprocal of the fraction instead. The reciprocal of 712\frac{7}{12} is 127\frac{12}{7}.

Multiplying by the Reciprocal

To divide by 712\frac{7}{12}, we can multiply by 127\frac{12}{7}. So, the expression becomes 13×127\frac{1}{3} \times \frac{12}{7}.

Multiplying Fractions

To multiply fractions, we need to multiply the numerators and denominators separately. Multiplying the numerators, we get 1×12=121 \times 12 = 12. Multiplying the denominators, we get 3×7=213 \times 7 = 21.

Simplifying the Result

Now that we have multiplied the fractions, we get 1221\frac{12}{21}. We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3.

Final Simplification

Dividing both the numerator and denominator by 3, we get 47\frac{4}{7}. So, the final simplified expression is 47\frac{4}{7}.

Conclusion

In this article, we simplified the given expression (12−16)÷712\left(\frac{1}{2} - \frac{1}{6}\right) \div \frac{7}{12} using various mathematical techniques. We started by evaluating the expression inside the parentheses, finding a common denominator, subtracting fractions, simplifying the fraction, dividing by a fraction, multiplying by the reciprocal, multiplying fractions, and finally simplifying the result. The final simplified expression is 47\frac{4}{7}.

Frequently Asked Questions

  • Q: What is the simplified form of the expression (12−16)÷712\left(\frac{1}{2} - \frac{1}{6}\right) \div \frac{7}{12}? A: The simplified form of the expression is 47\frac{4}{7}.
  • Q: How do I simplify a fraction? A: To simplify a fraction, you need to find the greatest common divisor of the numerator and denominator and divide both by it.
  • Q: What is the reciprocal of a fraction? A: The reciprocal of a fraction is obtained by swapping the numerator and denominator.

Further Reading

References

Introduction

In our previous article, we simplified the given expression (12−16)÷712\left(\frac{1}{2} - \frac{1}{6}\right) \div \frac{7}{12} using various mathematical techniques. In this article, we will answer some frequently asked questions related to the simplification process.

Q&A

Q: What is the simplified form of the expression (12−16)÷712\left(\frac{1}{2} - \frac{1}{6}\right) \div \frac{7}{12}?

A: The simplified form of the expression is 47\frac{4}{7}.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by it. The GCD is the largest number that divides both the numerator and denominator without leaving a remainder.

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

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

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

A: There are several ways to find the GCD of two numbers. One way is to list the factors of each number and find the largest common factor. Another way is to use the Euclidean algorithm, which is a step-by-step process for finding the GCD of two numbers.

Q: What is the Euclidean algorithm?

A: The Euclidean algorithm is a step-by-step process for finding the GCD of two numbers. It involves dividing the larger number by the smaller number and taking the remainder. Then, you divide the smaller number by the remainder and take the new remainder. You continue this process until the remainder is zero. The last non-zero remainder is the GCD.

Q: How do I divide fractions?

A: To divide fractions, you need to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping the numerator and denominator.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of 12\frac{1}{2} is 21\frac{2}{1}, and the reciprocal of 34\frac{3}{4} is 43\frac{4}{3}.

Q: How do I multiply fractions?

A: To multiply fractions, you need to multiply the numerators and denominators separately. Then, you simplify the resulting fraction by dividing both the numerator and denominator by their GCD.

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: How do I add fractions?

A: To add fractions, you need to find a common denominator and add the numerators. For example, to add 12\frac{1}{2} and 14\frac{1}{4}, you need to find a common denominator, which is 4. Then, you add the numerators: 24+14=34\frac{2}{4} + \frac{1}{4} = \frac{3}{4}.

Q: How do I subtract fractions?

A: To subtract fractions, you need to find a common denominator and subtract the numerators. For example, to subtract 12\frac{1}{2} from 14\frac{1}{4}, you need to find a common denominator, which is 4. Then, you subtract the numerators: 24−14=14\frac{2}{4} - \frac{1}{4} = \frac{1}{4}.

Conclusion

In this article, we answered some frequently asked questions related to simplifying fractions and dividing fractions. We also discussed the Euclidean algorithm and how to find the greatest common divisor (GCD) of two numbers. We hope that this article has been helpful in clarifying some common misconceptions and providing a better understanding of the simplification process.

Further Reading

References