The Rule for Dividing by a Fraction

Oct 13, 2025 by miodeya.com 78 views

Understanding the Problem

When dealing with division involving fractions, it's essential to understand the concept of dividing by a fraction. In this case, we have the expression 9 ÷ (9/7). To simplify this, we need to apply the rules of division and fractions.

The Rule for Dividing by a Fraction

The rule for dividing by a fraction is to multiply by its reciprocal. In other words, to divide by a fraction, we need to multiply by the fraction's reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.

Finding the Reciprocal of the Fraction

The given fraction is 9/7. To find its reciprocal, we swap the numerator and denominator, which gives us 7/9.

Applying the Rule for Dividing by a Fraction

Now that we have the reciprocal of the fraction, we can apply the rule for dividing by a fraction. To divide 9 by (9/7), we multiply 9 by the reciprocal of (9/7), which is 7/9.

Simplifying the Expression

To simplify the expression, we multiply the numerators and denominators separately. Multiplying 9 by 7 gives us 63, and multiplying 1 by 9 gives us 9. Therefore, the expression simplifies to 63/9.

Further Simplification

The fraction 63/9 can be further simplified by dividing both the numerator and denominator by their greatest common divisor (GCD). The GCD of 63 and 9 is 9. Dividing both 63 and 9 by 9 gives us 7/1.

Conclusion

In conclusion, the expression 9 ÷ (9/7) simplifies to 7. This is because dividing by a fraction is equivalent to multiplying by its reciprocal, and the reciprocal of (9/7) is 7/9. Multiplying 9 by 7/9 gives us 63/9, which further simplifies to 7/1.

Final Answer

The final answer is: 7

Explanation

To divide 9 by (9/7), we need to multiply 9 by the reciprocal of (9/7), which is 7/9. This gives us 63/9, which further simplifies to 7/1. Therefore, the final answer is 7.

Example Use Case

This concept can be applied to various real-world scenarios, such as calculating the cost of an item when the price is given as a fraction of the original price. For instance, if an item originally costs $9 and is now on sale for 9/7 of its original price, we can calculate the new price by multiplying 9 by the reciprocal of (9/7), which is 7/9.

Common Mistakes

One common mistake when dealing with division involving fractions is to forget to multiply by the reciprocal of the fraction. This can lead to incorrect answers. It's essential to remember that dividing by a fraction is equivalent to multiplying by its reciprocal.

Tips and Tricks

To simplify expressions involving division and fractions, it's essential to apply the rule for dividing by a fraction and to find the reciprocal of the fraction. Additionally, it's crucial to simplify the resulting fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).

Real-World Applications

This concept has various real-world applications, such as calculating the cost of an item when the price is given as a fraction of the original price, or determining the amount of time it takes to complete a task when the rate of work is given as a fraction of the total time.

Conclusion

In conclusion, dividing by a fraction is equivalent to multiplying by its reciprocal. To simplify expressions involving division and fractions, we need to apply the rule for dividing by a fraction and to find the reciprocal of the fraction. This concept has various real-world applications and is essential for solving problems involving division and fractions.

Frequently Asked Questions

Q: What is the rule for dividing by a fraction?

A: The rule for dividing by a fraction is to multiply by its reciprocal. In other words, to divide by a fraction, we need to multiply by the fraction's reciprocal.

Q: How do I find the reciprocal of a fraction?

A: To find the reciprocal of a fraction, we need to swap its numerator and denominator. For example, the reciprocal of 3/4 is 4/3.

Q: Can you give me an example of how to apply the rule for dividing by a fraction?

A: Let's say we want to divide 12 by (4/5). To do this, we need to multiply 12 by the reciprocal of (4/5), which is 5/4. Multiplying 12 by 5/4 gives us 60/4, which simplifies to 15.

Q: What is the difference between dividing by a fraction and multiplying by a fraction?

A: Dividing by a fraction is equivalent to multiplying by its reciprocal. For example, dividing 12 by (4/5) is the same as multiplying 12 by 5/4.

Q: Can you give me an example of a real-world application of dividing by a fraction?

A: Let's say we want to calculate the cost of an item when the price is given as a fraction of the original price. If an item originally costs $12 and is now on sale for 4/5 of its original price, we can calculate the new price by multiplying 12 by the reciprocal of (4/5), which is 5/4.

Q: What is the greatest common divisor (GCD) and how do I use it to simplify fractions?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. To simplify a fraction, we need to divide both the numerator and denominator by their GCD. For example, the GCD of 12 and 4 is 4. Dividing both 12 and 4 by 4 gives us 3/1.

Q: Can you give me an example of how to simplify a fraction using the GCD?

A: Let's say we want to simplify the fraction 12/4. To do this, we need to find the GCD of 12 and 4, which is 4. Dividing both 12 and 4 by 4 gives us 3/1.

Q: What are some common mistakes to avoid when dividing by a fraction?

A: One common mistake to avoid is forgetting to multiply by the reciprocal of the fraction. This can lead to incorrect answers. It's essential to remember that dividing by a fraction is equivalent to multiplying by its reciprocal.

Q: Can you give me some tips and tricks for simplifying expressions involving division and fractions?

A: To simplify expressions involving division and fractions, it's essential to apply the rule for dividing by a fraction and to find the reciprocal of the fraction. Additionally, it's crucial to simplify the resulting fraction by dividing both the numerator and denominator by their GCD.

Q: What are some real-world applications of dividing by a fraction?

A: This concept has various real-world applications, such as calculating the cost of an item when the price is given as a fraction of the original price, or determining the amount of time it takes to complete a task when the rate of work is given as a fraction of the total time.

Q: Can you give me some examples of how to apply dividing by a fraction in different scenarios?

Q: What is the final answer to the original problem 9 ÷ (9/7)?

A: The final answer to the original problem 9 ÷ (9/7) is 7.

Q: Can you explain the concept of dividing by a fraction in more detail?

A: Dividing by a fraction is equivalent to multiplying by its reciprocal. To simplify expressions involving division and fractions, we need to apply the rule for dividing by a fraction and to find the reciprocal of the fraction. This concept has various real-world applications and is essential for solving problems involving division and fractions.