Find The Quotient Of 9/10 And 2/3

Finding the Quotient of 9/10 and 2/3: A Comprehensive Guide

Dividing fractions can seem daunting, but with a clear understanding of the process, it becomes straightforward. This article will guide you through finding the quotient of 9/10 and 2/3, explaining the concept in detail and providing various approaches to solve the problem. We'll explore the underlying mathematical principles and offer practical examples to solidify your understanding. By the end, you'll be confident in tackling similar fraction division problems.

Understanding Fraction Division

Before diving into the specific problem, let's refresh our understanding of fraction division. When we divide fractions, we're essentially asking "how many times does one fraction fit into another?" This differs from multiplying fractions, where we find a portion of a portion.

The key to dividing fractions lies in the concept of the reciprocal. The reciprocal of a fraction is simply the fraction flipped upside down. For example, the reciprocal of 2/3 is 3/2.

The rule for dividing fractions is: To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

This can be expressed mathematically as:

(a/b) ÷ (c/d) = (a/b) x (d/c)

Where 'a', 'b', 'c', and 'd' are numbers, and 'b' and 'c' are not zero (division by zero is undefined).

Solving 9/10 ÷ 2/3

Now, let's apply this knowledge to solve our specific problem: finding the quotient of 9/10 and 2/3. This translates to the division problem: 9/10 ÷ 2/3.

Step 1: Find the reciprocal of the second fraction (the divisor).

The reciprocal of 2/3 is 3/2.

Step 2: Rewrite the division problem as a multiplication problem.

Our problem now becomes: 9/10 x 3/2

Step 3: Multiply the numerators (the top numbers) together.

9 x 3 = 27

Step 4: Multiply the denominators (the bottom numbers) together.

10 x 2 = 20

Step 5: Simplify the resulting fraction.

Our answer is now 27/20. This is an improper fraction (the numerator is larger than the denominator). We can convert this to a mixed number to make it easier to understand.

Converting the Improper Fraction to a Mixed Number:

To convert 27/20 to a mixed number, we divide the numerator (27) by the denominator (20).

27 ÷ 20 = 1 with a remainder of 7.

Therefore, 27/20 can be expressed as the mixed number 1 7/20.

Alternative Methods for Solving Fraction Division

While the reciprocal method is the most common and efficient way to divide fractions, there are alternative approaches you can use to reach the same solution:

Method 1: Using Decimal Conversion

One alternative approach is to convert the fractions into decimals before performing the division.

• 9/10 is equal to 0.9.
• 2/3 is approximately equal to 0.6667 (we'll use this approximation for simplicity, though the decimal expansion of 2/3 continues infinitely).

Now, divide 0.9 by 0.6667:

0.9 ÷ 0.6667 ≈ 1.35

This decimal value (1.35) is approximately equal to our mixed number solution (1 7/20). Keep in mind that this method relies on approximations due to the repeating decimal nature of 2/3. Therefore, the result is only an approximation and might not be perfectly precise.

Method 2: Finding a Common Denominator (Less Efficient)

Another, less efficient method, involves finding a common denominator for both fractions before dividing. While this works, it's generally less practical than the reciprocal method, particularly with larger or more complex fractions.

To illustrate, the common denominator for 10 and 3 is 30. We would then convert both fractions:

9/10 = 27/30 2/3 = 20/30

Then divide 27/30 by 20/30. This method still leads to the same solution, but it involves more steps compared to the reciprocal method.

Real-World Applications of Fraction Division

Understanding fraction division is crucial in various real-world scenarios. For instance:

• Cooking and Baking: If a recipe calls for 2/3 cup of flour and you only want to make half the recipe, you would divide 2/3 by 2 to determine the amount of flour needed.
• Construction and Engineering: Calculating material requirements often involves dividing fractional measurements.
• Sewing and Quilting: Cutting fabric to specific fractional lengths requires accurate fraction division calculations.
• Finance and Budgeting: Dividing a budget into fractional parts for different expenses requires a solid understanding of fraction division.

Practicing Fraction Division

The best way to master fraction division is through practice. Here are some practice problems:

1. 5/6 ÷ 1/3
2. 3/4 ÷ 2/5
3. 7/8 ÷ 3/4
4. 1/2 ÷ 5/8
5. 2/3 ÷ 1/6

By working through these problems using the reciprocal method (and checking your answers using other methods), you'll strengthen your skills and gain confidence in handling fraction division.

Conclusion

Finding the quotient of 9/10 and 2/3 ultimately boils down to understanding and applying the rule of multiplying the first fraction by the reciprocal of the second. The solution, 1 7/20, highlights the importance of converting improper fractions to mixed numbers for easier interpretation. While alternative methods exist, the reciprocal method offers the most efficient and direct path to solving fraction division problems. Remember to practice regularly to build proficiency and confidently tackle similar problems in various contexts. The more you practice, the more intuitive and comfortable you will become with fraction division. Remember that understanding the underlying concepts is key to mastering this important mathematical skill.