Find The Quotient Of 9/10 And 2/3

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

Finding the quotient of two fractions might seem daunting at first, but with a clear understanding of the process, it becomes straightforward. This article will guide you through calculating the quotient of 9/10 and 2/3, providing a step-by-step explanation and exploring the underlying mathematical concepts. We'll also delve into practical applications and related problems to solidify your understanding.

Understanding Division with Fractions

Before we tackle the specific problem of finding the quotient of 9/10 and 2/3, let's review the fundamental principles of dividing fractions. Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction is simply the fraction inverted; the numerator becomes the denominator, and the denominator becomes the numerator.

For example, the reciprocal of 2/3 is 3/2. Therefore, dividing by 2/3 is the same as multiplying by 3/2. This principle is crucial for solving division problems involving fractions.

Step-by-Step Calculation: 9/10 ÷ 2/3

Now, let's apply this principle to our specific problem: finding the quotient of 9/10 and 2/3, which can be written as:

9/10 ÷ 2/3

Step 1: Find the reciprocal of the second fraction.

The reciprocal of 2/3 is 3/2.

Step 2: Rewrite the division as multiplication.

Our problem now becomes:

9/10 × 3/2

Step 3: Multiply the numerators and the denominators.

Multiply the numerators together (9 x 3 = 27) and the denominators together (10 x 2 = 20). This gives us:

27/20

Step 4: Simplify the fraction (if possible).

In this case, 27 and 20 share no common factors other than 1, meaning the fraction is already in its simplest form.

Therefore, the quotient of 9/10 and 2/3 is 27/20.

Representing the Quotient as a Mixed Number

The quotient 27/20 is an improper fraction (where the numerator is larger than the denominator). It's often helpful to express improper fractions as mixed numbers (a whole number and a proper fraction).

To convert 27/20 to a mixed number, we perform the division:

27 ÷ 20 = 1 with a remainder of 7.

This means that 27/20 can be expressed as 1 7/20.

Visualizing the Division

While the mathematical process is precise, visualizing the division can enhance understanding. Imagine you have 9/10 of a pizza. You want to divide this 9/10 into portions that are each 2/3 of a whole pizza. The answer, 1 7/20, tells us that you can create one full portion of the size of 2/3 a pizza and a portion that is 7/20 of a whole pizza.

Practical Applications and Real-World Examples

Understanding fraction division is crucial in various real-world scenarios. Here are some examples:

• Cooking and Baking: Recipes often require dividing ingredients. For instance, if a recipe calls for 2/3 of a cup of flour, and you only want to make half the recipe, you need to divide 2/3 by 2 (or multiply by 1/2).

• Sewing and Tailoring: Calculating fabric requirements involves fraction division. If a pattern requires 9/10 of a yard of fabric and you need to divide this between 2 projects equally, then you'd need to calculate 9/10 ÷ 2.

• Construction and Engineering: Precise measurements are critical, often involving fractions. Dividing lengths or quantities of materials requires a solid understanding of fraction division.

Related Problems and Practice Exercises

To further solidify your understanding, let's explore some related problems:

Problem 1: Find the quotient of 5/6 and 1/4.

Solution: 5/6 ÷ 1/4 = 5/6 × 4/1 = 20/6 = 10/3 = 3 1/3

Problem 2: A piece of wood is 7/8 of a meter long. If you want to cut it into pieces that are each 1/4 of a meter long, how many pieces can you make?

Solution: 7/8 ÷ 1/4 = 7/8 × 4/1 = 28/8 = 7/2 = 3 1/2

Problem 3: What is the result of dividing 11/12 by 5/6?

Solution: 11/12 ÷ 5/6 = 11/12 × 6/5 = 66/60 = 11/10 = 1 1/10

Common Mistakes to Avoid

When working with fractions, several common mistakes can lead to incorrect answers. Here are a few to be mindful of:

• Forgetting to find the reciprocal: Remember that dividing by a fraction is the same as multiplying by its reciprocal. Failing to invert the second fraction is a common error.

• Incorrect multiplication: Ensure you multiply the numerators correctly and the denominators correctly. Double-check your calculations.

• Not simplifying the fraction: Always simplify your final answer to its lowest terms.

Conclusion

Finding the quotient of fractions, while initially appearing challenging, becomes manageable with a systematic approach. By understanding the concept of reciprocals and following the step-by-step process outlined in this article, you can confidently solve problems involving fraction division. Remember to practice regularly, working through various examples to reinforce your understanding and avoid common pitfalls. The ability to perform fraction division is a vital skill with numerous applications in diverse fields. Mastering this skill will undoubtedly enhance your mathematical proficiency and problem-solving abilities. Remember to always check your work and simplify your answers to their lowest terms for the most accurate and concise solutions.