Simplifying Fraction Division: The Quest for the Correct Quotient

Fraction division can sometimes be confusing, especially when trying to identify the correct divisor to achieve a specific quotient. The problem we'll explore in this article is: ldquo;By what number should -16/21 be divided so that the quotient is -4/7?rdquo; This article will break down the problem step-by-step and use various examples to help clarify the process.

Understanding the Problem

The problem states that we need to find a number to divide -16/21 by, such that the resulting quotient is -4/7. This type of problem can be approached using algebraic manipulation and understanding the relationship between the terms.

Example 1: By what number should -16/21 be divided to get -4/7?

Let's denote the unknown number by x. The equation then becomes:

-16/21 ÷ x -4/7 or -16/21 × 1/x -4/7 or -16/21 -4/7 × x

Solving for x, we get:

x -16/21 ÷ -4/7 x -16/21 × 7/-4 x (16 × 7) / (21 × 4) x (2 × 7) / (3 × 4) x 14/12 x 7/6 (simplified form)

However, the given example suggests that the number is 7/12, which means the correct approach involves simplifying the fraction further.

Example 2: Simplifying a Related Problem

A similar problem involves simplifying -15/36 to match -5/7. Let's break down this example step-by-step:

-15/36 ÷ x -5/7

The steps to solve are:

-15/36 -5/7 × x x -15/36 ÷ -5/7 x -15/36 × 7/-5 x (15 × 7) / (36 × 5) x (3 × 7) / (12 × 5) x 21/60 x 7/20 (simplified form, but the answer given is 7/12)

The provided solution suggests 7/12, which implies further simplification is required. Let's explore this further:

x -15/36 ÷ -5/7 x -15/36 × 7/-5 x 15/36 × 7/5 x 5/12 × 7/5 x 7/12

Therefore, the correct answer is 7/12, which simplifies the original fraction.

Alternative Approaches

Another example involves direct manipulation:

16/21 ÷ x -4/7 x 16/21 ÷ -4/7 x 16/21 × -7/4 x 16 × -7 / 21 × 4 x 4 × -1 / 3 × 1 x -4/3

This means -16/21 should be divided by -4/3 to yield -4/7.

A helpful tip from a student suggests a simpler approach:

Let x be the number to be divided. Then:

-15/36 ÷ x -5/7 -15/36 -5/7 × x x -15/36 ÷ -5/7 x -15/36 × 7/-5 x 15/36 × 7/5 x 7/12

Therefore, the correct answer is 7/12.

Conclusion

Through these examples, we can see that understanding the relationship between fractions and division can be crucial in solving such problems. The key is careful manipulation and checking the results. Whether through direct division, algebraic manipulation, or a student's simplistic approach, the correct answer to our problem is 7/12.

In summary, fraction division is a fundamental skill in mathematics and requires practice to master. Whether you're working on advanced division problems or simpler ones, the principles outlined in this article can help clarify the process and ensure accurate solutions.